Solar Radiation Forecasting Using Ad-Hoc Time Series Preprocessing and Neural Networks.

Transcription

1 Solar Raiation Forecasting Using A-Hoc Time Series Preprocessing an eural etworks. Christophe Paoli, Cyril Voyant, Marc Muselli, Marie Laure ivet To cite this version: Christophe Paoli, Cyril Voyant, Marc Muselli, Marie Laure ivet. Solar Raiation Forecasting Using A-Hoc Time Series Preprocessing an eural etworks.. Springer Berlin / Heielberg. International Conference on Intelligent Computing (ICIC 2009), Sep 2009, Ulsan, orth Korea. Lecture otes in Computer Science, 5754/2009, pp , 2009, Computer Science. <0.007/ _95>. <hal > HAL I: hal Submitte on 4 Dec 20 HAL is a multi-isciplinary open access archive for the eposit an issemination of scientific research ocuments, whether they are publishe or not. The ocuments may come from teaching an research institutions in France or abroa, or from public or private research centers. L archive ouverte pluriisciplinaire HAL, est estinée au épôt et à la iffusion e ocuments scientifiques e niveau recherche, publiés ou non, émanant es établissements enseignement et e recherche français ou étrangers, es laboratoires publics ou privés.

2 Solar raiation forecasting using a-hoc time series preprocessing an neural networks Christophe Paoli, Cyril Voyant,2, Marc Muselli, Marie-Laure ivet University of Corsica, CRS UMR SPE 634, (Vignola, Rte es Sanguinaires, Ajaccio Campus Grimali, Corte), France 2 Hospital of Castelluccio, Raiotherapy Unit, BP 85, 2077 Ajaccio, France {christophe.paoli, cyril.voyant, marc.muselli, marie-laure.nivet}@univ-corse.fr Abstract. In this paper, we present an application of neural networks in the renewable energy omain. We have evelope a methoology for the aily preiction of global solar raiation on a horizontal surface. We use an a-hoc time series preprocessing an a Multi-Layer Perceptron (MLP) in orer to preict solar raiation at aily horizon. First results are promising with nrmse < 2% an RMSE < 998 Wh/m². Our optimize MLP presents preiction similar to or even better than conventional methos. Moreover we foun that our ata preprocessing approach can reuce significantly forecasting errors. Keywors: Time Series, Preprocessing, Seasonality, Multi-Layer Perceptron Introuction Artificial intelligence techniques are becoming more an more popular in the renewable energy omain [], [2] an particularly for the preiction of meteorological ata such as solar raiation [3], [4] [5] [6]. Thereby many research works have shown the ability of Artificial eural etworks (As) to preict times series of meteorological ata. In this stuy an accoring to electricity suppliers, we focus on the preiction of global solar irraiation on a horizontal plane for aily horizon. In this way, we have investigate time series forecasting which is a challenge in many fiels. Because it has mae tremenous progress in the past twenty years in terms of theory, algorithms an applications, we have chosen to stuy As. Moreover, if we compare to conventional algorithms base on linear moels, As offer an attractive alternative by proviing nonlinear parametric moels. Through the propose stuy, we will particularly look at the Multi-Layer Perceptron (MLP) network which has been the most use of A architectures in the renewable energy omain [], [2]. The originality of our stuy is to a an a-hoc time series preprocessing step before using neural networks. Inee, as seen in [7] a ata preprocessing incluing eseasonalization an etrening can improve A forecasting performance. As A part of this research is foune by the Territorial Collectivity of Corsica.

