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6. Forecasting Methods for Different Forecast Horizons 101

The approach used in such methods consists in predicting the future solar irradi-
ance or irradiation (at different time scales) based on the past observed data [24].
Mathematically, the researched formulation is:

G tþh ¼ fðG t ; G t 1 ; G t 2 ; G t 3 ; .; G t p Þ (3.8)
In other terms, the future time step (t þ h) is forecasted based on the observed
data at the time (t, t 1., t p).
Some of these models are presented below.

6.2.3.1 Persistence and scaled persistence
As said in paragraph 5, the persistence model is often used as a reference for deter-
mining the skill factor. It is useful to know if a forecast model provides better results
than any trivial reference model, which is the persistence model [24].
The persistence model considers that the solar radiation at t þ 1 is equal to the
solar radiation at t. It assumes that the atmospheric conditions are stationary. It is
also called the naı ¨ve predictor.
(3.9)
G tþ1 ¼ G t
Its accuracy decreases with the time horizon and is generally not adequate for
more than 1 h.
An improved version of this model is the scaled persistence model. To take into
account the fact that the apparent position of the sun is not identical between t and
t þ 1, the persistence model is corrected with a clear-sky ratio term (see Eq. 3.5) [88]
and is then called scaled persistence
clear sky
G
tþ1
G tþ1 ¼ G t clear sky (3.10)
G t
6.2.3.2 Autoregressive Moving-Average Model
This model is well-known [89,90] for forecasting the future value of a time series.
An ARMA model is composed of an autoregressive part (AR) and a MA; p and q
are, respectively, the order of the AR and the MA (Eq. 3.11), and the ARMA
(Eq. 12) model is noted as ARMA(p, q) [91,92].
q
X
MAðqÞ/G t ¼ q i $ε t i ¼ qðLÞεðtÞ; ct˛ℤ (3.11)
i¼0
p
X
1 ε t ; ct˛ℤ
ARðpÞ/G t ¼ 4 $G t i þ ε t ¼½4ðLފ (3.12)
i
i¼1
This model uses the statistical properties of the time series and the BoxeJenkins
methodology. It has proved its efficiency for time series with a linear structure be-
tween the data mainly. To use ARMA model, the time series must be stationary,
which is not the case for solar radiation; therefore, a process must be used previously