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Nonlinear Analysis of Daily Global Solar Radiation Time
Series
G. Lpez1*, F.. !atlles"and .L. !os#$"1 Departamento de Ingeniera
Elctrica y Trmica, Escuela Politcnica Superior, Universidad de
Huelva,
211!, Huelva, Spain2 Departamento de "sica #plicada, Universidad
de #lmera, #lmera, Spain$%orresponding #ut&or,
ga'riel(lope)*die(u&u(es
Abstra#t
In t&is +or +e analyse a daily glo'al irradiance time series
into t&e -rame+or o- c&aotic dynamicsystems in order to
e.amine a possi'le underlying nonlinear 'e&aviour( Employed
met&ods are
'ased on a p&ase space reconstruction -rom t&e measured
data and are devoted to t&e calculation o-t&e properties o-
an underlying attractor, suc& as t&e /yapunov e.ponents(
0esearc&es on t&esedynamical system invariants +ill point
out t&e presence o- c&aos( e also use local lineal modelsas
a test -or nonlinearity(
T&e glo'al solar radiation data +ere measured at t&e
radiometric station o- t&e University o-#lmera Spain3 during
eig&t years( 0esults &ave s&o+n t&e non4e.istence
o- any attractor in t&e
p&ase space -or t&e glo'al irradiance time series(
5egative /yapunov e.ponents e.clude a c&aotic'e&aviour
t&at mig&t allo+ a 'etter s&ort term prediction
t&an autoregressive models, and t&e ideao- t&e
e.istence o- a nonlinear di--erential e6uation system( T&ese
results matc& +it& t&oseo'tained -rom applying local
linear models -or prediction, o- +&ic& estimations suggest
t&at t&edata are 'est descri'ed 'y a linear stoc&astic
process(
7ey+ords8 solar radiation, -orecasting, c&aos, nonlinear
time series(
1. %ntrodtion
In-ormation on t&e availa'ility o- solar radiation is needed
in many applications dealing +it& t&e
e.ploitation o- solar energy( Particularly, glo'al solar
radiation is one o- t&e most important inputparameters -or any
solar energy system and di--erent tec&ni6ues &ave 'een
developed to model and
-orecast it( 9nce t&e solar energy system lie concentrated
solar po+er plants3 is running,
prediction o- po+er load, normally done on an &ourly 'asis
+it& a prediction &ori)on 'et+een 1
and 2: &ours, is instrumental -or planning and operation o-
t&e total po+er system, e(g( -or 'uying
or selling po+er or -or solving t&e unit commitment and
dispatc& pro'lems(
Studies a'out solar radiation time series and ot&er
meteorological varia'les +it& autoregressive or
stoc&astic models ;14
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measured data and are devoted to t&e calculation o- t&e
properties o- an underlying attractor, suc&
as t&e /yapunov e.ponents( e also use local lineal models as
a test -or nonlinearity(
". '(perimental Data
T&e &ori)ontal glo'al solar radiation time series
sym'oli)ed 'y ?xt@, 'eing tt&e time step in days3consisted o-
2!:A daily values( T&ey +ere o'tained 'y integration -rom
e.perimental valuesaveraged every ten minutes during t&e years
1!!B41!!2 and every -ive minutes during t&e years1!!
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optimum time delay +e &ave used t&e met&od o- mutual
information;=( >utual in-ormation,
lie autocorrelation, tries to measure t&e e.tent to
+&ic& values Kxt+mare related to values o- Kxt, at
a given lag( Ho+ever, mutual in-ormation &as t&e
advantage o- using pro'a'ilities, rat&er t&an a
linear 'asis, to asses t&e correlation and t&us,
nonlinear correlations are taen into accounts( T&e
so-t+are implementation o- t&at algorit&m and o-
t&ose used &erea-ter3 is -rom t&e TISE#5pacage ;=(
42A 42B 41A 41B 4A B A 1B 1A 2B 2A42A
42B
41A
41B
4A
B
A
1B
1A
2B
2A
xt+
1>GHm
23
xt>GHm2
3
t41
a3
B AB 1BB 1AB 2BB 2AB
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1 2 < : A C I ! 1B 11 12B(B
B(2
B(:
B(C
B(
1(B
Percentage
o--alse
nearestnei
g&'ors
Em'edding dimension, d
"ig( :( Percentage o- -alse nearest neig&'ors as a -unction
o- t&e em'edding dimension -or t&e di--erencedtime
series(
)." Lo#ally linear predi#tion
T&e easiest nonlinear met&od o- local prediction +as
developed 'y /oren) ;:= and called method
of analogies( /et 'e xta point +it&in t&e d4dimensional
p&ase space, t&e predicted value a time T
later, xt+T, +ill 'elong to some ind o- interpolation 'et+een
no+n points xt1+T, (((, xtr+T, +&ere
xj1, (((, xtrare t&e nearest r points to xt( "armer and
Sidoro+ic& ;12= propose an en&anced met&od 'y
maing a least4s6uared -it o- a local linear map o- xtinto xt+T(
Doing TL 1, +e can o'tain t&e
predicted value as xt+1L atxtO bt'y minimising8
+ =
it Nx
tttt bxaxW
2
1 13
in regard to atand bt(Ntis t&e 4neig&'or&ood o-
xt(
%asdagli ;1B= suggests to use t&ese models as a test -or
nonlinearity prediction, o'taining t&e
average -orecast error given 'y8
>> ==
