An imp o ed model o he syn he ic gene a ion o high empo al esolu ion di ec no mal
i adia ion ime se ies
La añe a, M.1,Mo eno-Teje a, S.2, Sil a-Pé ez,M.A.2, Lillo-B a o,I.2
1 Andalusian Associa ion o Resea ch and Indus ial Coope a ion (AICIA).
2 Depa men o Ene gy Enginee ing, Uni e si y o Se ille.
Co esponding au ho :
Miguel La añe a, Andalusian Associa ion o Resea ch and Indus ial Coope a ion, Camino de
los Descub imien os s/n. 41092, Se ille, Spain.
Phone/Fax numbe : (+34)954487237/ (+34)954487233
E-mail: mla ane a@g e .es
Abs ac
Se e al s udies ha e con i med he ele an impac o he esolu ion and equency dis ibu ion
o sola adia ion da a on he esul s o de ailed p oduc ion models. Many o he a ailable di ec
no mal i adiance (DNI) da abases gene a ed om he sa elli e images ha e an hou ly
esolu ion. In he p esen wo k, we ha e p oposed imp o emen s o an exis ing model o he
gene a ion o 10-min syn he ic DNI da a om he hou ly a e age DNI alues. In he o iginal
model, he i adiance is di ided in o a de e minis ic and s ochas ic componen , i.e., he
con ibu ion om he hou ly mean and s ochas ic luc ua ion ob ained om he mean
depending on he sky condi ion, espec i ely. We ha e implemen ed se e al imp o emen s, and
he mos ele an is he consis ency o he syn he ic da a wi h he s a e o he sky. The
adap a ion and applica ion o he model o he loca ion o Se ille shows signi ican
imp o emen s o e i s p edecesso as i achie ed 7% RMSD in hou ly alues and 1% RMSD in
daily alues and p esen ed a ealis ic equency dis ibu ion in he 10-min esolu ion. In
compa ison o he o iginal model, he applica ion o he imp o ed model showed signi ican
pe o mance imp o emen s wi hou any u he adap a ions o o he loca ions wi h di e en
clima ological cha ac e is ics han Se ille.
Keywo ds
DNI; high equency; sola adia ion models; cloud ansien s
1 In oduc ion
The di ec no mal i adiance (DNI) ime se ies a e he basic inpu s o he simula ion o sola
he mal elec ici y (STE) plan s. The simula ion models ha e di e en equi emen s in e ms o
ime esolu ion o he DNI ime se ies depending on hei use o applica ion. S udies by Meye
e al. (2009) and Gall e al. (2010) ha e emphasized he need o use ime se ies wi h a ime s ep
sho e han 1 hou o de ailed pe o mance simula ions. Many pe o mance models used in
he comme cial p ojec s o assessing con ac ual pe o mance equi e a one-yea ime se ies
wi h a esolu ion ime o 5–15 min, as a easonable comp omise be ween he compu a ional
cos and accu acy.
Cop. 2015 Else ie . Licencia C ea i e Commons CC BY-NC-ND
On he o he hand, STE plan ope a o s equen ly use i adiance p edic ions based on he
sa elli e es ima es o me eo ological and sola adia ion models o ope a e in elec ici y ma ke s
o de ine subsequen ope a ional s a egies. Mos o he DNI p edic ion models a e based on
he sho - e m wea he o ecas s, which ha dly exceed an hou ly ime s ep (Vincen , 2013) e en
hough mos o he simula ion ools equi e highe equency.
