Short-term solar irradiance and irradiation forecasts \\ via different time series techniques: \\ A preliminary study
C�dric Join$^{1
2
3
This communication is devoted to solar irradiance and irradiation short-term forecasts, which are useful for electricity production. Several different time series approaches are employed. Our results and the corresponding numerical simulations show that techniques which do not need a large amount of historical data behave better than those which need them, especially when those data are quite noisy.
EFEA14.METEO.tex

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\title{Short-term solar irradiance and irradiation forecasts \\ via different time series techniques: \\ A preliminary study}


% author names and affiliations
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\author{C�dric Join$^{1,2,3}$, Cyril Voyant$^{4,5}$, Michel Fliess$^{2,6}$, \\ Marc Muselli$^5$, Marie-Laure Nivet$^5$, Christophe Paoli$^7$ and Fr�d�ric Chaxel$^1$
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1.- CRAN (CNRS, UMR 7039), Universit\'{e} de Lorraine \\ BP 239, 54506 Vand{\oe}uvre-l\`{e}s-Nancy, France
\\  {\tt \{cedric.join, frederic.chaxel\}@univ-lorraine.fr} \\
2.- AL.I.E.N. (ALg�bre pour Identification \& Estimation Num�riques) \\ 24-30 rue Lionnois, BP 60120, 54003 Nancy, France \\
{\tt \{cedric.join, michel.fliess\}@alien-sas.com} \\
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3.- Projet Non-A, INRIA Lille -- Nord-Europe, France \\
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4.- H�pital de Castelluccio, Unit� de Radioth\'erapie \\ BP 85, 20177 Ajaccio, France \\
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5.- SPE (CNRS, UMR 6134), Universit� di Corsica Pasquale Paoli \\ 20250 Corte, France \\
{\tt \{voyant, marc.muselli, marie-laure.nivet\}@univ-corse.fr} \\
6.- LIX (CNRS, UMR 7161), \'Ecole polytechnique \\ 91128 Palaiseau, France \\
{\tt Michel.Fliess@polytechnique.edu} \\
7.- D\'epartement de G�nie Informatique, Universit� Galatasaray  \\ No:36 34357 Ortak�y, Istanbul, Turkey \\
{\tt cpaoli@gsu.edu.tr}
}

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\begin{abstract}
This communication is devoted to solar irradiance and irradiation short-term forecasts, which are useful for electricity production. Several different time series approaches are employed. 
Our results and the corresponding numerical simulations show that techniques which do not need a large amount of historical data behave better than those which need them, especially when those data are quite noisy. 

\end{abstract}

\begin{IEEEkeywords}
Meteorological forecasts, solar irradiance and irradiation, electricity production, time series, trends, quick fluctuations, machine learning, multilayer perceptron, big data.

\end{IEEEkeywords}


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\section{Introduction}
\subsection{Generalities}\label{gener}
The aim of this communication is twofold:
\begin{enumerate}
\item It is devoted to solar irradiance and irradiation forecasts during rather short time intervals. As already noted in several publications (see, \textit{e.g.}, \cite{vi}, and \cite{notton1,notton2}) such predictions turn out to be very useful for electricity production by some systems like thermal and photovoltaic ones, where responses to solar variations are fast and complex. 
\item The above forecasts are achieved via different time series techniques which are compared. Although such an endeavor is of course not new (see, \textit{e.g.}, \cite{hossain,martin,rei}), this communication might be one of the very first which takes into account the implementation issues by evaluating quite closely their need of 
large historical data (see, also, \cite{hontoria}).
\end{enumerate}
Remember that times series (TS), which play an important r�le in numerical weather prediction  (see, \textit{e.g.}, \cite{duchon,hoca}, and the references therein), are also utilized in many other domains, like, for instance, econometrics and biology. We will not be employing here the classic setting for time series,\footnote{See \cite{meuriot} for a most interesting account and an analysis of the connections with econometrics and finance.} but
\begin{itemize}
\item  various topics from computer science, like artificial neural networks, computational intelligence and machine learning (see, \textit{e.g.}, \cite{beh,crone,de,kasa,kalo,mel,paoli,yona}).\footnote{The literature on the application to meteorological forecasts is already quite important. Combined with a lack of space it explains the absence of any review of those approaches in this communication.}
\item a new viewpoint on time series, which started in financial engineering (see, \textit{e.g.}, \cite{trend,agadir}), where
\begin{itemize}
\item no explicit mathematical model is needed,
\item the notions of \emph{trends} and \emph{quick fluctuations} are key ingredients.
\end{itemize}
\end{itemize}
Our time series were recorded every minute thanks to a meteorological station,\footnote{Vaisala Weather Transmitter WXT520 with an additive photosynthesis measuring probe head FLA613GS and the data loggers AHLBORN.}
which is on the roof of the \emph{Institut Universitaire de Technologie Nancy-Brabois}. In this paper, we are focusing on the solar irradiance in W/m$^2$ and irradiation in Wh/m$^2$. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Our multivariable meteorological TS stem from 
%munutes measurements  during 11 years in Ajaccio (Corsica). They contain four components, 
%\textit{i.e.}, the solar radiation (Sun in $Wh/m^2$), the humidity (Hum in $\%$), the temperature (Temp in $�C$) 
%and the wind speed (wind in Beaufort scale). Let us emphasize that the  solar radiation and the temperature exhibit very strong periodicities, whereas those features are much milder with the two other quantities.
%The prediction horizon is 1 hour which is also the time step of the TS. Those quite repetitive behaviors do
%necessitate to revisit techniques which work otherwise in a more or less satisfactory manner. 
%%%%%%%%%%%%%%%%%%%%%%

