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Energy 73 (2014) 978e986

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Energy

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A novel M-D (multi-dimensional) linear prediction filter approachfor hourly solar radiation forecasting

Emre Akarslan a, c, *, Fatih Onur Hocao�glu a, c, Rifat Edizkan b

a Afyon Kocatepe University, Department of Electrical Engineering, Afyonkarahisar, Turkeyb Osmangazi University, Department of Electrical and Electronics Engineering, Eskisehir, Turkeyc Solar and Wind Research & Application Center, Afyon Kocatepe University, Turkey

a r t i c l e i n f o

Article history:Received 13 January 2014Received in revised form27 June 2014Accepted 30 June 2014Available online 25 July 2014

Keywords:Solar irradianceForecastingLinear filterExtraterrestrial irradiance

* Corresponding author. Afyon Kocatepe UniversiEngineering, Afyonkarahisar, Turkey.

E-mail address: e.akarslan@gmail.com (E. Akarslan

http://dx.doi.org/10.1016/j.energy.2014.06.1130360-5442/© 2014 Elsevier Ltd. All rights reserved.

a b s t r a c t

2-D linear prediction filter approach is a well-known method in the literature. Such an approach thatuses past samples to predict present values is also applicable to solar irradiance data. Therefore, suchmodels have limited accuracy levels. In this study, a new approach for hourly solar radiation forecastingis developed. First, the data measured hourly throughout a year, i.e., ambient temperature and extra-terrestrial irradiance are converted into 2-D image forms. Then, the data points are evaluated as pixels ofthe images. Next, different M-D (Multi-Dimensional) linear prediction filter models are designed. Thenovelty of these models is that they utilize not only the solar irradiance image but also different imagesthat correlate with solar irradiance data. These filters are employed both to link the images with eachother and to predict solar irradiance data. It is shown that, to incorporate the temperature, extrater-restrial irradiance and the derivatives of these data with proposed M-D linear prediction filters, it ispossible to improve the prediction accuracy. The results are compared with previously developed linearprediction filter models and conventional methods. Experimental results illustrates that proposedapproach gives better prediction accuracies at a range 1%e30% for different M-D models compared with2-D ones.

© 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Renewable energy systems become more valuable withincreasing energy demands and environmental awareness. Energygeneration from PV (photovoltaic) panels attracts more attentiondue to the decreasing cost per Watt and eco-friendliness. Theproduction of a PV panel is directly related to the insolation at itslocation. Solar irradiance varies because of changes in the diurnaland seasonal position of the sun relative to the Earth. Furthermore,weather conditions affect solar irradiance that reaches the inclinedsurface of the PV panel.

The electricity generated from a PV panel is directly related tothe solar irradiance that falls on its surface. However, due toweather conditions, solar irradiance has a stochastic behavior overtime. To develop an accurate solar irradiance prediction model, onemust explore the possible effects of weather conditions on the data.Several methods are used for solar applications such as forecast of

ty, Department of Electrical

).

solar irradiance, including the ANN (Artificial Neural Network)[11,18,23,25,30,31]), Radial Basis Function network (RBF) [4], FuzzyLogic (FL) [5], Fuzzy-Genetic approach [13], Markov chains [2], theWavelet network [19], ANN-Wavelet [17], ARMA/ANN [32] and theAdaptive Neuro-Fuzzy Inference System (ANFIS) [20]. Suchnonlinear forecasting methods are successful in weather fore-casting since they are able to deal with nonlinear behaviors of thedata. [3] proposed a two-stage method in which a statisticalnormalization of solar power is first obtained using a clear skymodel with statistical smoothing techniques. Then, adaptive lineartime series models are employed to forecast the normalized solarpower. They used both AR (autoregressive) and ARX (AR withexogenous input) models and achieved an RMSE improvement ofapproximately 35%. AR models can determine current output, butthe disadvantage of the AR model is that they do not consider thepast disturbances and the process model. [16] used multiplicativeARMA models to generate an hourly series of global irradiation.ARMA models are very flexible because they can represent severaldifferent types of time series by using different orders. This methodcan predict when there is an underlying linear auto-correlationstructure in the time series. [34] proposed a new approach thatcontains two phases: the first phase is detrending, and the second