The need o high- equency DNI da a is e lec ed in he la es publica ions o models o
gene a ing syn he ic sola i adiance da a ocused on high empo al esolu ions (B igh e al.,
2015; Fe nández-Pe uchena e al., 2014, 2015; G an ham e al., 2013). Many o hese models
a e based p ima ily on au o eg essi e mo ing a e age (ARMA) o Ma ko ansi ion ma ix
(MTM) echniques, which p o ide ime se ies wi h inc eased empo al esolu ions. This leads o
he in oduc ion o imp o emen s ha helps in modeling he dynamic beha io o he sola
adia ion (Ngoko e al., 2014; Mo , 2013; Glasbey and Allc o , 2008). Ngoko e al. (2014)
p oposed a second-o de MTM model ha included s a is ical cha ac e is ics associa ed wi h
he a mosphe ic condi ion (clea , cloudy, o e cas ) leading o he imp o emen in he i s -o de
MTM model employed by Richa dson and Thomas (2011) o gene a ing 1-min alues. The use
o wa ele s and a i icial neu al ne wo k (ANN) echniques o he gene a ion o sola adia ion
alues is mainly ocused on he o ecas applica ions. Howe e , in mos o he cases, hese
models deal wi h daily o hou ly global ho izon al i adia ion (GHI) da a (Melli e al., 2005;
Lina es-Rod iguez e al., 2011).
The sola he mal ene gy concen a ing echnologies exploi only he di ec componen . This
componen has unique s a is ical p ope ies (Ska ei and Olse h, 1992), showing s eepe
g adien s han he global adia ion du ing he cloud ansien s. The e exis s a co ela ion
be ween he DNI and GHI ha helps in ob aining one om he o he wi h accep able esul s;
his is suppo ed by he model s udy conduc ed by Ska ei and Olse h (1992) on he syn he ic
gene a ion o i adiance alues a di e en ime in e als. Se e al au ho s ha e gene a ed DNI
alues syn he ically a a high equency om he global i adiance alues, bu only a ew au ho s
ha e ocused hei models on he gene a ion o high esolu ion DNI se ies om low- esolu ion
DNI alues.
Mo (2013) gene a ed sequences o ins an aneous global sola i adiance alues on a ho izon al
plane ha can be spli in o beam and di use componen s. The model was di ided in o a
s ochas ic and de e minis ic componen ela ed o he Angs öm-P esco eg ession. The cloud
co e was used as a s ochas ic d i e o he gene a ion o an on/o sequence o beam
i adiance (Mo , 2011); he p obabili y o beam i adiance is ep esen ed as he complemen
o one o he cloud co e s. B igh (2015) gene a ed minu e i adiance ime se ies on an a bi a y
plane om he p e ious es ima ion o bo h di ec and di use componen s. This me hodology
used wea he obse a ion da a o gene a e cloud ansien s sequences using he Ma ko chains.
Fe nández-Pe uchena p oposed he gene a ion o 1-minu e esolu ion DNI se ies om he daily
(Fe nandez-Pe uchena, 2014) and hou ly (Fe nández-Pe uchena, 2015) means o DNI alues.
The me hod was based on he p e ious gene a ion o a da abase o dimensionless high
equency daily cu es o se ies o DNI alues ob ained om he obse ed da a. The days we e
selec ed based on he closes Euclidean dis ance be ween he daily and hou ly means o he
gene a ed and measu ed se ies. The esul s ob ained wi h his model we e sa is ac o y in he
e ms o a iabili y and equency dis ibu ions; howe e , he au ho s p o ided no esul s in
ela ion o he de ia ion obse ed be ween he syn he ic and measu ed hou ly and daily means,
which is one o he majo a ge s o he p esen wo k.
Polo e al. (2011) p oposed a model ha was ela i ely indis inc o GHI and DNI. The model
gene a ed 10-min da a om he hou ly alues while main aining he s a is ical cha ac e is ics
o an obse ed da a se . Concep ually, he DNI was di ided in o a de e minis ic and s ochas ic
componen : he con ibu ion om he hou ly mean and s ochas ic de ia ion om he mean,
espec i ely. The main p oblems de ec ed wi h his model we e he di e ence be ween he
daily and hou ly cumula i e alues o he measu ed and syn he ic da ase s ha eached 2–4%
in daily o als and 15% o hou ly alues, and he misma ches in hei equency dis ibu ion in
he 10-min esolu ion.