3 global solar raiation has a eterministic part, we want to take into account this specificity. The paper is organize as follow: the section 2 escribes the physical phenomena we want to preict an introuces our a-hoc time series preprocessing. Section 3 presents the neural network architecture we esigne. Results are presente an iscusse in section 4 where several conventional methos for estimation an moeling of the meteorological ata are compare with our methoology. Section 5 conclues an suggests perspectives. 2 Data analysis an preprocessing There are two approaches that allow quantifying solar raiation: the physical moeling base on physical processes occurring in the atmosphere an influencing solar raiation [8], an the statistic solar climatology mainly base on time series analysis [8]. As alreay sai we have chosen to combine these two methos in a gray box approach to improve the quality of preiction. In this work, we have use the physical phenomena in an attempt to overcome the seasonality of the resource. 2. Meteorological ata Our stuy proposes to analyze the raiation time series (Wh.m-2) measure at the meteorological station of Ajaccio (METEO FRACE, Corsica, France, 4 55', 8 44'E). The ata representing the global solar raiation were measure on a aily basis from January 97 to December 989. Thus we have a X t time series to forecast for time t+; that is at horizon. To achieve this, we choose to use a gray box (or semi-physical) moel where time series preiction an moeling are mixe. For time series preiction only past values are use to forecast the future values at a given horizon. In the case of moeling, the ifferent physical processes involve are taken into account in orer to represent the variable however the horizon is. In the remainer of this paper, we choose the following naming convention: X t esignate the time series, an X,y is the moeling of the variables, where is the ay of the year y. In the next section, an explanation of the physical phenomenon is propose, an then we escribe our time series preprocessing. 2.2 Physical phenomenon We can observe in Figure that the global raiation consists of three types of raiation: irect raiation, iffuse raiation an groun-reflecte raiation [0]. The groun-reflecte raiation oes not concern us because we try to preict the raiation on a horizontal surface where the groun reflecte raiation oes not make sense. For clear sky, ie without clou cover, global raiation is relatively easy to moel because it is primarily ue to the istance from the sun sensor [9], [0], [], [2], [3]. It is not the same, when there are clous near the etectors. Inee, these are mostly stochastic phenomena, which epen on the weather site.

4 Fig.. Origin of the three types of raiation: irect raiation, iffuse raiation an grounreflecte raiation on a etector at groun level (left of the figure). Moification of the global irraiation profile accoringly to clous cover (right of the figure). The spectral analysis of the global raiation series highlights the high perioicity of the phenomenon (almost 365 ays perio). As propose in [7], it appeare wise to make the time series stationary as much as possible. Deseasonalize an etren a series allows to eliminate seasonal an tren components without changing the other information. In the present stuy, we have not consiere the solar irraiation like a tren process but only like a seasonal process. The choice of the methoology use epens on the nature of the seasonality. As in our case the seasonality is very pronounce an repetitive, so very eterministic an not stochastic. Moreover, it is possible to physically quantify the components of our irraiation series. Like we have seen in figure, the global solar irraiance on horizontal plane epens on irect an iffuse raiation. This specificity allows us to apprehen the perioicity of the phenomenon, an to reuce the non-stationarity of the series. In the next section, we propose to stationarize raiation ata to overcome the eterministic component which is easily quantifiable. Thus we evote to the preiction of clou cover inclue in the global raiation of the site. 2.3 A-hoc time series preprocessing In fact, by iviing the series by aily extraterrestrial raiation [4], we can quantify the annual perioicity. It is the first step of our stationarization process. In first approximation, it is possible to quantify the eterministic component of global raiation by the extraterrestrial solar raiation alone (H 0 ). Thus we apply on the original series X,y (where is the ay an y is the year) the ratio to tren metho. This leas to a new series (S,y ), known as series of inex clarity: St S, y = x, y / H 0 =. () After this step a new rigi seasonality is upate, we can lift it with the use of perioic seasonal factors [4]. This treatment aims to create a new istribution without perioicity. Although this pre treatment tens to stationarize the time series, a test of Fisher shows that seasonality was not optimal. Accoring Bourbonnais [4],

5 after using a ratio to tren metho (H o in this case) to correct rigi seasonalities, we can use a ratio to moving average. This secon ratio can be applie when there is no analytical expression of the tren. In our case, we fin that H o le a new seasonality which is ifficult to moel. That's why we mae a moving average ratio to overcome the seasonality. In the case of flexible seasonality, ie ranom in amplitue or perio, the filtering techniques by successive moving averages are recommene. /. m y, y = S, y. S + i, y 2. m + i= m (2) In our case as 2m+ = 365 ays, we obtain m = 82. To complete the process, then we use the 365 seasonal factors (y ). These are in fact coefficients which allow to overcome rigi seasonality by a moving average ratio escribe above. In orer to not istort the series, we have consiere that the total sum of the components of the series is the same before an after the report (final seasonal factors y * of the equation 5). The transition coefficients ( = 8, the number of years of history) an the average coefficients of the regular 365 ays are given by the equations 3 an 4. A new series seasonally ajuste that represents only the stochastic component of global raiation is given by the equation 6. y =., y y= y 365 y = y. 365 y= (3) (4) y * = y / y. (5) S = S y. (6) corr *, y, y / In the next section, we present the RA architecture use. 3 eural network architecture an esign The search for the ieal network structure is a complex an crucial task. We have aopte a fee forwar Multi-Layer Perceptron (MLP), which is the most commonly use in the renewable energy omain [], [2]. In orer to etermine the best network configuration, we have trie to stuy all the parameters available in this network architecture. To perform this optimization, we have consiere the practice hypothesis that parameters are orthogonal. We have optimize parameters by consiering each other constants. We use the Matlab software an its neural network toolbox to implement our network. The Matlab training an testing ata sets were set respectively to 80% an