Nomenclature

ANN Artificial Neural NetworkMBE Mean Bias ErrorAR AutoregressiveM-D Multi-DimensionalARMA Autoregressive and Moving Averagen Day of the yearARIMA Autoregressive Integrated Moving AveragePV PhotovoltaicARX AR with exogenous inputRMSE Root Mean Square Errorai Linear filter coefficients

rij Correlation value between the past valuesrk Correlation between the past and predicted pixelI DaySFRM Sunshine Fraction Radiation ModelJ HourTDNN Time Delay Neural NetworkI0 Extraterrestrial irradiance (Wm�2)ε Prediction errorIsc Solar constant (Wm�2)LPFM Linear Prediction Filter MethodẐiþ1,jþ1 Prediction pixelZi, j Pixel value

E. Akarslan et al. / Energy 73 (2014) 978e986 979

phase is prediction. They used the ARMA model to predict thestationary residual series and the controversial TDNN for the pre-diction. They tried to utilize the strength of the ARMA and theTDNN by combining them. For non-stationary time series one of themost popular statistical forecasting tools is a class of ARIMAmodels[6,7]. [27] showed that the ARIMAs work primarily because dif-ferencing at a 24-h horizon captures the sharp transitions in irra-diance associated with the diurnal cycle more accurately than othermethods. The use of ARMA and ARIMA provide a basis for manyproblems outside the realm of solar irradiance forecastingincluding economic and business planning, production planning,inventory and production control and optimization of industrialprocesses [6,10]. Solar irradiance data can be estimated usingmeteorological data such as temperature, wind speed and direc-tion, relative humidity, and extraterrestrial irradiance andgeographical data such as latitude and longitude[10,12,15,24,26,35,36]. Furthermore, past solar irradiance data canbe used to predict future solar irradiance values [8].

[8] used 2-D linear filters to forecast the hourly solar radiationdata. They proposed a novel representation of solar radiation data.First, the annual solar radiation data are transformed into a 2-Dimage. The rows and columns of the image correspond to daysand hours, respectively. Then, different types of linear filters withvarious filter-tap configurations are applied to the image, and theoptimal filter coefficients are calculated. Hocao�glu's study provedthat the current solar radiation data are highly correlatedwith thesenot only 1-h-before but also 1-day-before and 1-day-1-hour-before.However, in this study, the solar radiation data were considered tobe a time series, and the other variables that affect solar radiationswere not considered. Unlike Hocao�glu's study, in this study, a newapproach for hourly solar radiation forecasting is developed. First,each type of data is converted into a 2-D image. This approach linksthe images obtained from meteorological variables (solar irradi-ance, temperature, extraterrestrial irradiance and their derivativesover time), to each other using linear prediction filters. In addition,M-D (multi-dimensional) filter-tap configurations are formed. Tofind the optimal coefficients for the filters, all of the pixels in eachlinked pattern are scanned. Once the optimal coefficients aredetermined, eachmodel is tested. The results are compared not onlywith each other but also with the previously developed models.

2. Description of the data used

In this study, several data in image form, as explained in Section3, are employed for solar irradiance forecasting, such as tempera-ture, solar irradiance, extraterrestrial irradiance and the differences(approximate derivative for discrete data) of these variables overtime. These data, except for extraterrestrial irradiance, werecollected over period of 1-year (March 1, 2012eFebruary 28, 2013)

from the solar observation station in the Ahmet Necdet Sezer (ANS)campus area of the Afyonkarahisar region. In the solar observationstation, the parameters are measured every 10 min. The hourlyaverage values of these parameters were calculated and collected.Global solar irradiance measurements were made using theKipp&Zonen trademark CMP6 first class pyranometer.

The extraterrestrial irradiance is the intensity of the sun at thetop of the Earth's atmosphere and is calculated using solar geom-etry for the region. It varies throughout the year because of theEarth's elliptical orbit, which results in an Earth-Sun varying dis-tance during the year in a predictable way. To account for the ec-centricity of the Earth's orbit around the sun, an eccentricitycorrection factor is used, as in equation (1) [1]:

En ¼ Isc

�1þ 0:033 cos

�360n365

��(1)

where Isc is the solar constant and equal to 1367.7 Wm�2. n is theday of the year and for the Jan 1, n is 1 and for the Dec 31, n is 365.The extraterrestrial irradiance on a horizontal plane can beexpressed by:

Eh ¼ En cosðqzÞ (2)

qz is the zenith angle and can be calculated as:

cosðqzÞ ¼ cosðfÞcosðdÞcosðWÞ þ sinðfÞsinðdÞ (3)