\subsection{Overview of the various techniques}
Our TS analysis may be divided  into two groups, \textit{i.e.}, with or without a need of big data.
\subsubsection{Settings without large historical data}
\begin{itemize}
\item The setting \emph{without model} (WM), which was sketched in Section \ref{gener}, should be related from an engineering viewpoint to the success of \emph{model-free control} \cite{csm}.\footnote{It might be interesting here to emphasize here that this viewpoint is also useful for greenhouses \cite{sousse}.}
\item The \emph{persistence} (P) method \cite{paoli}, which is a trivial machine learning viewpoint, assumes no change between the forecast and the last measure.
\end{itemize}
\subsubsection{Settings with large historical data}
A rather precise modeling is needed, which is quite greedy. We are considering four cases:
\begin{itemize}
\item A \emph{MultiLayer Perceptron} (MLP), which is a standard artificial neural network, is among the most popular tool for analyzing meteorological TS (see, \textit{e.g.}, \cite{beh,crone,v1}).
\item If we take into account a probabilistic description of the TS, a stationary hypothesis is often necessary. It yields a CSI-MLP, \textit{i.e.}, a \emph{clear sky index} in connection with the MLP in order to deal with stationary TS \cite{v2}. 
The broadband Solis model \cite{solis} is used: it allows generating the global irradiation without clouds. The ratio between measurement and clear sky solar radiation defines the clear sky index.
\item The \emph{scaled persistence} (SP) \cite{v2} adds to the classical persistence P a clear sky procedure like the above one. This methodology is equivalent to clear sky index persistence
\end{itemize}
After a short presentation of time series without model in Section \ref{AI}, various experiments are reported in Section \ref{exp}. Some concluding remarks may be found in Section \ref{conclusion}.


\section{Time series without any explicit model}\label{AI}
\subsection{Nonstandard analysis and the Cartier-Perrin theorem}
Take the time interval $[0, 1] \subset \mathbb{R}$ and introduce as
often in \emph{nonstandard analysis}\footnote{See, \textit{e.g.}, \cite{diener1,diener2} and \cite{lobry} for an introduction to this fascinating domain which is stemming from mathematical logic.} the infinitesimal sampling
$${\mathfrak{T}} = \{ 0 = t_0 < t_1 < \dots < t_N = 1 \}$$
where $t_{\iota + 1} - t_{\iota}$, $0 \leq \iota < N$, is {\em
infinitesimal}, {\it i.e.}, ``very small'' (\cite{diener1,diener2}).  A \emph{time series} $X(t)$ is a function $X:
{\mathfrak{T}} \rightarrow \mathbb{R}$.
\begin{remark}
The reader, who is not familiar with nonstandard analysis, should not be afraid by the wording \textit{infinitesimal sampling}. It just means in plain words that the sampling time interval is ``small'' with respect to the total  
recording time. Let us also stress that several time scales are most natural within this formalism.
 
\end{remark}

The {\em Lebesgue measure} on ${\mathfrak{T}}$ is the function
$\ell$ defined on ${{\mathfrak{T}}} \backslash \{1\}$ by
$\ell(t_{i}) = t_{i+1} - t_{i}$. The measure of any interval $[c, d]$, $0 \leq c \leq d \leq 1$, is its length $d -c$.  The
\emph{integral} over $[c, d]$ of the time series $X(t)$ is the sum
$$\int_{[c, d]} Xd\tau = \sum_{t \in [c, d]} X(t)\ell(t)$$
$X$ is said to be $S$-{\em integrable} if, and only if, for any
interval $[c, d]$ the integral $\int_{[c, d]} |X| d\tau$ is
\emph{limited}, \textit{i.e.}, not infinitely large, and also infinitesimal, if $d - c$
is infinitesimal.

$X$ is $S$-{\em continuous} at $t_\iota \in {\mathfrak{T}}$ if, and
only if, $f(t_\iota) \simeq f(\tau)$ when $t_\iota \simeq
\tau$.\footnote{$a \simeq b$ means that $a - b$ is infinitesimal.}
$X$ is said to be {\em almost continuous} if, and only if, it is
$S$-continuous on ${\mathfrak{T}} \setminus R$, where $R$ is a {\em
rare} subset.\footnote{See \cite{cartier} and \cite{lobry} for this technical definition.}
$X$ is \emph{Lebesgue integrable} if, and only if, it is
$S$-integrable and almost continuous.

A time series ${\mathcal{X}}: {\mathfrak{T}} \rightarrow \mathbb{R}$
is said to be {\em quickly fluctuating}, or {\em oscillating}, if,
and only if, it is $S$-integrable and $\int_A {\mathcal{X}} d\tau$
is infinitesimal for any {\em quadrable} subset.\footnote{A set is
\emph{quadrable} \cite{cartier} if its boundary is rare.}

Let $X: {\mathfrak{T}} \rightarrow \mathbb{R}$ be a $S$-integrable
time series. Then, according to the Cartier-Perrin theorem
\cite{cartier}, the additive decomposition
\begin{equation}\label{decomposition}
\boxed{X(t) = E(X)(t) + X_{\tiny{\rm fluctuat}}(t)}
\end{equation}
holds where
\begin{itemize}
\item the \emph{mean} $E(X)(t)$ is Lebesgue integrable,
\item $X_{\tiny{\rm fluctuat}}(t)$ is quickly fluctuating.
\end{itemize}
The decomposition \eqref{decomposition} is unique up to an additive
infinitesimal.
\begin{remark}
See \cite{lobry} for a less demanding presentation of the Cartier-Perrin theorem.
\end{remark}
\begin{remark}\label{noise}
The above quick fluctuations should be viewed like corrupting noises in engineering  \cite{bruit}. Determining the trend is therefore similar to noise attenuation. According to the mathematical definition of quick fluctuations, this may be achieved by integrating on a short time window or, more generally, by any low pass filter. 
\end{remark}

