aial

M

R

py

Received 21 October 2009; received in revised form 17 July 2010; accepted 25 August 2010Available online 28 September 2010

2010 Elsevier Ltd. All rights reserved.

cal grids. In this study, we focus on the prediction of global et al., 2008). Thereby many research works have shown theability of Articial Neural Networks (ANNs) to predicttime series of meteorological data. Moreover, if we compareto conventional algorithms based on linear models, ANNsoer an attractive alternative by providing nonlinearparametric models. Through the proposed study, we will

Corresponding author. Address: Universite de Corse, UMR SPE 6134,Route des Sanguinaires, 20000 AJACCIO, France. Tel.: +33 4 95 52 41 30;fax: +33 4 95 45 33 28.

E-mail address: marc.muselli@univ-corse.fr (M. Muselli).

Available online at www.sciencedirect.com

Solar Energy 84 (2010)Keywords: Time series forecasting; Pre-processing; Articial Neural Networks; PV Plant Energy Prediction

1. Introduction and review

An optimal use of the renewable energy needs its charac-terization and prediction in order to size detectors or to esti-mate the potential of power plants. In terms of prediction,electricity suppliers are interested in various horizons toestimate the fossil fuel saving, to manage and dispatch thepower plants installed and to increase the integration limitof renewable energy systems on non-interconnected electri-

solar irradiation on a horizontal plane for daily horizon,which might interest electricity suppliers. In this way, wehave investigated both in time series forecasting, which isa challenge in many elds, and in articial intelligence tech-niques, which are becoming more and more popular in therenewable energy domain (Mellit and Kalogirou, 2008;Mellit et al., 2009) and particularly for the prediction ofmeteorological data such as solar radiation (Mubiru,2008; Mubiru and Banda, 2008; Kalogirou, 2001; HocaogluCommunicated by: Associate Editor Christian Gueymard

Abstract

In this paper, we present an application of Articial Neural Networks (ANNs) in the renewable energy domain. We particularly lookat the Multi-Layer Perceptron (MLP) network which has been the most used of ANNs architectures both in the renewable energydomain and in the time series forecasting. We have used a MLP and an ad hoc time series pre-processing to develop a methodologyfor the daily prediction of global solar radiation on a horizontal surface. First results are promising with nRMSE 21% andRMSE 3.59 MJ/m2. The optimized MLP presents predictions similar to or even better than conventional and reference methods suchas ARIMA techniques, Bayesian inference, Markov chains and k-Nearest-Neighbors. Moreover we found that the data pre-processingapproach proposed can reduce signicantly forecasting errors of about 6% compared to conventional prediction methods such as Mar-kov chains or Bayesian inference. The simulator proposed has been obtained using 19 years of available data from the meteorologicalstation of Ajaccio (Corsica Island, France, 41550N, 8440E, 4 m above mean sea level). The predicted whole methodology has been val-idated on a 1.175 kWc mono-Si PV power grid. Six prediction methods (ANN, clear sky model, combination. . .) allow to predict the bestdaily DC PV power production at horizon d + 1. The cumulated DC PV energy on a 6-months period shows a great agreement betweensimulated and measured data (R2 > 0.99 and nRMSE < 2%).Forecasting of preprocessed dusing neur

Christophe Paoli a, Cyril Voyant a,b,aUniversity of Corsica, CNRS UM

bHospital of Castelluccio, Radiothera0038-092X/$ - see front matter 2010 Elsevier Ltd. All rights reserved.doi:10.1016/j.solener.2010.08.011ly solar radiation time seriesnetworks

arc Muselli a,, Marie-Laure Nivet a

SPE 6134, 20250 Corte, France

Unit, BP 85, 20177 Ajaccio, France

www.elsevier.com/locate/solener

21462160

ergparticularly look at the Multi-Layer Perceptron (MLP) net-work which has been the most used of ANN architectures inthe renewable energy domain (Mellit and Kalogirou, 2008;Mellit et al., 2009) and time series forecasting (Crone, 2005;Faraway and Chateld, 1995; Hu and Hwang, 2002; Jainet al., 1996). We compare this tool with other classical pre-dictors like ARIMA, Bayesian inference, Markov chainsand k-Nearest-Neighbors predictors. Previous works (Paoliet al., 2009) have shown that a rst pre-processing treatmentbased on an simple extraterrestrial normalization allows toobtain a better correlation between simulated and measuredradiation signals. During this study, we use an ad hoc timeseries pre-processing step before using neural networks.Indeed, a data pre-processing including seasonal adjust-ment can improve ANN forecasting performances (Zhangand Qi, 2005). Two strategies of pre-treatment are com-pared: the rst based on the clearness index and the second

Nomenclature

x^t, x^d;y time series model at time t or at day d and year yxt, xd,y time series data at time t or at day d and year ykp, mq parameters of ARMA processKp, MQ parameters specic to seasonal ARMA processd, D nonseasonal and seasonal dierenceScorrd;y , St stationary time series (clearness index) for the

day d and the year y or for the time tScorrd;y , S

t stationary time series (clear sky index) for theday d and the year y or for the time t

b tting parameter of the Solis clear sky models global total atmospheric optical depthh solar elevation angleHgh,clear sky clear sky global horizontal irradiance

(MJ/m2)

C. Paoli et al. / Solar Enusing the clear sky index. Contrary to these previous studies(Paoli et al., 2009), we validate the simulator on a1.175 kWp mono-Si PV power grid connected on a wallof our laboratory. This validation step presents both addi-tional diculties: the PV modules are 80 tilted to horizon-tal and are localized at 10 km from the original station.

The paper is organized as follow: Section 2 describes thecontext in which this research was done and the data wehave used and introduces time series forecasting, presentsseveral conventional prediction and modeling methods ofdaily irradiation data including ANNs. Using both clear-ness and clear sky indexes pre-processing treatments forANNs, all these predictors are then tested and comparedwith the same 19 years data set in Section 3 (data usedfor training and validation of the ANN comes from themeteorological station of Ajaccio airport). All referencedand ANN original methods are compared on a seasonalpoint of view using classical statistical coecients (MAE,MBE, RMSE, nRMSE, R2). The global approach is vali-dated and discussed on real tilted PV modules(1.175 kWp) for the calculation of the DC energy. At last,the denition of a new parameter D that reects the ten-dency of the prediction (the rst derivative linked to thesignal predicted slope) is presented and discussed. The Sec-tion 4 concludes and suggests perspectives.

2. Methodology

2.1. Context

In this work, measured global daily radiation data takenfrom a meteorological ground station are used to forecastglobal solar irradiation for the next day. The global radia-tion consists of three types of radiation: direct, diuse andground-reected (Liu and Jordan, 1962). The ground-reected radiation does not concern us because we try topredict the radiation on a horizontal surface. For clearsky, global radiation is relatively easy to model because it

Hd0 extraterrestrial solar radiation coecient for dayd (MJ/m2)

Hdgh;clear sky clear sky global horizontal irradiance (MJ/m2) integrated on the day d

yd,y moving average of the stationary global radia-tion for the day d and the year y

yd intermediate seasonal factor for day dyd nal seasonal factor for day dEpv PV plant power (MJ)gPV PV plant eciency (%)Ib daily global radiation (tilt of b) (MJ/m

2)S surface of PV wall (m2)Dp, Dm tendency of prediction and tendency of measure

(MJ)

y 84 (2010) 21462160 2147is primarily due to the distance from the sun to the sensor(Hay and Davies, 1980; Ineichen et al., 1990; Liu andJordan, 1962; Perez et al., 1990; Reindl et al., 1990). Withcloudy sky, these are mostly stochastic phenomenon, whichdepend on the local weather.