So, extraterrestrial irradiance on a horizontal plane at any hourangle W is defined as:

Eh ¼ Isc

�1þ 0:033 cos

�360n365

��fcosðfÞcosðdÞcosðWÞ

þ sinðfÞsinðdÞg (4)

f is the latitude angle of the location in degree and forAfyonkarahisarf ¼ 38:450. The hour angle of the sun, W is formu-lated as:

W ¼�36024

�ðh� 12Þ (5)

where h is the time of the day. Declination angle of the sun isrepresented with d and calculated with:

d ¼ 23:45 sin�360

ð284þ nÞ365

�(6)

The solar irradiance values are smaller than that the extrater-restrial irradiance due to some atmospheric events such as clouds

E. Akarslan et al. / Energy 73 (2014) 978e986980

and rain. In this study, it is assumed that for the first-order deriv-ative in temperature, the extraterrestrial irradiance and solar irra-diance time series can carry important information for predictingsolar irradiances. The extraterrestrial irradiance features have onlypositive values, though the first-order derivatives in this featurehave both positive and negative deviations that correspond to aconsiderable amount of information about the behavior of solarirradiance over time. Therefore, it is assumed and shown that thefirst-order derivative of extraterrestrial irradiance improves themodel accuracy. The first-order derivatives in extraterrestrial irra-diance, temperature and solar irradiance time series are illustratedin Fig. 1 a, b and c respectively.

3. Representation of the data used

In this section, the preparation of the data with M-D filters isexplained. To obtain our new representation (hourly recordedduring March 1, 2012eFebruary 28, 2013) temperature, solar irra-diance and extraterrestrial irradiances are considered as 1-D timeseries. Then, the data are converted into a 2-D matrix. [9] firstproposed this 2-D representation of solar irradiance data. The 2-Dimage obtained from the solar irradiance data provides the toolfor the image processing techniques on solar irradiance data. Thisstudy, however, presents an innovative technique that improvesthe previously developed technique. This technique not only con-siders solar irradiance data to be an image but also converts othermeteorological parameters and their derivatives over time intoimages. To predict solar irradiance, these images are linked todifferent designs of optimal coefficient linear prediction filters. A

Fig. 1. a) First-order difference in extraterrestrial radiation b) First-order differ

linkage process is applied to these images. This process involvesconnecting different images that correspond to different parame-ters. In the linkage phase, the same pixel indices of different imagesare connected to each other by filter taps to calculate the predictionvalues (pixels) of the solar irradiance image. In this way, a combi-nation of different images in a Multi-layered form is obtained. Next,the pixel values of these data are evaluated for the predictionmodel. This technique enables us to represent different informationcompactly and in the same pattern. The combination of differentimages in the Multi-layered form is necessary when using our new3-D filter templates. These templates are 3-D and scan the image ina format 2-D. Therefore, they utilize the information on each layerof the Multi-layered form to predict solar irradiance. Here, theMulti-layered form means that two or more images are used andlinked to each other with the help of linear filters. In this study,with the help of the proposed technique, different inputs areincorporated into the forecast model. The accuracy of the forecast isthereby improved considerably.

In Fig. 2, the 2-D images of temperature, extraterrestrial irra-diance and solar irradiance are shown, and the 2-D image deriva-tions of temperature, extraterrestrial irradiance and solarirradiance over time are illustrated in Fig. 3. By inspecting the im-age version of the solar irradiance data in Fig. 2, it is easy tointerpret its daily and seasonal behavior. Dark blue regions (in theweb version) of the image indicate that there is no sunshine on thehorizontal surface. The transition fromblue to red indicates that thesolar irradiance, which falls on a horizontal surface, is increasing orvice versa. During wintertime, the dawn to dusk period is shorter,which produces a narrower protruding blob. Conversely, the red

ence in temperature data c) First-order difference in solar radiation data.

Fig. 2. From left to right 2-D images of temperature, extraterrestrial irradiance and solar irradiance.

E. Akarslan et al. / Energy 73 (2014) 978e986 981

blob is wider during summertime, which indicates that the day islonger. Hence, the width of the red blob indicates the seasonalchanges in the sunlight periods.