\subsection{On the numerical differentiation of a noisy signal}\label{ins} 
Let us start with the first degree polynomial time function $p_1 (t)
= a_0 + a_1 t$, $t \geq 0$, $a_0, a_1 \in \mathbb{R}$. Rewrite
thanks to classic operational calculus (see, \textit{e.g.},
\cite{yosida}) $p_1$ as $P_1 = \frac{a_0}{s} +
\frac{a_1}{s^2}$. Multiply both sides by $s^2$:
\begin{equation}\label{1}
s^2 P_1 = a_0 s + a_1
\end{equation}
Take the derivative of both sides with respect to $s$, which
corresponds in the time domain to the multiplication by $- t$:
\begin{equation}\label{2}
s^2 \frac{d P_1}{ds} + 2s P_1 = a_0
\end{equation}
The coefficients $a_0, a_1$ are obtained via the triangular system
of equations (\ref{1})-(\ref{2}). We get rid of the time
derivatives, i.e., of $s P_1$, $s^2 P_1$, and $s^2 \frac{d
P_1}{ds}$, by multiplying both sides of Equations
(\ref{1})-(\ref{2}) by $s^{ - n}$, $n \geq 2$. The corresponding
iterated time integrals are low pass filters which, according to Remark \ref{noise}, attenuate the
corrupting noises. A quite short time window is sufficient for
obtaining accurate values of $a_0$, $a_1$. Note that estimating $a_0$ yields the trend according to Remark \ref{noise}.

The extension to polynomial functions of higher degree is
straightforward. For derivatives estimates up to some finite order
of a given smooth function $f: [0, + \infty) \to \mathbb{R}$, take a
suitable truncated Taylor expansion around a given time instant
$t_0$, and apply the previous computations. Resetting  and utilizing
sliding time windows permit to estimate derivatives of various
orders at any sampled time instant.

\begin{remark}
See \cite{easy,NumDiff} for more details.
\end{remark}


\subsection{Application of the above calculations}
A ``good'' forecast $X_{\text{est}}(t+T)$, at time $t + T$, $T > 0$, of the time series $X(t)$ may be written
$$\boxed{X_{\text{est}}(t+T)=X_{\text{trend}}(t)+[\dot{X}_{\text{trend}}(t)]_eT}$$ 
where $[\dot{X}_{\text{trend}}(t)]_e$ denotes the derivative which is estimated as indicated in Remark \ref{noise} and Section \ref{ins}. In this paper,  
$T= 60$ minutes or $15$ minutes.

Let us stress that 
\begin{itemize}
\item $X_{\text{trend}}$ and $\dot X_{\text{trend}}$ are calculated with different time windows, \textit{i.e.}, respectively $10$ minutes and $75$ minutes.
\item What we predict is the trend and not the quick fluctuations (see also \cite{trend,agadir}).
\end{itemize}
%
\section{Experiments}\label{exp}

\subsection{Data}
We benefit from solar irradiance data, which were recorded during daylight every minute during three years, \textit{i.e.}, 2011, 2012 and 2013.
Two major types of experiments are distinguished:
\begin{enumerate}
\item irradiation is obtained according to the computation of the irradiance mean every hours, thus $\text{radiation}=\text{mean}(\text{radiance}(t-59), \text{radiance}(t-58),..., \text{radiance}(t))$ with $t=60 \times k$ minutes, $k=\{0, 1, 2, ...\}$.
\item Instantaneous measurements are used. 
\end{enumerate}
See the differences in Figure \ref{res}. Set $60$ minutes for the forecasting horizon.

\begin{figure}%[h]
\centering
\includegraphics[scale=.49]{radis.eps}
\caption{Irradiance in W/m$^2$ (blue $-$), Irradiation in Wh/m$^2$ (red $- -$) and Irradiance each hour in W/m$^2$ (black $*$) on 13 February 2013}\label{res}
\end{figure} 


\subsection{Irradiation forecasting}
All forecasting methods, namely P, SP, MLP and CSI-MLP, use directly irradiation measures, but not WM, for which records every minute permit better performances.
Figures \ref{ran} and \ref{ranZ} display the results for a $60$min forecast. More details may be found in the first lines of Tables \ref{TabL1} and \ref{TabL2}. The best method turns out to be SP. Even if the mathematical function ``clear sky'' plays a crucial r�le, his simplicity is remarkable. Note also that WM method is third with respect to the normalized $\text{L}^1$ norm. It becomes fourth with respect to the normalized $\text{L}^2$ norm.

\begin{figure}[h!]
\centering
\includegraphics[scale=.49]{radiation.eps}
\caption{Irradiation forecasting in Wh/m$^2$, year 2013}\label{ran}
\end{figure}

\begin{figure}[h!]
\centering
\includegraphics[scale=.49]{radiationZ.eps}
\caption{Zoom of Figure \ref{ran}}\label{ranZ}
\end{figure}

