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<ul><li><p>tioa</p><p>e K</p><p>lec</p><p>orm24</p><p>ciat</p><p>Abstract</p><p>1. Introduction</p><p>(Cao and Cao, 2006; Sfetsos and Coonick, 2000; Chenaet al., 2007; Cucumo et al., 2007; Aguiar and Collares-Pereira, 1992; Kaplanis andKaplani, 2007). Many dierent</p><p>models are used to predict the solar radiation time series</p><p>since they can represent several dierent types of time seriesby using dierent order. It has been proved to be competentin prediction when there is an underlying linear correlationstructure lying in the time series.</p><p>One major requirement for ARMA model is that thetime series must be stationary. However, from a stationary</p><p> Corresponding author.E-mail address: wuji0014@e.ntu.edu.sg (J. Wu).</p><p>Available online at www.sciencedirect.com</p><p>Solar Energy 85 (2011Solar radiation prediction is of great importance formany applications such as generation of electricity andproviding portable clean water (Sozen et al., 2004; Saylanet al., 2003; Dincer et al., 1996; Kaplanis, 2006; Rahmanand Chowdhury, 1988; Dinelli, 1995). Accurate predictioncan greatly improve the performance of these systems(Kaygusuz and Sari, 2003; Acock and Pachepsky Ya,2000; Fujioka, 1995; Tugay and Yilmaz, 2004). The solarradiation sequence can be treated as a time series, whichmeans that we can use mathematical models to t theunderlying random process and predict the next values</p><p>like the classic Auto-Regression, Auto-Regression andMoving Average (Box and Jenkins, 1970) and MarkovChain. Furthermore, adaptive methods such as the TimeDelay Neural Network (TDNN), which has been provento be reliable in predicting the future trend of a time series,can also be used to solve this problem.</p><p>One of themost popular andwidely used time series mod-els is the Autoregressive and Moving Average (ARMA)model (McKenzie, 1984). The popularity of the ARMAmodel is its ability to extract useful statistical propertiesand the adoption of the well-known BoxJenkins methodol-ogy (Box and Jenkins, 1970).ARMAmodels are very exibleIn this work, a new approach that contains two phases is used to predict the hourly solar radiation series. In the detrending phase,several models are applied to remove the non-stationary trend lying in the solar radiation series. To judge the goodness of dierent detr-ending models, the Augmented DickeyFuller method is applied to test the stationarity of the residual. The optimal model is used todetrend the solar radiation series. In the prediction phase, the Autoregressive and Moving Average (ARMA) model is used to predictthe stationary residual series. Furthermore, the controversial Time Delay Neural Network (TDNN) is applied to do the prediction.Because ARMA and TDNN have their own strength respectively, a novel hybrid model that combines both the ARMA and TDNN,is applied to produce better prediction. The simulation result shows that this hybrid model can take the advantages of both ARMAand TDNN and give excellent result. 2011 Elsevier Ltd. All rights reserved.</p><p>Keywords: Solar radiation prediction; ARMA; TDNN; Hybrid modelPrediction of hourly solar radiaof ARMA</p><p>Ji Wu , CheNanyang Technological University, School of E</p><p>Received 23 August 2010; received in revised fAvailable online</p><p>Communicated by: Asso0038-092X/$ - see front matter 2011 Elsevier Ltd. All rights reserved.doi:10.1016/j.solener.2011.01.013n using a novel hybrid modelnd TDNN</p><p>eong Chan</p><p>trical and Electronics Engineering, Singapore</p><p>26 January 2011; accepted 26 January 2011February 2011</p><p>e Editor Frank Vignola</p><p>www.elsevier.com/locate/solener</p><p>) 808817</p></li><li><p>method is used to check the stationarity (Dickey and</p><p>is adaptively formed based on the features presented by thedata. This data driven algorithm is suitable for many time</p><p>2</p><p>Enseries where no theoretical model is available. In this paper,TDNN is applied to predict the solar radiation as well.