In he p esen wo k, we p opose some imp o emen s o he abo e-men ioned model, based on
he knowledge o he unique ea u es o he DNI beha io . The implemen a ion o he
imp o emen s ha e been go e ned by wo condi ions: 1) he hou ly alues o he o iginal se ies
should be conse ed easonably in he syn he ic one, and 2) he dynamics o he luc ua ions o
he DNI mus be consis en wi h he s a e o he sky. The e o e, we p opose se e al
imp o emen s o ien ed owa d he adjus men o he model unde di e en sky condi ions, a
pa ame e o he dis inc ion o hazy days based on he hou ly DNI alues only, and an
imp o emen o es ic he alues o he s ochas ic componen o he model whene e
equi ed.
As a esul , we p esen an imp o ed model o he syn he ic gene a ion o 10-min DNI alues
om he hou ly DNI da a. The model, which sol es he weaknesses o i s p edecesso , has been
alida ed in i e loca ions wi h di e en clima ic condi ions, showing a sa is ac o y pe o mance
ega dless o he loca ion in which i was used. This esul was ob ained despi e he ac ha he
model has only been ained wi h da a om one o hese loca ions.
2 Me eo ological da abase
The da a se used o aining o he model consis s o he 10-min and 1-h a e ages o DNI
measu emen s egis e ed e e y 5 seconds du ing 13 yea s (2000–2012) a he me eo ological
s a ion o he G oup o The modynamics and Renewable Ene gy o he Uni e si y o Se ille. The
model imp o emen s ha e been checked wi h he co esponding alues o he yea 2013 and
la e alida ed in i e di e en loca ions in Spain co e ing la i udes om 37° N o 43° N. The
selec ed si es a e p esen ed in Table 1.
All he da a used in his wo k ha e been subjec ed o quali y-con ol p ocedu es ollowing he
BSRN ecommenda ions (Mo eno-Teje a e al., 2015).
3 Me hodology
The s udy ollowed he me hodology p oposed by Polo e al. (2011) o he gene a ion o 10-
min syn he ic i adiance alues om a gi en hou ly ime se . The DNI was di ided in o a
de e minis ic and s ochas ic componen .
The dynamics o he DNI a y conside ably depending on he a mosphe ic condi ions (clouds,
ae osols, e c.). The e o e, in he i s s ep, i is necessa y o pe o m a clus e ing o he 10-min
da a a ailable as a unc ion o he a mosphe ic condi ions. The clus e ing is pe o med by
calcula ing he no malized clea ness index, k’ (Pe ez e al., 1990), and g ouping he da ase s in o
ou k’ classes o in e als.
The second s ep is he calcula ion o he s anda d de ia ion o he 10-min DNI alues wi h
espec o he hou ly mean. The esul s a e no malized o he maximum alue o he comple e
da ase . This helps o gene a e he p obabili y densi y unc ion o he hou ly ime se s o
no malized s anda d de ia ions alues o each sky condi ion and i i o a be a dis ibu ion
cu e.
The p ocedu e o he gene a ion o syn he ic DNI alues di ides he sola adia ion in o a
de e minis ic and s ochas ic componen . The i s is gene a ed by he cubic in e pola ion o he
hou ly means calcula ed e e y 4 hou s in he 10-min ime scale. The s ochas ic componen is
dynamically ep oduced by using andom numbe s om he be a dis ibu ion cu e whose
cha ac e is ic pa ame e s ha e been i ed o each sky condi ion, in oducing a andom sign o
he luc ua ion. The p ocedu e is oughly desc ibed below:
i. Calcula ion o he cubic in e pola ion o he hou ly alues in a 10-minu es scale o
gene a e he shape whe e he luc ua ions will be added (𝐼10𝑚_𝑖3
𝑖).
ii. Gene a ion o andom numbe s om an uni o m dis ibu ion cu e [0,1] and
de e mina ion o he in e se be a alue co esponding o ha p obabili y and sky
condi ion. This alue is mul iplied by he maximum s anda d de ia ion o gene a e he
ampli ude o he luc ua ion (A).
iii. Gene a ion o andom numbe s om a no mal dis ibu ion cu e wi h ze o mean and
uni s anda d de ia ion o add o sub ac he ampli ude o o om he mean alue ( ).