4. Novel multi-dimensional linear prediction filter design

In this section a novel multi-dimensional prediction approach isproposed. This approach considers the history of the solar irradi-ances as well as the history and/or future (because the future of theextraterrestrial data are available) of other meteorological variablesrelated to solar irradiances. An important difference with the pre-vious study [8] is that the previous study can only use the pastsamples of solar irradiance. This study shows that the newapproach substantially improves the performance of the prediction.

In this approach, the multi-dimensional (M-D) linear filters areapplied to the overall image, which consists of the solar irradianceand correlated data; the optimal filter coefficients are determined.Consider the M-D linear prediction filter in Eq. (7).

(7)

The prediction pixel is determined using the past pixels withEq. (8).

�Ziþ1;jþ1;1 ¼ Zi;j;1$a1 þ Zi;jþ1;1$a2 þ Ziþ1;j;1$a3 þ Zi;j;2$a4

þ Zi;jþ1;2$a5 þ Ziþ1;j;2$a6 þ Zi;j;3$a7 þ Zi;jþ1;3$a8

þ Ziþ1;j;3$a9(8)

i,j and Zi,j are identified as the row and column number of thepixel and the pixel value, respectively. In M-D solar data, i denotesthe days, j denotes the hours and Zi,j denotes the solar irradiance orcorrelated data at day i and hour j. A prediction is made for allpossible (i,j) coordinates using past samples. The “linear prediction”

Fig. 3. As in Fig. 2, but for the first-ord

term, which is based on the prediction results, is derived as a linearfunction of the past samples.

In this approach, �Ziþ1,jþ1 is considered a linear combination ofpast samples, and the combination coefficients are optimized usinga minimum squared error between the estimation and the realvalue. The error at a particular coordinate (i,j) is calculated using Eq.(9):

εiþ1;jþ1 ¼ �Ziþ1;jþ1 � Ziþ1;jþ1 (9)

The energy of the total prediction error is, therefore, calculatedas:

3¼Xmi¼2

Xnj¼2

3i;j2 (10)

wherem and n identify the size of the image. The filter coefficients,which minimize this function, can be calculated using Eq. (11):

vε

va1¼ vε

va2¼ vε

va3¼ vε

va4¼ vε

va5¼ vε

va6¼ vε

va7¼ vε

va8¼ vε

va9¼ 0

(11)

The solution to Eq. (11) yields the following equation:

2666666666664

r11 r12 r13r21 r22 r23r31 r32 r33

r14 r15 r16r24 r25 r26r34 r35 r36

r17 r18 r19r27 r28 r29r37 r38 r39

r41 r42 r43r51 r52 r53r61 r62 r63

r44 r45 r46r54 r55 r56r64 r65 r66

r47 r48 r49r57 r58 r59r67 r68 r69

r71 r72 r73r81 r82 r83r91 r92 r93

r74 r75 r76r84 r85 r86r94 r95 r96

r77 r78 r79r87 r88 r89r97 r98 r99

3777777777775

2666666666664

a1a2a3a4a5a6a7a8a9

3777777777775

¼

2666666666664

r1r2r3r4r5r6r7r8r9

3777777777775

(12)

where rij is the correlation value between the past values, ai is thelinear filter coefficients and rk is the correlation between the pastand predicted pixel. This matrix equation can be expressed asR.a ¼ r, and the optimal filter coefficients can be found thus:

a ¼ R�1$r (13)

er derivatives of the parameters.

Fig. 4. An example of the multi-dimensional filters used in this study.

Fig. 5. Filters that use solar irradiance data only.

E. Akarslan et al. / Energy 73 (2014) 978e986982

In this study, 9 different filter-tap configurations are used. Threeof these configurations are same with the filter types used in studyof [8]. On the other hand, the rest six filter-tap configurations aredeveloped in this study in multi-dimensional form to utilize fromdifferent data to forecast solar irradiances. For instance, to illustratehow these filters are formed, 3-D visualization of three differentfilter-tab configurations are presented in Fig. 4. The pixels under theblue blocks (in the web version) identify the past samples that willbe used in the prediction. The pixels under the red block identifythe pixels that will be estimated. The bottom filter in Fig. 6 illus-trates the 3-D filter that is shown in Eq. (7). Each layer in Fig. 4

Fig. 6. Filters for triple comb

represents the image of the data used in forecasting. While thefirst filter-tap configuration (Top-left image in Fig. 4) provides theprediction of next hour solar irradiance with the actual value of itand two more data, with the second filter-tap configuration (Top-right image in Fig. 4) the next hour solar irradiance is predictedwith 1 day before solar irradiance and two more data, and so on.The filter-tap configurations determine the which time period ofdata (1-h before, 1 day before, etc.) will be used in prediction. Thefilter-tap configuration (Bottom image in Fig. 4) uses not only 1-h-before but also 1-day-before and 1-day-1-hour-before data forprediction.

ination of different data.