</p><p>The use of hybrid models has gained popularity as ittakes advantage of dierent models Makridakis et al.,1982). There are many approaches to combine dierentmodels (Reid, 1968; Bates and Granger, 1969; Clemen,1989). The basic idea of the model combination in forecast-ing is to use each models unique feature to capture dier-ent patterns in the data. Both theoretical and empiricalnding suggests that combining dierent models can bean ecient way to improve the forecast performance. Wehave chosen to use a hybrid model of ARMA and TDNNto improve the prediction accuracy.</p><p>The rest of the paper is organized as follows. In Sections2 and 3, we tried dierent detrending models on the solarradiation series to nd the optimal one to get the stationaryseries. In Sections 4 and 5, we simulated ARMA, TDNNand the hybrid model in the prediction of the solar radia-tion series and compared their performance.</p><p>2. The detrend models</p><p>As mentioned earlier, the solar radiation is non-station-ary and we need to detrend it. Because of the unpredictablenoise, it is not easy to nd the trend in a specic days ser-ies. Instead, we explored the trend in a much more stableseries namely the monthly average series. Below are somemodels that model the general distribution of hourly solarseries:</p><p>2.1. Jains model</p><p>Jains model (Baig et al., 1991) tries to t the solar radi-ation series with a Gaussian function. That is:</p><p>1 hm2 !Fuller, 1981;Harris, 1992).Also, auto correlation and partialcorrelation of the residual series and Akaike informationcriterion (AIC) of dierent orders are checked to congurethe optimal ARMA model in prediction (Akaike, 1974).</p><p>Time Delay Neural Network (TDNN) has been keenlyexplored and applied in predicting time series (Winyoftand Cander, 1999). The strength of this algorithm is itsability to model nonlinear series. With TDNN, there isno need to specify a particular model form, since the modeltest using Augmented DickeyFuller (ADF) test (Dickeyand Fuller, 1981; Harris, 1992), the solar radiation serieswas found to be non-stationary. Thus a phase of detrend-ing is needed to obtain the stationary series. In the rst sec-tion, we used dierent models to detrend the solarradiation. After detrending, the Augmented DickeyFuller</p><p>J. Wu, C.K. Chan / Solarrh r2p</p><p>p exp 2r2</p><p>1So is the sunshine hour of the day at a site with latitude uand suns declination could be calculated by:</p><p>S0 215</p><p>cos1 tanu tan d 3</p><p>d is the angle between the rays of the Sun and the plane ofthe Earths equator.</p><p>2.3. S. Kaplanis model</p><p>Another interesting model is proposed by Kaplanis(2006). That is:</p><p>rh a b cos 2ph m24</p><p> 4</p><p>The a and b in the equation should be decided according tothe actual situation of dierent area. m is the peak hour ofsolar radiation in this area.</p><p>2.4. Al-Sadahs model</p><p>Al-Sadah found that high order polynomial model isquite good in tting hourly global radiation on a horizontalsurface (Al-Sadah et al., 1990). This model is:</p><p>rh a1 a2h a2h2 5</p><p>3. The simulation of detrending models</p><p>In this section, these detrending models are simulatedwith actual data. The data collected is from the observationstation set up at Nanyang Technological University,Singapore. The sampling interval is 10 min. We ignoredthe data between 10:00 pm and 7:00 am as there is consis-tently no solar energy received during this period. The unitof solar radiation energy is watt-hour per square meter(W h/m2). The monthly mean series of February 2009 isshown in Fig. 1. Dierent models are applied successivelyto t the monthly mean series.</p><p>Using Jains model to t the monthly mean series, torh is the solar radiation of dierent time. h is time, r is thestandard deviation of the Gaussian curve. It is a parameterthat should be decided by the actual data. The parameter mrepresents the peak hour of a day.