The study proposes to analyze the radiation time series(MJ m2) measured at the meteorological station ofAjaccio (Corsica Island, France, 41550N, 8440E, 4 mabove mean sea level) maintained by the French meteoro-logical organization (Meteo-France). Ajaccio has a Medi-terranean climate, hot summers with abundant sunshineand mild, dry, clear winters. It is located near the sea butthere is also relief nearby (30 km). This geographical con-guration can make nebulosity dicult to forecast.

The data used represent the global horizontal solar radi-ation and are available on an hourly basis for a periodfrom January 1971 to December 1989. A compactXARIA-C station (DEGREANE HORIZON), certiedby Meteo-France, is used to record horizontal radiation(Kipp and Zonen CMP6) with a 3 s time sampling. Thesensor works in a range 090,000 J/m2 and requires a

ergquality control every year. Only average measures over1 hour are stored in the national database. These last val-ues are integrated to extract a daily time series of the globalhorizontal radiation.

2.2. Time series forecasting

A time series (Faraway and Chateld, 1995) is a collec-tion of time ordered observations xt, each one beingrecorded at a specic time t (period). Time series canappear in a wide set of domains such as nance, productionor control, just to name a few. In rst approximation, atime series model (x^t) assumes that past patterns will occurin the future. In fact a time series model could be used onlyto provide synthetic time series statistically similar to theoriginal one. The modeling of the series begins with theselection of a suitable mathematical model (or class ofmodels) for the data. Then, it is possible to predict futurevalues of measurements (Brockwell and Davis, 2006).

2.3. Most popular forecasting methods

We have identied in the time series forecasting literaturethe following methods: ARIMA (classical method), Bayes-ian inferences, Markov chains and k-Nearest-Neighborspredictors. We present in the next sub-sections each methodwith the model obtained after a specic optimization.

2.3.1. ARIMA techniques

The ARIMA techniques (Hamilton, 1994) are referenceestimators in the prediction of global radiation eld. It is astochastic process coupling autoregressive component(AR) to a moving average component (MA). This kindof model is commonly called SARIMA (p, d, q)(P, D, Q)and is dened as follows (p, d, q, P, D and Q are integersand B the lag operator):

kpBKpBS1 Bd1 BSDX t mqBMQBSut 1where Xt is a time series, kp are the parameters of the auto-regressive part, mq are the parameters of the moving aver-age part and ut is an error term distributed as a Gaussianwhite noise, Kp and MQ are parameters of seasonal autore-gressive and seasonal moving average part. The d and Dare nonseasonal and seasonal dierence and S is seasonalperiod. The optimization of these parameters must bemade depending on the type of the series studied. To helpus in this study, we chose to use Grocer, a very completetoolbox compatible with the Scilab software (Duboisand Michaux, 2008). The criterion adopted to considerwhen an ARMA model ts to the time series is the nor-malized root mean square error obtained by: nRMSE < x y2 > = < x2 >

qwhere x represents the measure-

ment and y the prediction, using averaged values. Thisparameter is generated by the prediction of 2 years of radi-

2148 C. Paoli et al. / Solar Enation not used during the ARMA parameters calculationstep. After several experiments the model best suited tothe study of global radiation is an ARMA (2,2), its repre-sentation is:

X t ut X2i1

ki X ti X2j1

mj utj 2

where k1 = 1.47, k2 = 0.47, m1 = 1.19 and m2 = 0.22.A Students T-test (introduced by William Sealy Gosset in1908) has been used to verify that these coecients were sig-nicantly dierent from zero and residual auto-correlogramtests have been computed to verify white noise error terms.

Moreover, we also wanted to study a simplest modelamong the dierent ARIMA techniques. We have chosento study the AR type and we have found that the mostinteresting in this case was the AR(8).

2.3.2. Bayesian inference

The second classical technique we have chosen is calledBayesian inference (Diday et al., 1982; Celeux andNakache, 1994; Pole et al., 1994). In this method, evidencesor observations are used to update or to newly infer theprobability that a hypothesis may be true. To estimate theprobability that the series is in the state yk at time t, it is use-ful to use the Bayes theorem. This ideology is expressedmathematically by the following formula (where @ repre-sents the initial and known measured, k is the class and ykthe value of this class, P is the conditional probability):

X t yk argmaxk

P X t ykj@

argmaxk

P X t yk P @ X t ykj 3

A practical way to solve this equation is to make theassumption of conditional independence as follows:

P@jX t yk YJj1

P X tjjX t yk 4

which leads to:

X t yk

argmaxk

P X t yk YJj1

P X tjjX t yk !

5

To use this kind of predictor, we must establish the con-ditional probability table of the series and so quantify thelast term of the equation. For the optimization we haveused Matlab software and we have been able to identifythat the best prediction was obtained with 50 classes(0 < k < 51) and an order of J = 3.

2.3.3. Markov chains

In forecasting domain, some authors have tried to useso-called Markov processes (Logofet and Lesnaya, 2000;Muselli et al., 2001), specically the Markov chains. AMarkov process is a stochastic process with the Markov

y 84 (2010) 21462160property. Markov property means that, given the presentstate, future states are independent of the past states. In

bility of state i to the state j is dened by pi,j. The family of

ergthese numbers is called the transition matrix of the Markovchain R, we have:

pi;j PX t j X t1 ij 6the formulation of the prediction can be resumed by:

X tn X t Rn 7The choice of the dimension of the transition matrix

(number of class), and order of the chain (determinationof the lag prediction) was done on Matlab. We obtainedan optimal prediction for 50 classes and an order of 3:

X t yk argmaxk

X t1 R X t2 R2 X t3 R3 ek

8where ek represents the set of basic vectors of the transitionmatrix.