Table 1Models used in this study.

The data used Formulation for 1-h-later data with currentdata (Filter 1-4-7)

Model 1 Solar irradiance R(iþ1) ¼ a*R(i)Model 2 Temperature-solar irradiance R(iþ1) ¼ a*R(i)þb*S(i)Model 3 Extraterrestrial irradiance-solar irradiance R(iþ1) ¼ a*R(i)þb*E(i)Model 4 Future extraterrestrial irradiance-solar irradiance R(iþ1) ¼ a*R(i)þb*E(iþ1)Model 5 First-order derivation of temperature-solar irradiance R(iþ1) ¼ a*R(i)þb*(vS(i))/vtModel 6 First-order derivation of extraterrestrial irradiance-solar irradiance R(iþ1) ¼ a*R(i)þb*(vE(i))/vtModel 7 First-order derivation of solar irradiance-solar irradiance R(iþ1) ¼ a*R(i)þb*(vR(i))/vtModel 8 First-order derivation of future extraterrestrial irradiance-solar irradiance R(iþ1) ¼ a*R(i)þb*(vE(iþ1))/vtModel 9 Temperature-extraterrestrial irradiance-solar irradiance R(iþ1) ¼ a*R(i)þb*E(i)þc*S(i)Model 10 Temperature-future extraterrestrial irradiance-solar irradiance R(iþ1) ¼ a*R(i)þb*E(iþ1)þc*S(i)Model 11 Past-current extraterrestrial irradiance-solar irradiance R(iþ1) ¼ a*R(i)þb*E(i-1)þc*E(i)Model 12 Current-future extraterrestrial irradiance-solar irradiance R(iþ1) ¼ a*R(i)þb*E(i)þc*E(iþ1)Model 13 First-order derivation of extraterrestrial irradiance and temperature-solar irradiance R(iþ1) ¼ a*R(i)þb*vE(i)/vt þ c*vS(i)/vtModel 14 First-order derivation of future extraterrestrial irradiance and temperature-solar

irradianceR(iþ1) ¼ a*R(i)þb*vE(iþ1)/vt þ c*vS(i)/vt

E. Akarslan et al. / Energy 73 (2014) 978e986 983

5. Experimental results and discussions

In this section, fourteen different models (Table 1) are builtusing the technique presented in the previous section. Modelsindicate the type of the data converted into an image whereas thefilters indicate the pixels that will be used to predict next value ofsolar irradiance. Model 1 (developed by Ref. [8] considers only apast sample of solar irradiance data in its prediction. For model 1three different filters of the previous study [8] are selected andapplied. While selection, computational complexity and predictionaccuracies are considered (please note that the whole comparisonwith all models proposed by Ref. [8] are available upon request).Model 2 considers temperature and solar irradiance data. In Model3, extraterrestrial irradiance and solar irradiance data are employedto predict future solar irradiance values. Model 4 uses the futuresample of extraterrestrial irradiance and past samples of solarirradiance in its prediction. Models 5e8 benefit from the first-orderderivatives, and Models 9e14 use a triple combination of images intheir prediction. All of the proposed models are summarized inTable 1.

The RMSE, RMSE (%) and MBE criteria are selected as a perfor-mance measure of the models. The RMSE is a commonly usedmeasure of the differences between the values extracted by aforecasting model or an estimator and the observed values. TheRMSE has the same units as the measured variable-parameter,which is forecasted by the model. The value of the RMSE providesinformation on the short term performance [22]. A lower RMSEindicates better performance. The MBE provides information in thelong term performance of the correlations by allowing a compari-son of the actual deviation between predicted andmeasured valuesterm by term. The MBE is used to describe whether a model over-(positive value) or under-(negative value) predicts the observation