6 W,,3 b,2 b,3 b2, 20%. As a result of this iterative process, the selecte network (see figure 2) has three neuron layers: input, hien an output layer. There was no significant ifference in the use of, 2 an 3 hien layer architectures. One hien layer was use in orer to minimize the complexity of the propose A moel. We trie several input layer configurations:, 2, 4, 8 an 5 variables. Best results were obtaine with 8 inputs which receive as input the enogenous entries S t-,.., S t-8 normalize on {0,}. The same process was use to fix the number of hien layer neurons. We foun that 3 neurons were sufficient. Finally, we have one neuron on the output layer Ŝ. t Concerning the transfer functions the best results were obtaine with the Gaussian (hien layer) an linear (output layer) function. Regaring the training algorithm, many experiences have enable us to choose the Levenberg Marquart optimization (secon-orer algorithm) with 5000 epochs an a ecrease factor (mu) of 0.5. Default values have been use for all other parameters. The early stopping technique was set to the maximum valiation failure (the parameter max_fail = 5). Bias St- b, W,, Bias St-2 W,2, W,,2 St-3 W2,, W,, St-4 W2,2, St-5 St-6 W2,3, Sˆ ( W, n, j. Sn + b, j ) n= t = W2, j,. e + j = b 2, St-7. St-8 Fig. 2. Architecture of the optimize MLP. The learning has concerne the years 97 to 987 an the performance function was mean square error MSE. 4 Results an iscussions Figure 3 summarizes the protocol that has allowe us to conuct our experiences an valiate our approach. A first treatment (step of the figure 3) allows to clean the series of atypical points. We have replace them by the average over the 9 years for the hours an the ays corresponing to the problems.

7 y measure ata ; VC=0,539 X, Global Raiation (W.h/m²) Time (Days) S, y clearness inex ; VC=0,326 0,9 0,8 clearness inex 0,7 0,6 0,5 0,4 0,3 0,2 0, Time (Days) corr S, y clearness inex, with mobil average an perioic coefficients ; VC=0,323 corr S ˆ +, y etrene ata (no unit),2 0,8 0,6 0,4 0, X ˆ +, y Time (Days) Fig. 3. Summarize of the protocol followe to obtain the preicte irraiation. Steps 2 an 3 have been escribe in the previous section an lea to a series correcte. In the step 4 we compare classical forecasting methos outline in the next section with our optimize MLP. Finally step 5 allows to reverse the preprocessing treatment an to obtain the preiction of global irraiation. 4. The classical forecasting methos In orer to measure the effectiveness of our approach, we have ecie to compare it with the following classical forecasting methos. The ARIMA techniques [5], [6] are reference estimators in the preiction of global raiation fiel. It is a stochastic process coupling autoregressive component (AR) to a moving average component (MA). After several experiments, we have obtaine an ecie to use an ARMA (2,2). Bayesian inference [7], [8] is another classical technique. In this metho eviences or observations are use to upate or to newly infer the probability that a hypothesis may be true. We have ientifie that the preiction was better if we ha 50 classes an an orer of 3. Some authors have trie to use so-calle Markov process [9], specifically the Markov chains, which is a stochastic process. The mean iea of this technique is that the escription of the present state fully captures all the information that coul influence the future evolution of the process. Future states will be reache through a probabilistic process instea of a eterministic one. In our case, we obtaine 50 for the imension of the transition matrix (number of class) an an orer of 3 for the chain (etermination of the preiction lag). The k-nearest neighbors algorithm (k-) [20] is a metho for classifying objects base on closest training examples in the feature space. k- is a type of instance-base learning, or lazy learning where the function is only approximate locally an all computation is eferre until classification. It can also be