2.3.4. K-Nearest Neighbors

The k-Nearest Neighbors algorithm (k-NN) (Sharif andBurn, 2006; Yakowitz, 1987) is a method for classifyingobjects based on closest training examples in the featurespace. k-NN is a type of instance-based learning, or lazylearning where the function is only approximated locallyand all computation is deferred until classication. It canalso be used for regression. Unlike previous models, thistool does not use a learning base. The method consists inlooking into the history of the series for the case the mostresembling to the present case. By considering a series ofobservations Xt, to determine the next term Xt+1, we mustnd among anterior information, which minimize thequantity dened on Eq. (9) (d is the quadratic error).

ro argminr

dX t;X tr dX t1;X tr1

dX tk;X trk 9In this study we have chosen a k equal to 10. After thisargument of the minimum search, the prediction can bewritten:

X t1 X tro1 10

2.4. Articial Neural Networks for prediction

Articial Neural Networks (ANN) are intelligent sys-tems that have the capacity to learn, memorize and createrelationships among data. We present in this section anoverview of this methodology in the context of time seriesother words, the description of the present state fully cap-tures all the information that could inuence the futureevolution of the process. Future states will be reachedthrough a probabilistic process instead of a deterministicone. The proper use of these processes needs to calculaterstly the matrix of transition states. The transition proba-

C. Paoli et al. / Solar Enprediction. An ANN is made up by simple processing units,the neurons, which are connected in a network by a largenumber of weighted links where the acquired knowledgeis stored and over which signals or information can pass.An input xj is transmitted through a connection, whichmultiplies its strength by a weight wij to give a productxjwij. This product is an argument to a transfer functionf, which yields an output represented as: yi = f(xjwij) wherei is an index of neurons in the hidden layer and j is an indexof an input to the neural network (Mubiru, 2008; Cortezet al., 2002). In a Multi-Layer Perceptron (MLP), neuronsare grouped in layers and only forward connections exist.This provides a powerful architecture, capable of learningany kind of continuous nonlinear mapping, with successfulapplications ranging from Computer Vision, Data Analysisor Expert Systems, etc. (Cortez et al., 2001; Faraway andChateld, 1995; Jain et al., 1996; Hu and Hwang, 2002;Crone, 2005). One of the most interesting characteristicof ANNs is their ability to learn and to model a phenom-enon. Fig. 1 gives the basic architecture for a MLP applica-tion to time series forecasting during the learning task. Toestablish the prediction, a xed number p of past values areset as inputs of the MLP, the output is the prediction of thefuture value of the time series. This method called sliding-window technique uses a moving time window to select Ntimes, p inputs data for training.

In order to determine the best network conguration,we have tried to study all the parameters available in thisnetwork architecture. The principal parameters witch inu-ence the number of local minima and the complexity of thenetwork and its learning are: the number of input (these arethe neurons which represent the input layer of the neuralnet); the number of hidden layer and their number of neu-rons; the activation (or transfer) function; the learningalgorithm and the comparison function used during thelearning phase. Additionally we have to think about thenormalization of data, the learning sampling size and datadistribution between the learning, test and validationphases. For convenience we have chosen to optimize theparameters separately with an intuitive order of preferencefor all the ANN used in this work. Thus the chosen param-eter optimization follows the pattern below:

(1) Number of input: 115 inputs.(2) Number of hidden layers and neurons: 13 hidden

layers, and 150 neurons by layer.(3) Transfer function of hidden layer: linear (with or

without saturating), exponential and sigmoid.(4) Normalization of data: size of the interval for stan-

dardization between 0.6 and 1.(5) Learning sample size: 5006000 samples.(6) Learning algorithm and parameters: all gradient des-

cent, conjugate gradient, and quasi-Newtonalgorithms.

(7) Comparison function: mean squared normalizederror (MSNE), mean square error (MSE), sum ofsquare error (SSE), mean absolute error (MAE).

y 84 (2010) 21462160 2149(8) Data distribution (learningtestvalidation): 602020 and 801010.

t-1

t-2

t-3

ergX

X

X

xtInput windows

Sliding window technique

2150 C. Paoli et al. / Solar En(9) Customization: new study on the number of hiddenneurons, and the hidden layer to complete the process.

As a result of this iterative process, the selected networkhas three layers: input, hidden and output layers. Therewas no signicant dierence in the use of 1, 2 and 3 hiddenlayers architectures. Considering this fact one hidden layerwas used in order to minimize the complexity of the pro-posed ANN model. We tried several input layer congura-tions, best results were obtained with eight inputs whichreceived the endogenous entries St1,. . ., St8 normalizedon {0, 1}. We found that three neurons on the hidden layerwere sucient. Finally, as we wanted to predict the solarradiation at horizon 1, we have one neuron on the outputlayer bS t. Concerning the activation functions the bestresults were obtained with the Gaussian (hidden layer)and linear (output layer) functions. Concerning the train-ing algorithm, many experiences enabled us to choose theLevenbergMarquardt optimization (second-order algo-rithm) with 5000 epochs and l decrease factor (learningrate) of 0.5, all other parameters have default values. TheANN conguration leads to an output signal correspond-ing to a nonlinear auto-regression to the data entry:

bSt X3j1

W 2;j;1 eP8n1

W 1;n;jStnb1;j

2 b2;1 11

Xt-p

Fig. 1. MLP application to time series foret

Error

Xt

tXy 84 (2010) 21462160where parameters Wi,j,k represent interconnection weightsbetween layers and bi,j bias correction coecients. Thereader can nd in the Appendix A the full list of these val-ues for the most ecient ANN predictor.

We used the Matlab software and its neural networktoolbox to implement the network. The Matlab learning,testing and validation were set respectively to 80%, 10%and 10%. These three steps constitute the training of thenetwork, after which the weights and bias values are opti-mized. The training has concerned the years 19711987 andthe performance function was the mean square error(MSE). In the learning step, the data are presented to theANN following a sequential order. The predicted valuesof global radiation are compared to the years 19881989,which were not used during training.

3. Results and discussion

3.1. Results with non-stationary time series

In this section we examine if the proposed network wasreally interesting in terms of daily irradiation prediction.Indeed, we have compared its performance with the fore-casting results obtained with a naive predictor (365 valuesrepresenting the average over 17 years of the consideredday), order three Markov chains, order three Bayesianinferences, an order 10 k-NN, an order 8 AutoRegressive

casting (bias nodes are not displayed).

AR(8). Table 1 and Fig. 2 present results obtained in thecase of prediction for a period of 2 years (19881989).The main comparison function used is the nRMSE fordaily global radiation prediction (d + 1). Moreover, classi-cal statistical parameters enable us to evaluate the predic-tion quality: the absolute error is measured by the rootmean square error (RMSE), the Mean Bias Error (MBE)witch gives an idea about the error in term of under or overestimation, determination coecient (R2) and mean abso-lute error (MAE).

As can be seen the predictors other than AR(8) andANN give the same results, slightly better than thoseobtained with a naive predictor. Although AR and ANNcould be considered as the best predictors, nRMSE areimportant considering from the point of view of energeticapplications such as grid management with decentralizedgrid connected systems. In order to improve these results,the possibility to determine a signal pre-treatment as inputof AR and ANN is discussed. As proposed in (Diday et al.,1982), it appeared interesting to make stationary the time

Table 1Evaluation of the prediction quality for all prediction methods, forecasting years 1988 and 1989, 1 day horizon.

nRMSE (%) 95% IC RMSE (MJ/m2) MAE (MJ/m2) R2 MBE (MJ/m2)

Nave predictor (persistence) 26.13 0.00 4.65 3.03 0.69 0.001Nave predictor (daily average) 22.52 0.00 4.01 3.11 0.75 0.39Markov Chain (order 3) 25.85 0.00 4.59 3.03 0.69 0.23Bayes (order 3) 25.57 0.00 4.55 3.01 0.69 0.13k-NN (order 10) 25.20 0.00 4.48 3.19 0.70 0.03AR(8) 21.18 0.00 3.77 2.85 0.78 0.61ANN[8,3,1] 20.97 0.15 3.73 2.82 0.79 0.58

C. Paoli et al. / Solar Energy 84 (2010) 21462160 2151Fig. 2. Comparison between measures and forecasts for the daily global horizo(d) Knn; (e) AR; (f) ANN. The thick line represents the graph y = x and thental radiation. (a) Average value; (b) Markov chain; (c) Bayesian inference;normal line, the linear regression.

series as much as possible. So we try to nd a treatment inorder to eliminate seasonal components without changingthe other information. This treatment is called a seasonaladjustment process. In the present study, we have consid-ered the solar irradiation like a seasonal phenomenon.The choice of the methodology used depends on the natureof the seasonality. In our case the seasonality is very pro-nounced and repetitive, so very deterministic and notstochastic. In the next sections, we present how we haveused the physical modeling of radiation to determine astationarization methodology for the daily signal.