Fig. 7. Filters for binary comb

and has the same units as the measured variable-parameter, whichis forecasted by the model. The ideal value of MBE is ‘zero’ [21,22].All data, except for the derivatives, are normalized in a 0 to 1 in-terval. The derived data are normalized according to the maximumvalue of the parameter in its entire time series. The derivatives,conversely, are normalized in a �1 to þ1 interval to avoid negativedeviations. Nine different filters with different sizes are designed inthis work. The first three filters (presented in Fig. 5) are the same asthe filters developed previously in the literature [8]. These filtersare applied to the multi-dimensional images of solar irradiancedata, and the future data are estimated. To compare the results ofthis study with those of the previous one, first, only the image ofsolar irradiance data is considered for the prediction. To utilizeadditional information from different data sets, multi-dimensionallinear filters, as illustrated in Figs. 6 and 7, are formed. These filtersare the multi-dimensional forms of the first three linear filters thatwere used in Hocaoglu's study. These filters were chosen for theirlow computational complexity and high prediction performance .

Filters 1e3 (Fig. 5) are used with Model 1 as a first step in thisstudy. This step is necessary to determine the success of the pro-posedmethod. The prediction performance of Filters 1e3 for Model1 is given in Table 2.

Filters 1 and 2 estimate the next hour's and next day's solarirradiance data using the current values of the data, whereas Filter3 uses current, 1-day-before, and 1-day-before-1-hour-later solarirradiance data to estimate the next hour's values.

Filters 7e9 (Fig. 9) are used for Models 2e8 and these filters aremulti-dimensional versions of Filters 1e3. As an example, considerFilters 1 and 7. Filter 1 is used to predict next hour's solar irradianceusing the current irradiance values, whereas Filter 7 is used topredict next hour's solar irradiance data using the current values ofirradiance and temperature. The results indicate that temperature

inations of different data.

Table 2The RMSE (Wm�2) and MBE (Wm�2) values obtained from Filters 1e3.

RMSE (Wm�2) MBE (Wm�2)

Filter-1 Filter-2 Filter-3 Filter-1 Filter-2 Filter-3

Model-1 103.056 106.13 71.617 �8.7688 �9.4984 �15.4211

Fig. 9. Correlation between the actual and predicted solar irradiance for Filter 6 withModel 13.

E. Akarslan et al. / Energy 73 (2014) 978e986984

provides additional information on the variations of the irradi-ances. The performance of Filters 7e9 with Models 2e8 is shown inTable 3.

In Table 3, the performance of Filters 7e9 is better than theperformance of Filters 1e3. In our model, the use of temperaturewith solar irradiance data can help fine-tune the solar irradianceprediction. Conversely, the prediction accuracies are furtherimproved when incorporating the future and present values of theextraterrestrial irradiance for the region. Filters 8 and 9, when usedwith Model 3, provides a significant improvement in predictionperformance. Furthermore, the use of future extraterrestrial irra-diance values gives better results than the use of current values(nearly 20% with Filter-7). Future extraterrestrial irradiance valuecarry information about the future solar irradiance since theextraterrestrial irradiance is the intensity of the sun at the top of theEarth's atmosphere. The solar irradiance values are smaller thanthat the extraterrestrial irradiance due to some atmospheric eventssuch as clouds and rain. Therefore, the use of future extraterrestrialirradiance improves the prediction accuracy. The accuracy of theprediction is improved, and the first-order derivatives of temper-ature and extraterrestrial irradiances are used instead of them-selves. For Filter 7, the best result is obtained from Model 8according to both RMSE and MBE, which employs the first-orderderivatives of extraterrestrial and solar irradiances over time.Negative MBE values indicate underestimation of the prediction.For this template (Filter 7), it is obvious that the usage of the firstorder derivative of future extraterrestrial irradiance (Model 8)instead of future extraterrestrial irradiance (Model 4) providesnearly 15% better RMSE for this dataset. The best results for pre-dicting solar irradiance data using the 1-day-before data are ob-tained from Model 3 according to RMSE values; the positive MBEindicates overestimation of the prediction. This result indicates thatthe extraterrestrial data are important for a prediction using 1-day-before data. Furthermore, it is obvious that a variation in extrater-restrial 1-day-before data does not carry as much information as 1-h-before data. The best result for Filter 9 is obtained from Model 6.Filter 9 outperforms the other filters (Filters 7 and 8) not only withthe use of 1-h-before data but also with the use of 1-day-before and1-day-1-h-before data. As seen in Table 3, derivatives of the tem-perature and extraterrestrial carry more significant informationthan themselves in forecasting. By using the same filter template

Fig. 8. Mesh plot of the prediction error for Filter 6 with Model 13.