8 use for regression. Unlike previous ones this tool oes not use a learning base. The metho consists in looking into the history of the series for the case the most resembling to the present case. In our stuy we choose a k equal to 0. The following section presents the results obtaine. 4.2 Results To etermine whether our network was really interesting in terms of aily preiction of solar raiation, we compare its performances with the forecasting results obtaine with a naive preictor (average over 8 years of the ay consiere), orer 3 Markov chains, orer 3 Bayesian inferences, an orer 0 k-, an orer 8 AR without preprocessing, an ARMA(2,2) with preprocessing. The preicte results for each combination were compare statistically using three parameters: the Root Mean Square Error (RMSE), the normalize RMSE (nrmse), an the Mean Bias Error (MBE): RMSE = 2 ( C i M i ). i+ (6) nrmse = 2 2 ( Ci Mi ) / ( Mi ). i + i = (7) MBE = ( C i M i ). i+ (8) Table presents results we have obtaine in the case of an annual error for aily preiction of global solar raiation. Table. Annual error for all preiction methos, years 988 an 989, 8 simulations Preiction methos nrmse Confience interval aïve preictor 26 % 0% Markov Chain (orer 3) 25, % 0% Bayes (orer 3) 25,6 % 0% k- (orer 0) 25,20 % 0% AR(8) without preprocessing 2,8 % 0,2% ARMA(2,2) with preprocessing 20,3 % <0,% A[8,3,] without preprocessing 20,97 % 0,5% A[8,3,] with preprocessing 20,7 % 0,% We highlight that the preictors other than ARMA an A give the same results, slightly higher than those obtaine with a naive preictor. Even without preprocessing, ARMA an A are the best preictors. The preprocessing improves the quality of preiction an allows access to the 20% error. C i an M i are respectively the i th calculate an measure values an is the total number of

9 observation. Table 2 etails in the MLP case, the annual preiction errors obtaine for the years 988 an 989. Table 2. Annual preiction error for the years 988 an 989 with our MLP Arithmetic Mean 95% confience interval nrmse 20,2 % 0, % RMSE 997,97 6,333 MBE -04,239 28,794 R² 0,80 0,002 Monthly average error 3,9 % 0,4 % The confience interval is calculate after 0 simulations. Given the small size of the confience intervals, we can say that there are very few local minimums. The monthly average error represents the error for the value of the irraiation. We obtain that the combination of the preiction of global raiation receive after month is ifferent from an average of 4% of the aggregate measure. The negative MBE means that we unerestimate the solar potential on average over the year. Since we have an atypical ay of low irraiation then there is a tenency to overestimate. The etermination coefficient R² is greater than 0,8. Figure 4 shows the errors of preiction an istinguishes the seasons for the years 988 an 989. As we can see best results; i.e. less important error, in term of forecast are obtaine in summer. These results can be use for example by energy managers who nee to avoi using hyraulic power plants in ry season. Fig. 4. Seasonal errors for the aily preiction of the years 988 an 989 (mean with 95% confience interval). There shoul be a compromise between RMSE an nrmse. The nrmse are useful for comparison an optimization. But for the interpretation of energy, we must look at the RMSE. The spring season is the most ifficult to preict. The absolute error is consistent. However, we fin that in summer the error oes not excee 900 Wh / m², while the irraiation is important. MBE are foun negative, which inicates an unerestimation. The MBE is not significantly ifferent from one season to another. Thus we will always have the same preiction error, whatever the season. Finally, Figure 5 compares the real ata of solar raiation with the results obtaine with our MLP with preprocessing. The error of preiction is also rawn. The increase in errors at the beginning of the cycle correspons to the spring when the clou isturbances are very important. We can see it is very ifficult to preict the raiation cause of very noisy variable. Figure 6 shows the correlation between experimental an simulate global irraiation.

10 Fig. 5. Real ata of solar raiation an results obtaine with our MLP with preprocessing an error of preiction. Dashe line is the error; soli line is the real ata of solar raiation an re points are the preiction. Fig. 6. Correlation between experimental an simulate global irraiation. We systematically overestimate the ays when the irraiation is minimal (winter). The points that lie at the very top of the line y = x shows that it is very ifficult to preict the ays when the irraiation ha to be theoretically important. We woul unoubtely have improve the results optimizing an A by season, but it woul complicate the proceure, an ten to ecrease the robustness of the proceure. The next section conclues this paper an suggests prospects. 6 Conclusion an perspectives This paper has evelope an A preictor approach to etermine global irraiation at aily horizon in orer to help electrical managers. We have use an a-hoc time series preprocessing an a time series preiction esigne MLP. Although the location was very specific, with the proximity to the sea an the mountain that can greatly affect nebulosity, we have obtaine relevant results. Seasonal RMSE are less than 998 Wh/m-2 (nrmse < 2%). A processes presents a great interest compare to classical stochastic preictor like ARIMA. Moreover we foun that our ata preprocessing approach reuce significantly forecasting errors. The next step of our work will be to valiate our preictor on real photovoltaic system. It was recently installe in our laboratory an we are awaiting ata. In the future, it seems important to stuy shorter time horizons. As matter of fact, electrical

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