\subsection{Irradiance forecasting}
We try now to forecast the irradiance, \textit{i.e.}, the instantaneous hourly values (see the stars $*$ in Figure \ref{res}). Contrarily to the previous calculations, averages are no more useful. Corrupting noises play therefore
a much more important r�le. 
Only instantaneous values are used for P, SP, MLP and CSI-MLP methods. For MLP and CSI-MLP, three learning data sizes were considered, \textit{i.e.}, one, two, and three years. The errors (normalized values) are related to the lower values computed among seven runs. Figures \ref{rad} and \ref{radZ} demonstrate that not only WM yields the best results but is also more robust with respect to the noises. WM is able moreover 
to take advantage of minute data: this a true advantage. Tables \ref{TabL1} and \ref{TabL2} confirm those statements.

\begin{figure}[h!]
\centering
\includegraphics[scale=.49]{radiance.eps}
\caption{Radiance forecasting in W/m$^2$, year 2013}\label{rad}
\end{figure}
\begin{figure}[h!]
\centering
\includegraphics[scale=.49]{radianceZ.eps}
\caption{Zoom on figure \ref{rad}}\label{radZ}
\end{figure}



%\subsection{Quantitative result and conclusions}

%\begin{table*}[h!]
%  \centering
%  \begin{tabular}{c|c|c||c|ccc|ccc}
%\hline
% & P & WM & SP & \multicolumn{3}{c}{MLP} & \multicolumn{3}{c}{MA-MLP}\\
%%	\hline
% & & & & 1 year& 2 years& 3 years&1 year&2 years& 3 years\\
%	\hline
%Radiation& 0.4724& 0.3688& 	{\bf 0.3311}& 0.3524 &0.3502 &	0.3476 & 	0.3623&0.3598&0.3558\\
%Radiance& 0.7475& {\bf 0.3619}& 	0.6558& 	0.5803 & 0.5770 & 0.5748& 	0.6456&0.6226&0.6185\\
%\hline
%\end{tabular}
%  \caption{Normalized $\mathcal{L}^2$ of 5 predictors. In bold the best results}\label{Tab1}
%\end{table*}

%{\LARGE 
\begin{table*}%[h]
  \centering
  \large
  \begin{tabular}{c|c|c||c|ccc|ccc}
\hline
 & P & WM & SP & \multicolumn{3}{c}{MLP} & \multicolumn{3}{c}{CSI-MLP}\\
%	\hline
 & & & & 1 year& 2 years& 3 years&1 year&2 years& 3 years\\
	\hline
Irradiation& 0.3373& 0.2393& 	{\bf 0.1962}& 0.2445 &0.2512 &0.2432 & 	0.2251&0.2240&0.2236\\
Irradiance& 0.4700& {\bf 0.2665}& 	0.3449& 	0.4705 & 0.4153 & 0.4177& 	0.3654&0.3544&0.3575\\
\hline
\end{tabular}
  \caption{Normalized $\text{L}^1$ norm of the five predictors for a $60$ minutes horizon and a time step of $60$ minutes. Best results in bold}\label{TabL1}
\end{table*} %}

\begin{table*}%[h]
  \centering
    \large
  \begin{tabular}{c|c|c||c|ccc|ccc}
\hline
 & P & WM & SP & \multicolumn{3}{c}{MLP} & \multicolumn{3}{c}{CSI-MLP}\\
%	\hline
 & & & & 1 year& 2 years& 3 years&1 year&2 years& 3 years\\
	\hline
Irradiation& 0.6200& 0.4952& 	{\bf 0.44276}& 0.4763 &0.4751 &0.4690& 	0.4654&0.4626&0.44291\\
Irradiance& 1.0177& {\bf 0.5126}& 	0.8943& 	0.8860 & 0.8487 & 0.8447& 	0.7917&0.7872&0.7842\\
\hline
\end{tabular}
  \caption{Normalized $\text{L}^2$ norm for the five predictors for a $60$ minutes horizon and a time step of $60$ minutes. Best results in bold}\label{TabL2}
\end{table*}




%radiation
%radiance (instantan�e)

\subsection{Extension to shorter time forecasts}
Switch now to a $15$ minutes forecast for the irradiation. The two most efficient methods are compared, \textit{i.e.}, SP and WM.
The normalized norm $\text{L}^2$ gives, for a time step of $5$ minutes and a $15$ minutes time horizon, respectively for SP and WM, $0.5624$ and $0.5399$. The superiority of the behavior of WM  may be given by the percentage $4.2 \%$.
Figures \ref{w} and \ref{s} show the predictor behavior predictor during one day in winter and in summer with slightly different time scales.



\begin{figure}[h!]
\centering
\includegraphics[scale=.49]{stW.eps}
\caption{Winter day (W/m$^2$)}\label{w}
\end{figure}

\begin{figure}[h!]
\centering
\includegraphics[scale=.49]{stS.eps}
\caption{Summer day (W/m$^2$)}\label{s}
\end{figure}




\section{Conclusion}\label{conclusion}
It is noteworthy to stress that methods with no need of a large amount of historical data give often better results, especially with noisy time series. Further investigations will hopefully confirm and precise those preliminary results. 
They should moreover yield new elements for epistemological discussions on the nature of ``good'' weather forecasts (see, \textit{e.g.}, \cite{murphy}).
\newpage
\section*{Acknowledgment}
The installation of the meteorological station was made possible by the project  E2D2, or \textit{\'Energie, Environnement \& D\'eveloppement Durable}, which is supported by the \emph{Universit\'e de Lorraine} and
the \textit{R\'egion Lorraine}.

\vspace{4cm}

\begin{thebibliography}{1}

%\bibitem{IEEEhowto:kopka}
%H.~Kopka and P.~W. Daly, \emph{A Guide to \LaTeX}, 3rd~ed.\hskip 1em plus
%  0.5em minus 0.4em\relax Harlow, England: Addison-Wesley, 1999.