(Filter-7), in case of using derivate of the temperature instead ofitself, prediction accuracy increased nearly 10%. Similarly, in case ofusing derivate of extraterrestrial instead of itself, prediction accu-racy increased nearly 30%. Therefore, it can be said that the deri-vations of the extraterrestrial radiations is a better feature than thealteration of the data itself in prediction of solar irradiance. Theusage of future extraterrestrial irradiances instead of present valuesprovided 20% better prediction results.

Filters 4e6 are used with Models 9e14. These models areformed from a triple combination of images. The prediction per-formances of these filters are presented in Table 4.

The best result for the Filter 4 case is obtained from Model 12according to RMSE value. The use of current and future extrater-restrial values together provides better results than does usingthese values separately. Finally, the overall best result is obtainedfrom Filter 6 with Model 13, where this multi-dimensional hierar-chy improves the prediction accuracy considerably. All results ob-tained using the multi-dimensional hierarchy are better than thoseobtained in the Hocaoglu's study. It is obvious from the experi-ments that the first-order derivatives of temperature and extra-terrestrial irradiance carry significant information for solarirradiance prediction. Furthermore, the next hour's solar irradiancedata are correlated with the actual 1-day-before and 1-day-before,1-h-later data. If we consider the physical phenomena underlyingthe process, we know that if there is no cloud effect the data willbehave in a same manner with extraterrestrials. Therefore the datahas an intention to behave such. Furthermore, to use of first orderderivative provide important information about the solar irradi-ance. For instance, first order derivative of temperature explainsomething about the weather condition and the weather conditionaffect the solar irradiance that reaches the inclined surface of the PVpanel. On the other hand, in case a stochastic event occurs (such as

Table 3The RMSE (Wm�2) and MBE (Wm�2) values obtained from Filters 7e9.

RMSE (Wm�2) MBE (Wm�2)

Filter-7 Filter-8 Filter-9 Filter-7 Filter-8 Filter-9

Model-2 101.896 104.941 69.974 5.9120 0.7507 �1.0896Model-3 98.102 92.387 61.064 0.8737 5.1521 4.1404Model-4 77.766 98.786 63.871 �8.9170 0.6794 7.7531Model-5 90.512 105.799 68.820 7.1283 �8.9790 �1.6505Model-6 65.858 105.838 60.990 4.2371 �9.5205 0.4748Model-7 85.346 104.909 67.558 13.5672 �8.3142 �5.5875Model-8 65.556 105.815 61.780 �0.7808 �9.5960 1.2491

Table 4The RMSE (Wm�2) and MBE (Wm�2) values obtained from Filters 4e6.

RMSE (Wm�2) MBE (Wm�2)

Filter-4 Filter-5 Filter-6 Filter-4 Filter-5 Filter-6

Model-9 97.922 92.387 60.842 1.3655 4.9215 4.7141Model-10 76.603 98.607 62.504 �7.1460 4.8505 1.3850Model-11 65.187 92.107 N/A 0.5263 5.7969 N/AModel-12 63.390 93.261 N/A �5.5989 1.3302 N/AModel-13 65.667 105.247 60.739 4.1965 �0.7403 3.9799Model-14 64.767 105.685 61.118 �1.0290 �8.8602 6.2795

N/A: Not available due to unidentified inverse matrix in Eq. (13).

Table 6Comparison between the best result obtained in this study and conventionalmethods.

Model type Location of data used RMSE(%)

[8] LPFM Turkey(Eskisehir) 36.90[24] GSRHS USA (Austin) 39.00[33] Clear Sky France (Ajaccio) 41.80[33] Persistance France (Ajaccio) 35.30[14] SFRM South Korea (Seoul) 34.21[14] Angstr€om type equations South Korea (Seoul) 43.09Present

StudyProposed approach(Model-13, Filter-6)

Turkey (Afyonkarahisar) 31.29

E. Akarslan et al. / Energy 73 (2014) 978e986 985

partial cloudy situation) the temperature will decrease. Therefore,Filter 6 with Model 13 gives the best result. The mesh plot of theprediction error for this model is presented in Fig. 8. The errorpixels are almost uncorrelated with each other. The uncorrelatedsamples indicate that the prediction almost totally exploits thepredictable part of the data by producing a residue correspondingto a white noise. Therefore, the model works with very good ac-curacy. The same conclusion can be extracted from the correlationsbetween themeasured and predicted values of solar irradiance thatare illustrated in Fig. 9. It is clear from Fig. 9 that prediction resultsare closely matching the actual data along the diagonal axis.Therefore, the slope of the linear fit for Fig. 9 is close to 45�, and thescatter is narrow along the diagonal axis. These observations indi-cate a high forecasting success for the proposed algorithm.

