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102 CHAPTER 3 Forecasting of Intermittent Solar Energy Resource

in view to obtain a stationary solar radiation time series by removal of the season-
ality and trend.
The seasonality is generally deleted by using a clear-sky model, which contains
all the seasonal variability. Several models exist to detrend the hourly solar radiation;
a good overview of these models is given in [24].

6.2.3.3 Autoregressive Integrated Moving-Average Model
The ARIMA model [93] is a reference model for forecasting solar radiation and is
derived from AR and MA models. ARIMA models are fitted to time series data
either to better understand the data or to predict future points in the series (fore-
casting). They are applied in some cases where data exhibit evidence of nonstatio-
narity, where an initial differencing step (corresponding to the “integrated” part of
the model) can be applied one or more times to eliminate the nonstationarity.
Nonseasonal ARIMA models are generally denoted ARIMA (p, d, q) where
parameters p, d,and q are nonnegative integers, p being the order (number of time
lags) of the AR, d is the degree of differencing (the number of times the data
have had past values subtracted), and q is the order of the moving-average model.
When two out of the three terms are zeros, the model may be referred to based on
the nonzero parameter, dropping “AR”, “I,” or “MA” from the acronym describing
the model. For example, ARIMA(1,0,0) is AR(1), ARIMA(0,1,0) is I(1), and
ARIMA(0,0,1) is MA(1).
A regression in log has been applied to the inputs of the ARIMA model to predict
solar radiation [94]. At 24-h horizon, this model is more accurate than the others.
6.2.3.4 Artificial Neural Network
ANNs are particularly predominant in the field of time series forecasting with
nonlinear methods. Indeed, the availability of historical data from the meteorolog-
ical utility databases added to the fact that NNs are data-driven approaches capable
of performing a nonlinear mapping between sets of input and output variables make
this modeling tool very attractive. ANNs with d inputs, m hidden neurons, and a sin-
gle linear output unit define a nonlinear parameterized mapping from an input vector
x to an output vector y (see the relationship given in Eq. (3.13) and illustrated in
Fig. 3.15).
!
m d
X X
y ¼ yðx; wÞ¼ w j f w ji x i (3.13)
j¼1 i¼1
The m hidden neurons are related to the tangent hyperbolic function
x
x
x
x
fðxÞ¼ ðe e Þ=ðe þ e Þ. The parameter vector w ¼ðfw j g; fw ji gÞ governs
the nonlinear mapping and is estimated during a phase which is called the training
or learning phase. During this phase, the NN is trained using the data set D that con-
tains a set of n input and output examples. The second phase, called the generaliza-
tion phase, consists of evaluating the accuracy of the model on another data set, so