\bibitem{beh}
M. A. Behrang, E. Assareh, A. Ghanbarzadeh, A. R. Noghrehabadi, ``The potential of different artificial neural network (ANN) techniques in daily global solar radiation modeling based on meteorological data,'' Solar Energy, vol. 84, pp. 1468--1480, 2010.

\bibitem{cartier}P. Cartier, Y. Perrin, ``Integration over finite
sets,'' in  F. \& M. Diener, Eds: Nonstandard Analysis in Practice, Springer, 1995, pp. 195-204.

\bibitem{crone}
S.F. Crone, ``Stepwise selection of artificial neural networks models for time series prediction,'' J. Intelligent Syst., vol. 14, pp. 99--122, 2005.

%\bibitem{5min}
%K. Dahmani, R. Dizene, G. Notton, C. Paoli, C. Voyant, M.-L. Nivet, Estimation of 5-min time-step data of tilted solar global irradiation using ANN (Artificial Neural Network) model, \emph{Energy}, DOI: http://dx.doi.org/10.1016/j.energy.2014.04.011, Elsevier, 2014.

\bibitem{de}
J. G. De Gooijer, R. J. Hyndman, ``25 years of time series forecasting,'' Int. J. Forecasting, vol. 22, pp. 443--473, 2006.

\bibitem{diener1}F. Diener, M. Diener, 
Tutorial,  F. \& M. Diener, Eds:
Nonstandard Analysis in Practice, Springer, 1995, 1--21.

\bibitem{diener2}F. Diener, G. Reeb,
Analyse non standard, Hermann, 1989.

\bibitem{duchon}
C. Duchon, R. Hale, Time Series Analysis in Meteorology and Climatology: An Introduction, Wiley-Blackwell, 2012. 

\bibitem{bruit}
M. Fliess, ``Analyse non standard du bruit,''
C.R. Acad. Sci. Paris Ser. I, vol. 342, pp. 797--802, 2006

\bibitem{trend} M. Fliess, C. Join, ``A mathematical proof of the existence of trends
in financial time series,'' in A. El Jai, L. Afifi, E. Zerrik, Eds: Systems Theory: Modeling,
Analysis and Control, Presses Universitaires de Perpignan, 2009, pp. 43--62. Available at \newline
{\tt \tiny http://hal.archives-ouvertes.fr/inria-00352834/en/}

\bibitem{csm}
M. Fliess, C. Join, ``Model-free control,'' Int. J. Control, vol. 86, pp. 2228--2252, 2013. Available at \newline
{\tt \tiny http://hal.archives-ouvertes.fr/hal-00828135}

%\bibitem{ifrane}
%M. Fliess, C. Join, Towards a new viewpoint on the causality for time series, Colloque Mod�lisation Contr\^ole Analyse Syst�mes en l'honneur Prof. A. El Jai, Ifrane, 2014. Available at 
%{\tt \tiny http://hal.archives-ouvertes.fr/hal-00991942/en/}

\bibitem{agadir}
M. Fliess, C. Join, F. Hatt, ``A-t-on vraiment besoin d'une approche
probabiliste en ing\'{e}nierie financi\`{e}re?,'' in Proc. Conf. M\'{e}dit.
Ing\'{e}nierie S\^{u}re Syst\`{e}mes Complexes, Agadir, 2011. Available at
{\tt \tiny http://hal.archives-ouvertes.fr/hal-00585152/en/}

\bibitem{easy}
M. Fliess, C. Join, H. Sira-Ram\'{\i}rez, ``Non-linear
estimation is easy,'' Int. J. Modelling Identif.
Control, vol. 4, pp. 12--27, 2008. 
Available at {\tt \tiny
http://hal.archives-ouvertes.fr/inria-00158855/en/}

%\bibitem{hsr}
%M. Fliess, H. Sira-Ram\'{\i}rez, ``Closed-loop parametric identification for continuous-time linear systems via new algebraic techniques,'' in H. Garnier, L. Wang, Eds: 
%Closed-Loop Parametric Identification for Continuous-Time Linear Systems via New Algebraic Techniques, Springer, 2008, pp. 362-391.

\bibitem{hoca}
F. O. Hocaoglu, F. Karanfil, ``A time series-based approach for renewable energy modeling,'' Renew. Sustain. Energy Rev., vol. 28, pp. 204--214, 2013.

\bibitem{hontoria}
L. Hontoria, J. Aguilera, P. Zufiria, ``An application of the multilayer perceptron: Solar radiation maps in Spain,'' Solar Energy,
vol. 79, pp. 523--530, 2005.

\bibitem{hossain}
M. Hossain, A. Oo, A. Ali, ``Hybrid prediction method for solar power using different computational intelligence algorithms,'' Smart Grid Renew. Energy, vol. 4, 2013, pp. 76--87; 2013.

\bibitem{kasa} N.K. Kasabov, Q. Song, ``DENFIS: Dynamic evolving neural-fuzzy inference
system and its application for time-series prediction,'' IEEE Fuzzy Syst., vol. 10, pp. 144--154, 2002.

\bibitem{kalo}
S. Kalogirou, ``Artificial neural networks in renewable energy systems applications: A review,'' Renew. Sustain. Energy Rev., vol. 5, pp. 373--401, 2001.

\bibitem{sousse}
F. Lafont, J.-F. Balmat, N. Pessel, M. Fliess,
``Model-free control and fault accommodation for an
  experimental greenhouse,'' in Proc. Int. Conf. Green Energy
  Environ. Engin., Sousse, 2014. Available at 
 {\tt \tiny http://hal.archives-ouvertes.fr/hal-00978226/en/}

\bibitem{lobry}
C. Lobry, T. Sari, ``Nonstandard analysis and representation of reality,'' Int. J. Control, vol. 81, pp. 519--536, 2008.