The experimental results show that the use of the derivativesover time improves the prediction accuracies. The derivative withrespect to 1-h-before works better than the derivative with respectto this hour on the previous day. This result was expected becausethe 1-h change from the previous day cannot carry as much sig-nificant information as the data obtained 1-h later. Furthermore, inall experiments, filter templates which use the 1-hour before, 1-daybefore and 1-hour 1-day before data (Filter-3, Filter-6 and Filter-9)were outperformed than the others. It means that the use of 1-h, 1-day and 1-h 1-day before data with together provide better pre-dictions than the use of filter templates which utilize from 1-hourbefore or 1-day before data.

In order to place the work in perspective with other publishedwork in the field the prediction accuracy of previously developedmodels with proposed approach are compared in the sense ofRMSE (%) errormeasure in this section. The RMSE (%) rates obtained

Table 5The RMSE (%) rates obtained in this study.

RMSE(%)

Filter-1 Filter-2 Filter-3

Model-1 53.10 54.68 36.90

Filter-7 Filter-8 Filter-9

Model-2 52.50 54.07 36.05Model-3 50.54 47.60 31.46Model-4 40.07 50.90 32.91Model-5 46.63 54.51 35.46Model-6 33.93 54.53 31.42Model-7 43.97 54.05 34.81Model-8 33.78 54.52 31.83

Filter-4 Filter-5 Filter-6

Model-9 50.45 47.60 31.35Model-10 39.47 50.80 32.20Model-11 33.59 47.46 N/AModel-12 32.66 48.05 N/AModel-13 33.83 54.23 31.29Model-14 33.37 54.45 31.49

in this study are shown in Table 5. Best result is obtained fromModel 13 with Filter 6. Table 6 illustrates the comparison of con-ventional methods and proposed approach according to RMSE (%)criterion. It can be seen from the Table 6 that proposed approachoutperforms the conventional methods.

6. Conclusions

In this work, a new approach for hourly solar radiation fore-casting is developed. This technique evaluates annual hourly solarirradiances, temperatures, extraterrestrials and their derivativesover time as multi-dimensional images. These images are linkedwith each other with the help of the proposed optimal coefficientlinear prediction filters. These filters scan the overall pattern, andthe pixel value corresponding to next hour's solar irradiance dataare predicted. In this work, 14 different models and 9 differentfilter-tap configurations were designed and tested. The perfor-mance of each model was compared with each other and withpreviously developed models. The experimental results showedthat the proposed approach provides significant improvement tothe prediction accuracy. The contribution of this paper to therelevant literature should be itemized as follows:

1 The multi-dimensional visualization of the data makes theprocess much clearer.

2 The use of the present, past and future values of extraterrestrialcalculations considerably improves solar irradiance predictionaccuracy (at rates ranging from 10% to 20%).

3 Optimal coefficient linear prediction filters can link proposedmulti-dimensions (i.e., image versions of solar irradiances anddata correlated with solar irradiances) and provide improvedaccuracy for solar irradiance prediction (at rates ranging from 3%to 40%).

4 The first-order derivatives in temperature, extraterrestrial andsolar irradiance data provide considerable information for theprediction process. Thismeans that variations onmeteorologicaldata carry more information than the irradiance itself to predictthe next value of solar irradiance.

5 The extraterrestrial irradiance carry very important informationabout the solar irradiance since it is the theoretical limit of thesolar irradiances. Moreover, it is possible to use the future valuesof the extraterrestrials since they are deterministic for a region(they can be calculated theoretically). Besides, the variations oftemperature data and its derivations carry considerable infor-mation about the weather conditions. To sum up, in this paper,all of these beneficial features are combined in proposed MDmanner. Since the MD filters, inherently takes the differentweather conditions into account, the proposed approach isrobust and can be applied to predict solar irradiances accurately,for any region in the world having different seasonal effects.

E. Akarslan et al. / Energy 73 (2014) 978e986986

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