3.2. Result with stationary time series

3.2.1. Pre-processing proposed

In this section we present all the steps of an ad hoc timeseries pre-processing. Fig. 3 summarizes the dierent stepswe want to experiment. The classical methods like k-NN,Bayesian inference and also nave predictor dont requirethe use of pre-processing. In the literature, very few refer-ences relate ways the data stationary. Concerning the Mar-kov chains, we assumed that the simplest model withoutpre-processing was sucient in comparison with othersclassical methods. For the pre-processing, a rst treat-ment (step 1) allows us to clean the series of non-typicalpoints related to sensor maintenances or absence of mea-

surement. 4.1% of measurements were missing and replac-ing by the hourly average over the 19 years for the givenday.

Steps 2 and 3 will be described in the next section andlead to a series corrected utilizing clear sky or extraterres-trial normalizations. In case of stationary transformationby clear sky model the term Scorrd;y will be replaced byScorrd;y . The step 4 consists to use one of the forecastingmethods which have been outlined in the previous sectionwith the optimized MLP. Finally, step 5 allows to reversethe pre-processing treatment and to obtain the predictionof global irradiation. The VC parameter represents the sig-nal dispersion. In the remainder of this paper, we choosethe following naming convention: Xt designates the timeseries, and Xd,y the modeling of the variables, where d isthe day of the year y. To remove the seasonal dependencies,we chose to study both the clearness index and clear skyindex. According to Bird and Hulstrom (1981), the globalradiation on a horizontal plane under clear sky (dened asEarths standard atmosphere without cloud) is connectedto the extraterrestrial radiation and solar elevation angle.In daily case, the periodic phenomenon of irradiation isonly due to the extraterrestrial solar radiation coecientfor day d (Hd0) (Badescu, 2008). Thus we apply on the ori-ginal series Xd,y the ratio to trend method (Bourbonnaisand Terraza, 2008). This leads to a new series Sd,y, known

2152 C. Paoli et al. / Solar Energy 84 (2010) 21462160Fig. 3. Summarize of the protocol followed to obtain the predicted irradiation. VC is the variation coecient (ratio to standard deviation to the means).

Y d 1NXNy1

Y d;y

!16

yd 1365

X365d1

yd

!17

yd ydy

18

ergy 84 (2010) 21462160 2153as clearness index series (often noted as K in scienticpress):

Sd;y Xd;yHd0

12

In parallel, we wanted to test another method to removeseasonal eects, which is done by using the clear sky modelfor clear sky index. There are a lot of methods to determinethis model. In our case, we preferred the simplied Solisclear sky model (Ineichen, 2008) based on radiative trans-fer calculations and the LambertBeer relation (Muelleret al., 2004). In this case the clear sky global horizontalirradiance (Hgh,clear sky) reaching the ground is dened by:

Hgh;clear sky H 0 es= sinbh sinh 13where s is the global total atmospheric optical depth (0.37in our case), h is the solar elevation angle and b is a ttingparameter (0.35 for us). The daily integration of theHgh;clear sky parameter allows us to determine the daily solarratio for modeling Hdgh;clear sky . A series of test (not pre-sented in this paper) has allowed to validate the Solis modelon horizontal and 80 tilted daily global radiation We ob-tain a relation equivalent to Eq. (12):

Sd;y Xd ;y

Hdgh;clear sky14

These treatments aim to create a new distribution with-out periodicity. Although the previous pre-treatment tendsto make the time series stationary, a test of Fisher showsthat seasonality was not optimal. According to Bourbon-nais and Terraza (2008), after using a ratio to trend method(Eqs (12) and (14)) to correct rigid seasonalities, a ratio tomoving average can be used. This second ratio can beapplied when there is no analytical expression of the trend.In this case, we nd that Ho led a new seasonality which isdicult to model. That is the reason why we considered amoving average ratio to overcome this exible seasonality:

Y d;y Sd;y12m1

PmimSdi;y

15

In the present case, 2m + 1 = 365 days (period of origi-nal time series), we obtain m = 182. The term Sd,y is equiv-alent to Sd;y , they refer only to the chosen stationarizationmethodology. To complete the process, we use the 365 sea-sonal factors (yd). These are coecients which allow over-coming seasonality by a moving average ratio describedabove (Bourbonnais and Terraza, 2008). In order not todistort the series, we have considered that the total sumof the components of the series is the same before and afterthe report (nal seasonal factors yd) of the Eq. (18). Thetransition coecients (N = 18, the number of years of his-tory) and the average coecients of the regular 365 daysare given by Eq. (16). A new series seasonally adjusted that

C. Paoli et al. / Solar Enrepresents only the stochastic component of global radia-tion is given by Eq. (19).d

Scorrd;y Sd;yyd

and Scorr

d;y Sd;yyd

19

Thereafter, the juxtaposition of dierent values of Scorrd;yor Scorr

d;y will be resumed on stationary time series noted

St (extraterrestrial stationarization) and St (clear sky sta-

tionarization). After this step we needed to verify the eec-tiveness of pre-treatment on the time series. To achieve thiswe used a Fast Fourier Transform (FFT). The spectrum ofthe original series shows an important peak for the value365 days that disappears after the pre-treatment.

3.2.2. Comparison of ARIMA and ANNIn Table 2, the statistical errors nRMSE obtained in the

case of pre-treatment with the two best predictors (AR/ARMA and ANN) are shown. The results are slightly bet-ter with pre-processing. The reduction of the error predic-tion is low, but the dierence between the two methods issignicant, given the average and the condence interval.A nRMSE reduction from 21.0% to 20.2% represents again of 1%.

Table 3 details in both ANN pre-processing normaliza-tions (clear sky and clearness indexes) the annual predic-tion errors obtained for the years 1988 and 1989. In thisstudy (global horizontal radiation) no signicant dier-ences between the both stationarization methodologies(clearness index and clear sky index) are noted.

The condence interval is calculated after eight training-simulations providing information on the predictionrobustness. The weights are initiated before each simula-tion. With small condence intervals, it can be said thatthere are very few local minimums. The ANN learning steperror is obtained for RMSE < 104 (xed before the treat-ment), leading to the stop of the learning. The monthlyaverage error is the error (nRMSE) generated by monthlyaverage values of the global radiation prediction. It is infact the error of the monthly sum of global radiation. Ascan be seen the prediction is dierent from an average of4% of the measured data. The negative MBE (0.37 MJ/

Table 2Annual errors for ARMA and ANN prediction methods (nRMSE 95%CI), forecasting years 1988 and 1989, 1 day horizon.