Finally, o es ima e he 10 minu e- alue o he ins an ,
i
, he nex ope a ion is pe o med by
he ollowing equa ion:
𝐼10𝑚
𝑖=𝐼10𝑚_𝑖3
𝑖+𝑠𝑖𝑔𝑛(𝑟)∙𝐴 (1)
Whe e, he subsc ip 10
m
ep esen s he ime scale and
i
3 ep esen s he cubic in e pola ed
alue, i ep esen s he ime ins an , is he andom numbe om he be a dis ibu ion, and A is
he ampli ude o he luc ua ion.
4 Imp o emen s
In his wo k, we p opose he ollowing imp o emen s o he o iginal model by Polo e al. (2011):
- Classi ica ion o sky condi ion based on he use o kb ins ead o k’ .
- No maliza ion o he de ia ions o each p oposed kb ange o he maximum alue in
he in e al, ins ead o using he maximum alue o he comple e da a se .
- Es ablishmen o a minimum h eshold o hou ly DNI o he conside a ion o
luc ua ions.
- Use o a pe u ba ion coe icien o model hazy days.
- I e a i e p ocedu e o ma ch he daily i adia ion o he measu ed and syn he ic da a.
4.1 Sky condi ion classi ica ion
Polo e al. (2011) used k’ o he classi ica ion o he sky condi ion. The clouds ansien do no
ha e he same impac on he componen s o he sola adia ion. Dis u bances in DNI we e ound
o be s eepe han hose o GHI o ce ain ypes o sky condi ions. k’ is based on GHI and does
no p ope ly de ine he sky condi ions o his applica ion. Ins ead, we p opose he use o kb
(Ska ei and Olse h, 1992) di ided in in e als o 0.1 as desc ibed in Table 2 ins ead o he
o iginal di ision in o ou in e als.
𝑘𝑏=𝐼𝑏𝑛/𝐼𝑏𝑛𝑐𝑠 (2)
Whe e, Ibn is he obse ed di ec no mal i adiance and 𝐼𝑏𝑛𝑐𝑠is he clea -sky DNI.
The clea -sky DNI is calcula ed based on he model AB p oposed by Sil a-Pe ez (2002):
𝐼𝑏𝑛𝑐𝑠 =𝐼𝑐𝑠 ∙𝐸0∙𝐴
1+𝐵∙𝑚𝑅 (3)
Whe e, 𝑚𝑅 is he ela i e ai mass de e mined acco ding o he exp ession o Kas en and Young
(1989), Ics is he sola cons an , and E0 is he co ec ion due o Ea h-Sun dis ance. A and B a e
empi ical pa ame e s in ended o model he s a e o anspa ency o u bidi y o he
a mosphe e.
4.2 S anda d de ia ion no maliza ion
In he analysis o he a iabili y o he i adiance da a o he o iginal model, he s anda d
de ia ions o he 10-min se s a e no malized by di iding i by he maximum de ia ion o he
comple e da ase . This p ocedu e assumes ha he ampli ude o luc ua ions emains simila
unde all sky condi ions. Howe e , his assump ion may no hold ue in case o DNI. The e o e,
we sugges he no maliza ion o he de ia ions o each p oposed kb ange o he maximum
alue in he in e al.
The la ges luc ua ions occu du ing he passage o clus e s o low clouds wi h high densi y like
cumulus o s a ocumulus. The maximum de ia ions a e ound in he cen al in e als o he
clea ness index as shown in Table 3.
The highes di e ences a e ound o low alues o he clea ness index because o he lowe DNI
luc ua ions ha occu due o he p esence o dense and compac clouds du ing he mos ly-
co e ed sky condi ions.
4.3 Minimum alue o luc ua ions
In case o absence o clouds o o e cas sky wi h DNI hou ly alues lowe han 90 W/m2, no
luc ua ions a e obse ed in he DNI alues. Al hough, he la e case is no ele an o he
pe o mance o a STE plan , his case is s ill conside ed impo an o he de elopmen o an
accu a e model. The calcula ion o he syn he ic DNI is implemen ed by neglec ing he
luc ua ions o hou ly alues below 90 W/m2. Hence, he syn he ic DNI is equal o he cubic
in e pola ion o he hou ly alues (de e minis ic componen ).