</p><p>2.2. Baigs model</p><p>Baigs model (Bevington, 1969) is developed based onthe Jains model. Baig modied Jains model to better tthe record data at the start and end of series. The model is:</p><p>rh 1r2p</p><p>p exp h m2</p><p>2r2</p><p> ! cos 180 h m</p><p>2</p><p>So 1</p><p>" #( )</p><p>ergy 85 (2011) 808817 809estimate the parameter r and m, the method of least squaremethod was applied to train the model to nd those</p></li><li><p>Fig. 1. The monthly mean series of February 2009.Fig. 3. The comparison of Baigs model and actual series of February2009.</p><p>Fig. 4. The comparison of S. Kaplanis model and actual series of</p><p>810 J. Wu, C.K. Chan / Solar Enparameters. The comparison of Jains model and actualmonthly mean series is presented in Fig. 2.</p><p>The solid line is the actual series and dot line is the esti-mated series of Jains model.</p><p>We can see that Jains model generally ts the mid-rangeof the monthly mean series. But it fails badly at the begin-ning and ending of the model.</p><p>Baigs model aims to improve the accuracy at the begin-ning and ending of the actual series, as shown in Fig. 3.</p><p>The solid line is the actual series and the dot line isBaigs model. Since the Baigs model is based on Jainsmodel, the general prole is the same. However, the modelprovides a much better t at the beginning and endingstage of the actual series (see Fig. 4).</p><p>S. Kaplanis model is dierent from the previous twomodels. This model is based on cosine function. The exper-imental data provides the estimate for the parameter a andb in the function.Fig. 2. The comparison of Jains model and actual series of February2009.ergy 85 (2011) 808817The solid line the actual series and dot line is theS. Kaplanis model.</p><p>Using Al-Sadahs model also requires us using actualdata to train the model to get the value of these unknowncoecients. We used the least square method and theactual data as input. Fig. 5 is the result achieved aftertraining.</p><p>To evaluate the stationarity of the solar radiation afterdetrending by these models, we use the AugmentedDickeyFuller (ADF) test (Dickey andFuller, 1981;Harris,1992). The ADF test is a test for unit root in a time series.If there is a unit root in time series, the time series is notstationary; otherwise, it should be stationary.</p><p>The model of ADF test is:</p><p>@Y t l bt qY t1 @1Y t1 @pY tp et 6The l in this equation is a constant, b represent the trend.And p is the order of autoregressive process. {et} is a</p><p>February 2009.</p></li><li><p>From the table we can see that the test results for allthese models are below the critical value. Hence, we cansurmise that the residual series of all these models can beregarded as stationary series. The statistical power ofAl-Sadahs model is the highest one, which means theresidual series detrended by the Al-Sadahs model has thelowest probability to contain a unit root.</p><p>Another key factor that should be taken into consider-ation is the accuracy of tting the actual series. To evaluatethis accuracy, we use the indicator: root mean square error(RMSE) and the normalized root mean square error(NRMSE):</p><p>RMSE 1n</p><p>XN1</p><p>ei mi 2 !1</p><p>2</p><p>7</p><p>1N</p><p>P1N ei mi 2</p><p> 12Fig. 5. The comparison between Al-Sadahs model and the actual series of</p><p>February 2009.</p><p>J. Wu, C.K. Chan / Solar Energy 85 (2011) 808817 811sequence of independent normal random variables withmean zero and variance r2 = 1.</p><p>ADF test applies Eq. (6) to construct a statistics analo-gous to the regression t statistics for the test of hypothesisthat the series has unit root (or q = 1).