Clearness indexmethodology (%)

Clear sky indexmethodology (%)

AR(8) without pre-processing 21.18 0 21.18 0ARMA(2,2) with pre-processing 20.31 0 20.32 0

ANN[8,3,1] without pre-processing 20.97 0.15 20.97 0.15ANN[8,3,1] with pre-processing 20.17 0.1 20.25 0.1

Campo delOro). Two additional diculties have to beovercome: the PV modules are tilted 80 to horizontaland are located at 10 km from the meteorological stationof Ajaccio-Campo delOro.

3.3.1. System description

The system has a nominal power of 6.525 kWp com-posed by 1.8 kWp and 4.725 kWp amorphous and mono-crystal PV modules respectively built in 6 independentpower subsystems (Fig. 5). PV power predictions from

Fig. 4. Seasonal errors for the daily prediction of the years 1988 and 1989(mean with 95% condence interval).

ergm2 and 0.32 MJ/m2) means that the solar potential isunderestimated over the year. With non-ordinary low irra-diation days, a tendency to overestimate is noted. The con-clusion is based on the simulations of global radiationduring these special days. We dened non-ordinary lowirradiation days as days with thick cloud cover causing avery low sunshine duration. The determination coecientR2 is greater than 0.8 that is good in relation to the noisepresent in measurements. There should be a compromisebetween RMSE and nRMSE. The nRMSE, computedfrom the mean global radiation obtained on the season,are useful for comparison and optimization (20.17% forclearness index and 20.25% for clear sky index). But forthe absolute interpretation of received energy we must lookat the RMSE. In order to better understand the predictorperformances, Fig. 4 shows the errors of prediction anddistinguishes the seasons for the years 1988 and 1989 fora clear sky pre-processing. Best results in term of forecastare obtained in summer. These results can be used forexample by energy managers who need to avoid usinghydraulic power plants in dry season.

The spring season is the most dicult to predict withaccuracy given the climate instabilities of this period. Theabsolute error is consistent. However, we nd that in sum-mer the error does not exceed 3.24 MJ/m2 (900 Wh/m2),while the irradiation is important. MBE are found nega-tive, which indicates an underestimation. The MBE is notsignicantly dierent from one season to another. Thuswe always have the same prediction error, whatever theseason. We systematically overestimate the days when theirradiation is minimal (winter). Moreover it is very dicultto predict the days when the irradiation had to be theoret-ically important. We would undoubtedly have improved

Table 3Annual prediction error for the years 1988 and 1989 with our MLP.

Clearness indexmethodology

Clear sky indexmethodology

nRMSE (%) 20.17 0.1 20.25 0.1RMSE (MJ/m2) 3.59 3.60MAE (MJ/m2) 2.65 2.67MBE (MJ/m2) 0.37 0.32R2 0.801 0.790Monthly average error (%) 3.9 4.1

2154 C. Paoli et al. / Solar Enthe results optimizing an ANN by season, but it wouldcomplicate the process, and tend to decrease the procedurerobustness.

After presenting the pre-treatment to be performed onthe series and after verifying its eectiveness, we proposein the next section to validate this process on a facadePV system.

3.3. Frontage PV

In order to validate the approach proposed, we decidedto use the previous simulator on a real frontage PV systeminstalled recently in our laboratory (Vignola near Ajaccio-y 84 (2010) 21462160ANN methodology described in this paper have been com-puted from the central part of the PV plant on the frontside exposed to the south (azimuth zero) and tilted at 80(Index PV on Fig. 5). The PV system consists of 9 SUN-TECH 175S-24Ac with 1.175 kWp nominal power con-nected to a 1.85 kW SUNNY BOY SMA inverter for PVproduction on the grid. The irradiance sensor used(Fig. 5) is an INGENIEURBURO SI-12TC calibrated bythe PTB Braunschweig (German national metrology Insti-tute): scale range between 0 and 1200 W/m2 requiring anannual quality control for calibration.

For the daily PV power calculation, we use in rstapproximation, a classical linear production based on a

with

inclined clear sky radiation. This software is very recog-nized in the eld of renewable energies (Bouhouras et al.,2009; Qoaider and Steinbrecht, 2009; Spanos and Duckers,2004). Eq. (21) has been built to convert the horizontal glo-bal radiation where b is the angle of inclination and I theglobal radiation. On the basis of a 6-months period, thisequation leads to a nRMSE = 14% from measured andsimulated daily tilted irradiation between 80 and 0(RMSE = 2.4 MJ/m2). To validate this methodology, wehave compared the tilt results with a methodology moresophisticated based on the Climed-2 and Klucher models.The rst model is used to determine the diuse and beam

PV

I

C. Paoli et al. / Solar Energy 84 (2010) 21462160 2155EPVMJ gPVIbS 20where Ib is the daily global irradiation on the PV system(MJ m2, b = 80), S is the usable surface of the PV systemunder consideration (S = 10.125 m2).

We propose to predict the DC electrical energy pro-duced by PV modules, at 1 day horizon between January15th 2009 and June 15th 2009. Unfortunately, no historicalirradiation data (on synoptic Campo delOro station andon our laboratory) are available for b = 80. So as it isshown in Fig. 6 we planed to use an ANN trained on thesite of Campo delOro (10 km from the PV modules) withhorizontal global radiation data. The training phasedescribed in Section 3 is performed; only the predictionphase is dierent because we need to transpose the tilteddata to horizontal data for use the ANN (and the reverseprocess as output).

3.3.2. Global radiation on a tilted planeconstant PV plant eciency gPV 15.3% (R2 = 0.9967),

Fig. 5. Frontage PV system (6.525 kWp). The PV plant under study ismentioned by the PV mark. Tilted (80) daily solar radiation Ib ismeasured on the top of the system facing south.Before continuing, it is necessary to develop a methodol-ogy to convert the horizontal data for the tilt angle of thePV. This approach used the PVSYST software version 4(www.pvsyst.com) to determine the global horizontal and

Preprocessing

Data horizontal and stationnarized

PreANN,

Measured Global

radiation (80)

Measureen

Xt

Fig. 6. PV energy prediction methodology based on3.3.3. Predictor models and results

In order to conduct an objective study, several forecast-ing experiments have to be completed in parallel. Theircomparisons allow to generate interpretable results of PVpower prediction (Eq. (20)). We have chosen to use modelsthat have shown the best results in the horizontal case pre-sented in previous sections, and also a nave predictor forquantify the prediction quality. The selected models are:

(A) Prediction omitting cloud cover: clear sky modeling

The Solis model (Section 3.2.1) is used and coupled withthe PVSYST process (Eq. (21)) to determine the tilted clearsky global radiation on Vignola site.

dictorARMA...