\bibitem{martin}L. Mart�n, L.F. Zarzalejo, J. Polo, A. Navarroa, R. Marchante, M. Cony, ``Prediction of global solar irradiance based on time series analysis: Application to solar thermal power plants energy production planning,'' Solar Energy, 
vol. 84, pp. 1772--1781, 2010

\bibitem{NumDiff}
M. Mboup, C. Join, M. Fliess, ``Numerical differentiation with annihilators in noisy environment,'' Num. Algo., vol. 50, pp. 439--467, 2009.

\bibitem{mel}
A. Mellit, S. Kalogirou, ``Artificial intelligence techniques for photovoltaic applications: A review,'' Prog. Energy Combustion Sci., vol. 34, pp. 574--632, 2008.

\bibitem{meuriot}V. Meuriot, Une histoire des concepts des s\'{e}ries temporelles, Harmattan--Academia, 2012.

\bibitem{solis}
R.W. Mueller, K.F. Dagestad, P. Ineichen, M. Schroedter-Homscheidt, S. Cros, D. Dumortier, R. Kuhlemann, J.A. Olseth, G. Piernavieja, C. Reise, L. Wald, D. Heinemann, ``Rethinking satellite-based solar irradiance modelling: The SOLIS clear-sky module,'' Remote Sensing Environ., vol. 91, pp. 160--174, 2004.

\bibitem{murphy} A.H. Murphy, ``What is a good forecast? An essay on the nature of goodness in weather forecasting,'' Weather Forecast., vol. 8, pp. 281--293, 1993.

\bibitem{notton1}
G. Notton, C. Paoli, S. Diaf, ``Estimation of tilted solar irradiation using artificial neural networks,'' Energy Proc., vol. 42, pp. 33--42, 2013.


\bibitem{notton2}
G. Notton, C. Paoli, L. Ivanova, S. Vasileva, M.-L. Nivet, ``Neural network approach to estimate 10-min solar global irradiation values on tilted planes,'' Renewable Energy, vol. 50, pp. 576--584, 2013.


\bibitem{paoli}
C. Paoli, C. Voyant, M. Muselli, M.-L. Nivet, ``Forecasting of preprocessed daily solar radiation time series using neural networks,'' Solar Energy, vol. 84, pp. 2146--2160, 2010.

\bibitem{rei}
G. Reikard, ``Predicting solar radiation at high resolutions: A comparison of time series forecasts,'' Solar Energy, vol. 83, pp. 342--349, 2009.

\bibitem{vi}
G. Vijayakumar, M. Kummert, S.A. Klein, W.A. Beckman, ``Analysis of short-term solar radiation data,'' Solar Energy, vol. 79, pp. 495--504, 2005.

\bibitem{v1}
C. Voyant, M. Muselli, C. Paoli, M.-L. Nivet, ``Optimization of an artificial neural network dedicated to the multivariate forecasting of daily global radiation,'' Energy, vol. 36, pp. 348--359, 2011.

\bibitem{v2}
C. Voyant, C. Paoli, M. Muselli, M.-L. Nivet, ``Multi-horizon solar radiation forecasting for Mediterranean locations using time series models,'' Renew. Sustain. Energy Rev., vol. 28, pp. 44--52, 2013.

\bibitem{yona} A. Yona, T. Senjyu, T. Funabashi, P. Mandal, C.-H. Kim, ``Decision technique of solar radiation prediction applying recurrent neural network for short-term ahead  power output of photovoltaic system,'' 
Smart Grid Renew. Energy, vol. 4, pp. 32--38, 2013.

\bibitem{yosida}
K. Yosida, Operational Calculus (translated from the
Japanese), Springer, 1984.




\end{thebibliography}




% that's all folks
\end{document}



\section{Multilayer perceptron}\label{mlp}
An \underline{A}rtificial \underline{N}eural \underline{N}etwork, or  ANN, is an information processing paradigm that is inspired by the way biological nervous systems, such as the brain, process information. They have 
already been utilized quite often for meteorological purposes. This summary will therefore be quite short.

Define the the time series $X(t)$ by the following regression, which might be nonlinear,
\begin{equation*}
X(t+1)=f_n (X(t),X(t-1),\cdots,X(t-p+1))+\epsilon(t+1) 
\label{c7} 				
\end{equation*}
where 
\begin{itemize}
\item $p$ (resp. $n$) is the number of past observations (resp. model parameters),
\item $\epsilon(t+1)$ is some additive zero-mean perturbation. 
\end{itemize}
Although a large range of different architectures of artificial neural networks is available, the \underline{M}ulti\underline{L}ayer \underline{P}erceptron, or MLP, remains the most popular. 
Feedforward MLP networks with two layers, one hidden layer and one output layer, are often used for modeling and forecasting, and have been validated. To forecast the time series, a fixed number $p$ of past values are set as inputs of the MLP, the output is the prediction of the future value. Considering the initial time series equation (Equation above), we can transform this formula to the non-linear case of one hidden layer MPL with $b$ related to the biases, $f$ and $g$ to the activation function of the output and hidden layer, and $\omega$ to the weights (see equation \eqref{c8}). The number of hidden nodes ($H$) and the number of the input nodes ($In$) allow detailing this transformation. 
\begin{equation}
\hat X(t+1)=f(\Sigma_{i=1}^H\circ_i\omega_i^2+b^2)%?_(i=1)^H?o_i  ?_i^2+b^2) with o_i=g(?_(j=1)^In??X(t-j+1)? ?_ij^1+b_i^1)		
\label{c8} 	
\end{equation}
with $\circ_i=g(\Sigma_{j=1}^{in}X(t-j+1)\omega_{ij}^1+b_i^1)$.\\
In the presented study, the MLP has been computed with the Matlab$^{�}$ software and its Neural Network toolbox. The characteristics chosen related to previous works are the following: one hidden layer, the activation functions are the continuously and differentiable hyperbolic tangent (hidden) and linear (output), the Levenberg-Marquardt (approximation to the Newton?s method) learning algorithm with a max fail parameter before stopping training equal to $3$. The optimization of the number of input nodes is done with the mutual information (the first minimum of this function defines the deepest lag considered, i.e. if the minimum concerns the lag $3$, only  $X(t) X(t-1)$ and $X(t-2)$ are taken as inputs of the MLP). The number of hidden neurons is taken equal to the input nodes number. The results shown in the next are related to the best networks among six different trainings coupled with a random weight initialization.






