𝐼10𝑚
𝑖=𝐼10𝑚_𝑖3
𝑖 (4)

Whe e, 𝐼10𝑚
𝑖 is he gene a ed syn he ic i adiance and 𝐼10𝑚_𝑖3
𝑖 is he cubic in e pola ion o he
hou ly alues.
Figu es 1, 2, and 3 compa e he esul s o he imp o emen s in he daily DNI.
The luc ua ions o he imp o ed model a e mo e consis en wi h he pe o mance o he 10-min
DNI measu ed da a han he o iginal model.
4.4 Pe u ba ion coe icien
DNI modeling in hazy days is a ask o high ele ance (Gueyma d, 2005) and has been only
pa ially sol ed. While hou ly and daily kb alues sugges pa ly cloudy days (kb≈0.55), he DNI
may show negligible luc ua ions om he cubic in e pola ion. A his poin , we ha e de ined a
new coe icien “P2” o de e mining he luc ua ions ha occu due o DNI.
The coe icien is based on he concep o o uosi y, commonly used in he di usion o po ous
media (Eps ein, 1989) ha can be de ined in a simpli ied manne as he ela ionship be ween
he leng h o a cu e, L, and a s aigh segmen (cho d) ha joins i s ends, X.
τ=𝐿𝑋
⁄ (5)
Figu e 4 ep esen s he di usion in a po ous medium whe e he o uosi y is la ge in case B
han in case A because he cho d is smalle e en o he same leng h.
Many ma hema ical equa ions ha e been p oposed o e he yea s o he accu a e es ima ion
o he alue o o uosi y o a ce ain cu e ( ). Pa asius e al. (2005) p oposed es ima ion as
he in eg al o he squa ed de i a i e o he cu e di ided by he cu e leng h, L.
τ=∫(𝑓′(𝑡))2𝑑𝑡
𝑡2
𝑡1 𝐿 (6)
Wi h ega d o sola adia ion, Muselli e al. (2000) p oposed a coe icien o es ima e he
pe u ba ion s a e o he hou ly clea ness index cu e du ing he day om he in eg al o he
second de i a i e.
S2=∑{𝑘𝑡ℎ+2 −(2∙𝑘𝑡ℎ+1)+𝑘𝑡ℎ}2
ℎ (7)
In his s udy, we p opose a coe icien o es ima e he pe u ba ion s a e o he hou ly DNI o
es ima e days wi hou luc ua ions in he 10-min scale simila o he S2 coe icien by using he
di ec no mal i adiance p o ile ins ead o he clea ness index p o ile.
P2=∑{𝐼𝑏𝑛ℎ+2 −(2∙𝐼𝑏𝑛ℎ+1)+𝐼𝑏𝑛ℎ}2
ℎ (8)
In an expe imen al app oach, we ha e obse ed ha luc ua ions wi h high ampli ude and
equency occu ed unde a iable sky condi ions. Fo daily kb index lowe han 0.3, egula
luc ua ions we e obse ed on he DNI. Thus, we ha e calcula ed he pe u ba ion coe icien
o daily alues o kb> 0.3 only. Obse ing he daily g aphs o he DNI, we ha e iden i ied a alue
o P2=215·103W2/m4 ha de ines a bounda y be ween days, wi h and wi hou he luc ua ions,
wi h an 82% o success.
Figu e 5 illus a es he e ec o his imp o emen by compa ing he esul s o he o iginal model,
modi ied by using kb ins ead o k’ , o he da a classi ica ion (o iginal model + imp o emen 1,
le ) wi h he imp o ed model ( igh ) o a hazy day.
4.5 Simila i y in he daily sums
The applica ion o he o iginal model in he gene a ion o syn he ic se ies o en esul s in
signi ican di e ences be ween he cumula i e daily alues o he o iginal hou ly and syn he ic
se ies. This p oblem can be sol ed by means o an i e a i e p ocedu e whe e he daily syn he ic
se ies a e ecalcula ed un il bo h cumula i e daily alues di e in less han 2%. A daily
unce ainly o a 2% is accep ed since ha ep esen s he unce ain y o mos o he i s class
py heliome e s.