Data horizontal and stationnarized

forecasting

Postprocessing

d Electrical ergy

Electrical conversion

Comparator

Xt+1

Electrical approximationfraction, and the second to tilt the diuse component. Inhourly case, the Climed-2Klutcher methodology is veryconvincing, (Noorian et al., 2008; Notton et al., 2006a,2006b) but in daily case (integration of hourly data), theresults are similar with the methodology proposed in thisarticle. For both methodologies the nRMSE values arealmost identical. Because the ClimedKlutcher methodol-ogy is only applicable on some locations near the Mediter-ranean Sea, we have chosen to develop in this article onlythe PVSYST method which is applicable everywhere. Inthe Eq (21), the diuse and ground-reected radiationsare implied in the bracketed term:

Ib Ib0: IbIb0

clear sky21a daily horizontal irradiation ANN simulator.

(B) Prediction by average values

Daily solar radiation averages (b = 0) are computedfrom the 19 years of historical data available on the siteof Campo delOro, then inclined at 80 (Eq. (21)).

(C) ANN without pre-processing

ANN is used with time series not made stationary,directly on the raw data.

(D) ANN with the clearness index

This method follows the schematic of Fig. 6. The timeseries is made stationary with the processing based onextraterrestrial radiation.

(E) ANN with clear sky index

Equivalent to (D) but the processing used is the ratio toclear sky model (Solis).

(January 15June 15, 2009), the ANN with clear sky index(E) methodology slightly outperforms the other. Indeed,all results computed are consistent (nRMSE = 34.1%,RMSE = 0.81 MJ,MBE = 0.03 MJ andMAE = 0.62 MJ).But the dierences are small among (C)(F) experiments i.e.between the use of ANN or ARMA predictor, with or with-out pre-processing. The max value of RMSE is 0.85 MJ((C), ANN without pre-processing), the min is 0.81 MJ((E), ANN clear sky) and the dierence is only 0.04 MJ/day. Only clear sky (A) and average (B) models are theworst forecasters (respectively nRMSE = 46.6% and47.6%). Bi-monthly errors show dierent behavior for pre-diction processes. The methods with clear sky index arethe most relevant for the months for which the signal is verynoisy (JanuaryFebruary prediction with (E), ANN, andMarchApril with (F), ARMA). In the case of MayJuneprediction, the simple method ANN without pre-processing(C) is the most relevant with nRMSE = 15.5% while thesophisticate methods with pre-processing (D)(F) generatenRMSEP 16.2%. The MBE factor is very weak for (C)(F) (max = 0.12 MJ in (F) case is very acceptable). Inthe case of the wall-PV power system with 80 tilt, the max-

ricatics

2156 C. Paoli et al. / Solar Energy 84 (2010) 21462160(F) ARMA(2,2) with clear sky index

Equivalent to (E) but the ANN are replaced by the lin-ear methodology (ARMA).

Identical irradiation data from Campo delOro meteoro-logical station are used as inputs for the processes (B)(F)except model A where a physical model Solis computesclear sky conditions. Table 4 presents errors obtained onthe period of prediction (JanuaryJune) and details of thebi-monthly errors. As can be seen on the 6-months period

Table 4Bi-monthly prediction errors and 6-months prediction errors on the electJanuary 2009 to 15 June 2009. The mean and variance are the characteris

A clear sky B average C ANN

JanuaryFebruary Mean: 2.39 MJ/m2

nRMSE (%) 56.4 57.1 41.7RMSE (MJ) 1.47 1.49 1.09MBE (MJ) 1.04 1.02 0.13MAE (MJ) 1.06 1.12 0.98

MarchApril Mean: 2.27 MJ/m2

nRMSE (%) 49.3 45.1 38.3RMSE (MJ) 1.22 1.11 0.95MBE (MJ) 0.79 0.58 -0.01MAE (MJ) 0.79 0.76 0.76

MayJune Mean: 1.95 MJ/m2

nRMSE (%) 17.9 18.9 15.5RMSE (MJ) 0.35 0.37 0.31MBE (MJ) 0.20 0.08 0.03MAE (MJ) 0.66 0.79 0.63

JanuaryJune Mean: 2.19 MJ/m2

nRMSE (%) 46.6 47.6 35.6RMSE (MJ) 1.11 1.13 0.85

MBE (MJ) 0.65 0.64 0.05MAE (MJ) 0.66 0.79 0.63imum radiation value is obtained during the winter monthsconcluding that the E process is the most relevant. ANNand ARMA perform almost similar, denoting the stochasticnature of the time series and thus the impossibility to pre-dict the cloud eect on solar radiation.

These two processes present no signicant dierences;only these two predictors are considered in the following.Fig. 7 shows the cumulative daily predicted PV power vs.measured ones from January 15 to June 15, 2009. This typeof chart is very interesting to check the gap between mea-surement and simulation over the long-term (seasonal vari-

l energy from PV plant (1.175 kWp, b = 80) for 1 day horizon from 15of the global radiation time series on the period.

D ANN clearness E ANN clear sky F ARMA clear sky

Std. dev.: 0.06238.7 37.9 38.41.01 0.99 1.01

0.01 0.02 -0.040.90 0.87 0.89

Std. dev.: 0.05938.3 37.9 37.70.95 0.94 0.93-0.03 -0.02 0.010.77 0.77 0.75

Std. dev.: 0.03116.2 16.4 16.40.32 0.32 0.33

0.07 0.09 0.120.63 0.62 0.62

Std. dev.: 0.05534.7 34.1 34.20.82 0.81 0.810.03 0.03 0.050.63 0.62 0.62

ergy 84 (2010) 21462160 2157C. Paoli et al. / Solar Enations and under or overestimations of the PV energy). Allgures show a good correlation between measured and pre-dicted data (R2 > 0.99). We understand easily that (C)(F)methods are similar and although nRMSE are important,the accumulated predictions are almost identical to themeasures (2.1% for ANN without pre-processing, 1.5%for ANN with clearly sky and clarity index and also forARMA with clear sky index).

It is important to understand why the daily errors aresignicant. The rst hypothesis is based on the high fre-quency noise series. In fact, the sampling frequency is thesame as the frequency of the noise. It seems very unlikelythat the ANN can predict extra-ordinary days at leastif the previous days cloud cover is ordinary. The secondhypothesis is that ANNs (and ARMA) do not take risksand propose irradiation value centered, on a mean valuewith a small standard deviation. The output of the networkis then an improved average that ts the precursor trendof previous lags. Table 5 presents the decomposition for theANN prediction errors (E process).

The decomposition of the error is quite signicant. Weunderstand easily that the total error committed is impor-tant: the combination of pre-processing and modeling data(tilting, normalization for ANN) induces an error of about20%. Furthermore the year 2009 (JanuaryJune) is not rep-resentative, there was an escalation of 13% error due to

Fig. 7. Cumulative predictions and measures. (a) ANN without pre-processinARMA with clear sky index.specic meteorological conditions (rainy period). More-over, the 80 module tilt is very destabilizing, because thewinter is a period of high climatic instability, but also a per-iod where irradiation (80) is the most important.