1409.7476.extracted_bib
{IEEEhowto:kopka %H.~Kopka and P.~W. Daly, A Guide to \LaTeX, 3rd~ed.\hskip 1em plus % 0.5em minus 0.4em\relax Harlow, England: Addison-Wesley, 1999.
1409.7476.extracted_bib
{beh M. A. Behrang, E. Assareh, A. Ghanbarzadeh, A. R. Noghrehabadi, ``The potential of different artificial neural network (ANN) techniques in daily global solar radiation modeling based on meteorological data,'' Solar Energy, vol. 84, pp. 1468--1480, 2010.
1409.7476.extracted_bib
{cartierP. Cartier, Y. Perrin, ``Integration over finite sets,'' in F. \& M. Diener, Eds: Nonstandard Analysis in Practice, Springer, 1995, pp. 195-204.
1409.7476.extracted_bib
{crone S.F. Crone, ``Stepwise selection of artificial neural networks models for time series prediction,'' J. Intelligent Syst., vol. 14, pp. 99--122, 2005. %
1409.7476.extracted_bib
{5min %K. Dahmani, R. Dizene, G. Notton, C. Paoli, C. Voyant, M.-L. Nivet, Estimation of 5-min time-step data of tilted solar global irradiation using ANN (Artificial Neural Network) model, Energy, DOI: http://dx.doi.org/10.1016/j.energy.2014.04.011, Elsevier, 2014.
1409.7476.extracted_bib
{de J. G. De Gooijer, R. J. Hyndman, ``25 years of time series forecasting,'' Int. J. Forecasting, vol. 22, pp. 443--473, 2006.
1409.7476.extracted_bib
{diener1F. Diener, M. Diener, Tutorial, F. \& M. Diener, Eds: Nonstandard Analysis in Practice, Springer, 1995, 1--21.
1409.7476.extracted_bib
{diener2F. Diener, G. Reeb, Analyse non standard, Hermann, 1989.
1409.7476.extracted_bib
{duchon C. Duchon, R. Hale, Time Series Analysis in Meteorology and Climatology: An Introduction, Wiley-Blackwell, 2012.
1409.7476.extracted_bib
{bruit M. Fliess, ``Analyse non standard du bruit,'' C.R. Acad. Sci. Paris Ser. I, vol. 342, pp. 797--802, 2006
1409.7476.extracted_bib
{trend M. Fliess, C. Join, ``A mathematical proof of the existence of trends in financial time series,'' in A. El Jai, L. Afifi, E. Zerrik, Eds: Systems Theory: Modeling, Analysis and Control, Presses Universitaires de Perpignan, 2009, pp. 43--62. Available at \newline {\tt \tiny http://hal.archives-ouvertes.fr/inria-00352834/en/
1409.7476.extracted_bib
{csm M. Fliess, C. Join, ``Model-free control,'' Int. J. Control, vol. 86, pp. 2228--2252, 2013. Available at \newline {\tt \tiny http://hal.archives-ouvertes.fr/hal-00828135 %
1409.7476.extracted_bib
{ifrane %M. Fliess, C. Join, Towards a new viewpoint on the causality for time series, Colloque Mod�lisation Contr\^ole Analyse Syst�mes en l'honneur Prof. A. El Jai, Ifrane, 2014. Available at %{\tt \tiny http://hal.archives-ouvertes.fr/hal-00991942/en/
1409.7476.extracted_bib
{agadir M. Fliess, C. Join, F. Hatt, ``A-t-on vraiment besoin d'une approche probabiliste en ing\'{enierie financi\`{ere?,'' in Proc. Conf. M\'{edit. Ing\'{enierie S\^{ure Syst\`{emes Complexes, Agadir, 2011. Available at {\tt \tiny http://hal.archives-ouvertes.fr/hal-00585152/en/
1409.7476.extracted_bib
{easy M. Fliess, C. Join, H. Sira-Ram\'{\irez, ``Non-linear estimation is easy,'' Int. J. Modelling Identif. Control, vol. 4, pp. 12--27, 2008. Available at {\tt \tiny http://hal.archives-ouvertes.fr/inria-00158855/en/ %
1409.7476.extracted_bib
{hsr %M. Fliess, H. Sira-Ram\'{\irez, ``Closed-loop parametric identification for continuous-time linear systems via new algebraic techniques,'' in H. Garnier, L. Wang, Eds: %Closed-Loop Parametric Identification for Continuous-Time Linear Systems via New Algebraic Techniques, Springer, 2008, pp. 362-391.
1409.7476.extracted_bib
{hoca F. O. Hocaoglu, F. Karanfil, ``A time series-based approach for renewable energy modeling,'' Renew. Sustain. Energy Rev., vol. 28, pp. 204--214, 2013.
1409.7476.extracted_bib
{hontoria L. Hontoria, J. Aguilera, P. Zufiria, ``An application of the multilayer perceptron: Solar radiation maps in Spain,'' Solar Energy, vol. 79, pp. 523--530, 2005.
1409.7476.extracted_bib
{hossain M. Hossain, A. Oo, A. Ali, ``Hybrid prediction method for solar power using different computational intelligence algorithms,'' Smart Grid Renew. Energy, vol. 4, 2013, pp. 76--87; 2013.