</p><p>Fig. 6 shows the residual of monthlymean solar radiationseries after detrending by the models introduced above. Thetest is carried out on the residual series under the null hypoth-esis that the true underlying process contains a unit root. Ifthe test result is above the critical value, that means weshould accept the null hypothesis and the time series as notstationary; otherwise we should reject the null hypothesisand the time series as stationary.After this testing procedure,we can attain the statistical power, which is the probabilitythat the test rejects a false null hypothesis.In the test procedure, we use the same signicance level0.05 for all the four models (see Figs. 7 and 8).</p><p>Fig. 6. The residual series after detrenNRMSE 1N</p><p>PN1 mi</p><p>8</p><p>In the above two equations, {ei} is the model tted seriesand {mi} is the actual series (see Table 1).</p><p>The RMSE and NRMSE of dierent models are shownin Table 2.</p><p>From Table 2 we see that Al-Sadahs model yields thebest result in tting accuracy.</p><p>Since Al-Sadahs model renders the best performance inboth detrending and tting, we chose it to detrend the solarradiation series for further prediction.</p><p>4. The prediction model</p><p>4.1. ARMA model</p><p>The Autoregressive Moving Average (ARMA) model isusually applied to auto correlated time series data (Box andJenkins, 1970; McKenzie, 1984). This model is a great toolding by the four dierent models.</p></li><li><p>En812 J. Wu, C.K. Chan / Solarfor understanding and predicting the future value of a spec-ied time series. ARMA is based on two parts: autoregres-sive (AR) part and moving average (MA) part. Also, thismodel is usually referred as ARMA (p, q). In which pand q are the order of AR and MA respectively.</p><p>Autoregressive (AR) models are based on the idea thatthe current value of the series, xt can be explained as afunction of p past values: xt1; xt2; . . . ; xtp where p deter-</p><p>Fig. 7. The partial correlation of the residual series.</p><p>Fig. 8. The auto correlation of the residual series.</p><p>Table 1The ADF test for the detrending models.</p><p>Detrendingmodel</p><p>Statisticalpower</p><p>Signicantlevel</p><p>Testresult</p><p>Criticalvalue</p><p>Jains model 0.9961 0.05 3.9002 2.8994Baigs model 0.9941 0.05 3.7176 2.8994S. Kaplanis model 0.9961 0.05 3.8936 2.8994Al-Sadahs model 0.9990 0.05 6.3815 2.8994</p><p>Table 2The RMSE and NRMSE of dierent models.</p><p>Model Error (RMSE) Error (NRMSE)</p><p>Jains model 0.0459 0.1407Baigs model 0.0434 0.1330S. Kaplanis model 0.0444 0.1360Al-Sadahs model 0.0231 0.0706mines the number of steps into the past needed to predictthe current value. An autoregressive model of order p,abbreviated AR (p), is of the form:</p><p>xt b1xt1 b2xt2 bqxtq wt 9where xt is stationary. b1; . . . ; bq are constants (q 0). wt isa Gaussian white noise series with mean zero.</p><p>The moving average (MA) model is an alternative to theautoregressive representation in which the xt on the left-hand side of the equation are assumed to be combinedlinearly, while the moving average model of order q,abbreviated as MA (q), assumes the white noise wt on theright-hand side of the dening equation are combinedlinearly to form the observed data.</p><p>The moving average model of order q, or MA (q) model,is dened to be:</p><p>xt wt h1wt1 h2wt2 hqwtq 10Now we mix the Autoregressive and Moving Average to</p><p>get the ARMA model. The denition for the model is asfollow.</p><p>A time series {xt; t = 0, 1, 2. . .} is ARMA (p, q) if itis stationary and:</p><p>xt b1zt1 bqztq wt h1wt1 hqwtq 11The parameters p and q are called the autoregressive andthe moving average orders, respectively. {wt; t = 0, 1,2. . .} is a Gaussian white noise sequence.</p><p>To determine the order of ARMA model, we need tocalculate partial correlation and auto correlation of the ser-ies. The plot for partial correlation and auto correlationwith dierent lags are as follows.</p><p>The gures above apparently show that both the partialcorrelation and auto correlation decrease sharply after 1lag. Therefore, the p and q order for ARMA should bothbe 1.</p><p>The Akaike information criterion (AIC) can also beapplied to...</p></li></ul>