To try to compare all prediction methods (AF) in a laststep, we tried to present a new factor that reects the ten-dency of prediction (the rst derivative linked to the signalpredicted slope). In the case of the measured signal, we usethe Dm coecient dened as follows (I is the measured val-ues of PV electricity energy): Dm It 1 It, and in thecase of prediction using the coecient Dp (bI represents thevalues predicted for the DC PV energy):Dp bI t 1 It.

It is interesting to note that a Students T-test betweenthe coecient Dm and Dp for ANN and ARMA shows thatpredictions with the ANN process better represent the real-ity (t = 0.364 and p = 0.71 for ANN vs. t = 0.584 andp = 0.56 for ARMA), recall that p-value is the probability

Table 5Decomposition of the nRMSE for electrical energyprediction (E process).

nRMSE (%)

Pre-processing and modeling 20Specicity of year 2009 13Electrical conversion 1Total observed error 34

g, (b) ANN with clearness index, (c) ANN with clear sky index and (d)

m p

[0;1 e [and] 1 + e; +1[: area where one commits an

tors as Markov chains. The choice of a pre-processing builton clearness index (with Hd0) of clear sky index (H

dgh;clear sky)

leads to comparable results.These simulation tools have been successfully validated

on the DC energy prediction of a 1.175 kWp mono-Si PVpower system connected to the grid. On a seasonal pointof view, ANN with clear sky pre-processing, representsfor winter months (January, February) an adequate solu-tion (nRMSE 37%). For summer months, ANN withoutpre-processing gives the best results (nRMSE 15%).Cumulative simulated and measured PV productions arein very good agreement (R2 > 0.99) validating the wholeprediction process (nRMSE < 2% for a 6-months period).

Finally, a new dierential variable D has been intro-duced to study predicted tendency errors based on the rstderivative predicted signal. According to a Student test,ANN and ARMA simulated daily irradiation proles con-rm the good accuracy for the predicted tendency at d + 1.These two methodologies are similar.

ergerror of prediction; take e = 0.2. In the rst interval ona Dp > Dm and the other interval Dp < Dm,

[1 e; 1 + e]: good prediction zone, the sign and theabsolute value is correct.

Table 6 presents the D values and its results for thefour classes considering for each interval, the number ofevents obtained on the studied period.

With this methodology we must recognize that ANNand ARMA are a little dierent. (A) and (B) processesare not ecient for daily DC PV energy predictions(Fig. 8a). It must be said that it is very dicult to distin-guish the quality of prediction made with ANN or ARMA(Fig. 8b). Only the Student test used with the coecient Dmand Dp allows separating the prediction quality of the twomodels with ANN slightly higher than ARMA.

4. Conclusion and perspectives

In this paper, an ANN prediction approach developedto determine global irradiation at daily horizon (d + 1)which can help electrical managers with grid-PV power sys-tems connected is presented. This prediction model hasbeen compared to other prediction methods (AR, ARMA,k-NN, Markov Chains, etc.). We have used an ad hoc timeseries pre-processing (based on clearness or clear skyindexes) and a time series prediction designed MLP.that the averages between the forecasting and measuredvalues are identical. So according to this indicator, on aver-age, ANN is more representative than ARMA for the trendestimation. Table 6 presents the coecient comparing thequality of prediction based on the ratio D = Dm/Dp for allprediction methods (A)(F). This type of parameter Dallows determining four interpretation intervals:

0, D < 0, and vice versa,

Table 6Tendency of prediction with D-values distributions for all processes.

D-Values A B C D E F

1.2 38 44 38 46 43 43

2158 C. Paoli et al. / Solar EnAlthough the location was very specic, with the proximityto the sea and the mountain that can greatly aect nebulos-ity, we have obtained relevant results.

Without pre-processing, AR(8) and ANN models pre-sents better daily RMSE of about 3.6 MJ/m2 (998 Wh/m2) and nRMSE 21% compared to Markov chain,Bayes, k-NN methods where nRMSE 2526%. However,annual pre-processing ANN methods based on clearnessindex and clear sky index reduce forecasting errors ofabout 56% (nRMSE 20%) compared to classical predic-0

10

20

30

40

50

-5 -4 -3 -2 -1 0 1 2 3 4 5

-Value

Freq

uenc

y

AB

0

10

20

30

40

50

Freq

uenc

y

CDEF

(a)

(b)

-5 -4 -3 -2 -1 0 1 2 3 4 5

-Value

Fig. 8. Distribution of D-values (X-axis) representing the quality ofprediction. (a) AB processes not relevant, (b) CF processes correspond-ing to best ANN and ARMA simulators.

y 84 (2010) 21462160In the future, this tool (ANN) could eventually help thesystem manager, for implanting new PV systems especiallyon isolated electrical grid where 30% for renewable energypower system represents an integration limit. With opera-tional prediction tools, this threshold can be increased to50% allowing to limit the utilization of fossil plants to sup-ply electricity.

Moreover, it seems important to study shorter time hori-zons (hour). As a matter of fact, electrical managers arealso interested to horizons that can range from hour

Meteorological Organization (Meteo-France) which has

0:90

ergB1 0:531:02

B@ CAB2 0:13

References

Badescu, V., 2008. Modelling Solar Radiation at the Earth Surface,Recent Advances. Viorel Ed., ISBN: 978-3-540-77454-9.

Bird, E., Hulstrom, R.L., 1981. A Simplied Clear Sky Model for Directand Diuse Insolation on Horizontal Surfaces, SERI/TR-642761.Solar Energy Research Institute, Golden, USA, Colorado.

Bouhouras, A.S., Marinopoulos, A.G., Labridis, D.P., Dokopoulos, P.S.,2009. Installation of PV systems in Greece reliability improvement inthe transmission and distribution system. Electric Power SystemsResearch.

Bourbonnais, R., Terraza, M., 2008. Analyse Des Series Temporelles.Dunod Ed., Paris, 318p, ISBN: 9782100517077.

Brockwell, P.J., Davis, R.A., 2006. Time Series: Theory and Methods,second ed. Springer series in statistics, USA, 577p, ISBN: 0387974296.supervised the data collection from their data bankfor Campo delOro synoptic station. We thank JeanPANIGHI (University of Corsica) for the supply of irradi-ance sensors technical characteristics.

Appendix A

MLP weight values after training with clarity index forthe global radiation (most ecient conguration). W1 cor-responds to the matrix where each element W1n,j, is theweight related to the number n of input neurons (from 1to 8) and the number j of hidden neurons (from 1 to 3).W2 corresponds to a vector with three elements represent-ing the weights between the three hidden nodes and theoutput. B1 is the bias vector related to each hidden neuron(three values) and B2 the output bias value.

W 11:26 0:02 0:26 0:001 0:08 0:34 0:12 0:130:02 0:19 0:29 0:89 0:41 0:64 0:78 0:850:21 0:50 0:14 0:09 0:61 0:48 0:49 0:02

0@ 1AW 2 0:47 0:07 0:07 0 1to several hours: from 3 h to 24 h. In this new congura-tion, other ANN architecture types have to be studied: timedelay, recurrent ANNs, etc.

In the long-term, it would be also very interesting tostudy a network trained on an urban data, and used onother site with equivalent geographical feature, and maybecombine both ANN and Geographic Information Systems(GIS) approaches.