1409.7476.extracted_bib
{kasa N.K. Kasabov, Q. Song, ``DENFIS: Dynamic evolving neural-fuzzy inference system and its application for time-series prediction,'' IEEE Fuzzy Syst., vol. 10, pp. 144--154, 2002.
1409.7476.extracted_bib
{kalo S. Kalogirou, ``Artificial neural networks in renewable energy systems applications: A review,'' Renew. Sustain. Energy Rev., vol. 5, pp. 373--401, 2001.
1409.7476.extracted_bib
{sousse F. Lafont, J.-F. Balmat, N. Pessel, M. Fliess, ``Model-free control and fault accommodation for an experimental greenhouse,'' in Proc. Int. Conf. Green Energy Environ. Engin., Sousse, 2014. Available at {\tt \tiny http://hal.archives-ouvertes.fr/hal-00978226/en/
1409.7476.extracted_bib
{lobry C. Lobry, T. Sari, ``Nonstandard analysis and representation of reality,'' Int. J. Control, vol. 81, pp. 519--536, 2008.
1409.7476.extracted_bib
{martinL. Mart�n, L.F. Zarzalejo, J. Polo, A. Navarroa, R. Marchante, M. Cony, ``Prediction of global solar irradiance based on time series analysis: Application to solar thermal power plants energy production planning,'' Solar Energy, vol. 84, pp. 1772--1781, 2010
1409.7476.extracted_bib
{NumDiff M. Mboup, C. Join, M. Fliess, ``Numerical differentiation with annihilators in noisy environment,'' Num. Algo., vol. 50, pp. 439--467, 2009.
1409.7476.extracted_bib
{mel A. Mellit, S. Kalogirou, ``Artificial intelligence techniques for photovoltaic applications: A review,'' Prog. Energy Combustion Sci., vol. 34, pp. 574--632, 2008.
1409.7476.extracted_bib
{meuriotV. Meuriot, Une histoire des concepts des s\'{eries temporelles, Harmattan--Academia, 2012.
1409.7476.extracted_bib
{solis R.W. Mueller, K.F. Dagestad, P. Ineichen, M. Schroedter-Homscheidt, S. Cros, D. Dumortier, R. Kuhlemann, J.A. Olseth, G. Piernavieja, C. Reise, L. Wald, D. Heinemann, ``Rethinking satellite-based solar irradiance modelling: The SOLIS clear-sky module,'' Remote Sensing Environ., vol. 91, pp. 160--174, 2004.
1409.7476.extracted_bib
{murphy A.H. Murphy, ``What is a good forecast? An essay on the nature of goodness in weather forecasting,'' Weather Forecast., vol. 8, pp. 281--293, 1993.
1409.7476.extracted_bib
{notton1 G. Notton, C. Paoli, S. Diaf, ``Estimation of tilted solar irradiation using artificial neural networks,'' Energy Proc., vol. 42, pp. 33--42, 2013.
1409.7476.extracted_bib
{notton2 G. Notton, C. Paoli, L. Ivanova, S. Vasileva, M.-L. Nivet, ``Neural network approach to estimate 10-min solar global irradiation values on tilted planes,'' Renewable Energy, vol. 50, pp. 576--584, 2013.
1409.7476.extracted_bib
{paoli C. Paoli, C. Voyant, M. Muselli, M.-L. Nivet, ``Forecasting of preprocessed daily solar radiation time series using neural networks,'' Solar Energy, vol. 84, pp. 2146--2160, 2010.
1409.7476.extracted_bib
{rei G. Reikard, ``Predicting solar radiation at high resolutions: A comparison of time series forecasts,'' Solar Energy, vol. 83, pp. 342--349, 2009.
1409.7476.extracted_bib
{vi G. Vijayakumar, M. Kummert, S.A. Klein, W.A. Beckman, ``Analysis of short-term solar radiation data,'' Solar Energy, vol. 79, pp. 495--504, 2005.
1409.7476.extracted_bib
{v1 C. Voyant, M. Muselli, C. Paoli, M.-L. Nivet, ``Optimization of an artificial neural network dedicated to the multivariate forecasting of daily global radiation,'' Energy, vol. 36, pp. 348--359, 2011.
1409.7476.extracted_bib
{v2 C. Voyant, C. Paoli, M. Muselli, M.-L. Nivet, ``Multi-horizon solar radiation forecasting for Mediterranean locations using time series models,'' Renew. Sustain. Energy Rev., vol. 28, pp. 44--52, 2013.
1409.7476.extracted_bib
{yona A. Yona, T. Senjyu, T. Funabashi, P. Mandal, C.-H. Kim, ``Decision technique of solar radiation prediction applying recurrent neural network for short-term ahead power output of photovoltaic system,'' Smart Grid Renew. Energy, vol. 4, pp. 32--38, 2013.
1409.7476.extracted_bib
{yosida K. Yosida, Operational Calculus (translated from the Japanese), Springer, 1984.