Acknowledgements

This work was partly supported by the TerritorialCollectivity of Corsica. We thank the French National

C. Paoli et al. / Solar EnCeleux, G., Nakache, J.P., 1994. Analyse Discriminante sur VariablesQualitatives. Polytechnica, Paris, 270 p, ISBN: 2840540274.Cortez, P., Rocha, M., Neves, J., 2001. Evolving time series forecastingneural network models. In: Proceeding of International Symposium onAdaptive Systems: Evolutionary Computation and ProbabilisticGraphical Models.

Cortez, P., Sollari Allegro, F., Rocha, M., Neves, J., 2002. Real-timeforecasting by bio-inspired models. In: Proceeding (362) ArticialIntelligence and Applications.

Crone, S.F., 2005. Stepwise selection of articial neural networks modelsfor time series prediction journal of intelligent systems. Department ofManagement Science Lancaster University Management School Lan-caster, United Kingdom.

Diday, E., Lemaire, L., Pouget, J., Testu, F., 1982. Elements Danalyse deDonnees. Dunod, Paris.

Dubois, E., Michaux, E., 2008. Grocer: an econometric toolbox for Scilab..

Faraway, J., Chateld, C., 1995. Times series forecasting with neuralnetworks: a case study. Research report 95-06 of the statistics group,University of Bath.

Hamilton, J.D., 1994. Times series analysis. ISBN 0-691-04289-6.Hay, J.E., Davies, J.A., 1980. Calculation of the solar radiation incident

on an inclined surface. In: Proceedings if the First Canadian SolarRadiation Workshop, pp. 5972.

Hocaoglu, F.O., Gerek, O.N., Kurban, M., 2008. Hourly solar forecastingusing optimal coecient 2-D linear lter and feed-forward neuralnetworks. Solar Energy 82 (8), 714726.

Hu, Y., Hwang, J., 2002. Handbook of neural network signal processing.ISBN 0-8493-2359-2.

Ineichen, P., 2008. A broadband simplied version of the Solis clear skymodel. Solar Energy 82 (8), 758762.

Ineichen, P., Guisan, O., Perez, R., 1990. Ground-reected radiation andalbedo. Solar Energy 44 (4), 207214.

Jain, K., Jianchang, M., Mohiuddin, K.M., 1996. Articial neuralnetworks: a tutorial. IEEE Computer 29 (3), 3144.

Kalogirou, S.A., 2001. Articial neural networks in renewable energysystems applications: a review. Renewable and Sustainable EnergyReviews 5, 373401.

Liu, B.Y.H., Jordan, R.C., 1962. Daily insolation on surfaces tiltedtowards the equator. Trans SHRAE 67, 526541.

Logofet, D.O., Lesnaya, E.V., 2000. The mathematics of Markov models:what Markov chains can really predict in forest successions. EcologicalModelling 126, 285298.

Mellit, A., Kalogirou, S.A., 2008. Articial intelligence techniques forphotovoltaic applications: a review. Progress in Energy and Combus-tion Science 1 (1), 5276.

Mellit, A., Kalogirou, S.A., Hontoria, L., Shaari, S., 2009. Articialintelligence techniques for sizing photovoltaic systems: a review.Renewable and Sustainable Energy Reviews 13 (2), 406419.

Mubiru, J., 2008. Predicting total solar irradiation values using articialneural networks. Renewable Energy 33 (10), 23292332.

Mubiru, J., Banda, E., 2008. Estimation of monthly average daily globalsolar irradiation using articial neural networks. Solar Energy 82 (2),181187.

Mueller, R.W., Dagestad, K.F., Ineichen, P., Schroedter-Homscheidt, M.,Cros, S., Dumortier, D., Kuhlemann, R., Olseth, J.A., Piernavieja, G.,Reise, C., Wald, L., Heinemann, D., 2004. Rethinking satellite-basedsolar irradiance modelling: the SOLIS clear-sky module. RemoteSensing of Environment 91, 160174.

Muselli, M., Poggi, P., Notton, G., Louche, A., 2001. First order Markovchain model for generating synthetic Typical Days series of globalirradiation in order to design PV stand alone systems. EnergyConversion and Management 42 (6), 675687.

Noorian, A.M., Moradi, I., Kamali, G.A., 2008. Evaluation of 12 modelsto estimate hourly diuse irradiation on inclined surfaces. RenewableEnergy 33, 14061412.

Notton, G., Poggi, P., Cristofari, C., 2006a. Predicting hourly solarirradiations on inclined surfaces based on the horizontal measure-

y 84 (2010) 21462160 2159ments: performances of the association of well-known mathematicalmodels. Energy Conversion and Management 47, 18161829.

Notton, G., Cristofari, C., Poggi, P., 2006b. Performance evaluation ofvarious hourly slope irradiation models using Mediterranean experi-mental data of Ajaccio. Energy Conversion and Management 47, 147173.

Paoli, C., Voyant, C., Muselli, M., Nivet M.L., 2009. Solar radiationforecasting using ad-hoc time series preprocessing and neural net-works. ICIC2009, Korea, Paper ID: 1812, pp. 898907, doi: 10.1007/978-3-642-04070-2_95.

Perez, R., Ineichen, P., Seals, R., 1990. Modelling daylight availability andirradiance components from direct and global irradiance. Solar Energy44 (5), 271289.

Pole, A., West, M., Harrison, J., 1994. Applied Bayesian Forecasting andTime Series Analysis. Chapman and Hall/CRC, ISBN: 0412044013.

Qoaider, L., Steinbrecht, D., 2009. Photovoltaic systems: a cost compet-itive option to supply energy to o-grid agricultural communities inarid regions. Applied Energy.

Reindl, D.T., Beckman, W.A., Due, J.A., 1990. Evaluation of hourlytilted surface radiation models. Solar Energy 45 (1), 917.

Sharif, M., Burn, D.H., 2006. Simulating climate change scenarios usingan improved K-nearest neighbor model. Journal of Hydrology 325 (14), 179196.

Spanos, I., Duckers, L., 2004. Expected cost benets of building-integrated PVs in UK, through a quantitative economic analysis ofPVs in connection with buildings, focused on UK and Greece.Renewable Energy 29, 12891303.

Yakowitz, S., 1987. Nearest neighbors method for time series analysis.Journal of Time Series Analysis 8, 235247.

Zhang, G.P., Qi, M., 2005. Neural network forecasting for seasonal andtrend time series. European Journal of Operational Research 160, 501514.

2160 C. Paoli et al. / Solar Energy 84 (2010) 21462160

Forecasting of preprocessed daily solar radiation time series using neural networksIntroduction and reviewMethodologyContextTime series forecastingMost popular forecasting methodsARIMA techniquesBayesian inferenceMarkov chainsK-Nearest Neighbors

Artificial Neural Networks for prediction

Results and discussionResults with non-stationary time seriesResult with stationary time seriesPre-processing proposedComparison of ARIMA and ANN

Frontage PVSystem descriptionGlobal radiation on a tilted planePredictor models and results

Conclusion and perspectivesAcknowledgementsAppendix AReferences