Figu e 6 shows he block diag am o he implemen ed model.
5 Resul s
The common p ac ices we e ollowed o benchma king o he modeled i adiance da ase s
(Beye e .al, 2008). We used he oo mean squa ed di e ence (RMSD) as he main s a is ical
me hod o compa ison o hese obse a ions and gene a e da a syn he ically. In he analysis,
only dayligh hou s we e conside ed.
RMSD=√1
𝑁∑(𝐼𝑚𝑒𝑎𝑠
𝑖−𝐼𝑠𝑦𝑛𝑡ℎ
𝑖)2
𝑁
𝑖=1 (9)
Whe e, N is he numbe o da a pai s, Isyn h is he syn he ic DNI and Imeas is he measu ed DNI
(yea 2013 o he loca ion o Se ille).
The co esponding ela i e di e ences a e calcula ed as ollows:
𝑟𝑅𝑀𝑆𝐷=𝑅𝑀𝑆𝐷 𝐼𝑚𝑒𝑎𝑠
⁄ (10)
The no malized oo mean squa ed de ia ion (NRMSD) is calcula ed as ollows:
𝑁𝑅𝑀𝑆𝐷=𝑅𝑀𝑆𝐷 (𝐼𝑚𝑎𝑥 −𝐼min)
⁄ (11)
Whe e, Imax and Imin a e he maximum and minimum alues o he obse ed da ase , espec i ely.
The analysis has been made in wo ime scales, daily and hou ly. In o de o quan i y he e ec
o each implemen ed imp o emen , we ha e gene a ed he ollowing in e media e models:
• O iginal: This is he o iginal model ha uses he k’ index o he classi ica ion o he sky
condi ions.
• O iginal_kb: Includes he sky classi ica ion using he kb index.
• M1: Expands he ange o he kb index o 0.1 o each sky condi ion.
• M2: Implemen s he no maliza ion o he s anda d de ia ion (pa . 4.2).
• M3: Includes a DNI h eshold below which luc ua ions a e neglec ed (pa . 4.3).
• M4: Includes he pa ame e P2 o iden i ica ion o hazy days (pa . 4.4).
• M5: Implemen s an i e a i e p ocedu e o es ic he di e ences in daily cumula i e alues
(pa . 4.5).
• I3: Does no include any imp o emen , bu only he cubic in e pola ion o he hou ly alues a
a 10-min scale.
Each MX model includes he imp o emen s o he p e ious ones. Thus, model M5 includes all
he imp o emen s. We ha e also included model I3, which implemen s only he de e minis ic
componen wi h no s ochas ic con ibu ion.
The esul s o he benchma king in he daily scale a e p esen ed in Table 4 and Figu e 7; Table 5
and Figu e 8 show he esul s in he hou ly scale.
I is ema kable ha he use o he clea ness index, kb led o a educ ion o he daily and hou ly
e o s o mo e han 50% compa ed o he o iginal model. The imp o emen s M1, M2, and M3
do no signi ican ly educe hese e o s because hey mainly a ec he low i adiance alues
ha ha e less weigh in he global compu a ion. The e ec o imp o emen 4 a e only no iceable
in hazy days, which is an in equen condi ion and canno be quan i ied on hese scales, while
imp o emen 5 educes he e o by less han 0.8% and 7% in a daily and hou ly scale,
espec i ely.
6 Discussion and alida ion
The cubic in e pola ion o he hou ly alues a a 10-min scale (I3) and he model ha includes
all he imp o emen s (M5) show a simila pe o mance in e ms o RMSD, bu he i s one is
much simple . To compa e i he espec i e syn he ic da ase s a e s a is ically ep esen a i e,
we analyzed he p obabili y densi y unc ion (PDF) and cumula i e dis ibu ion unc ion (CDF) a
he 10-min esolu ion, which a e conside ed o ha e a signi ican impac on he pe o mance o
he STE plan s. The use o da a wi h un ealis ic equency dis ibu ions as inpu o STE
simula ion so wa e leads o un ealis ic ene gy yields (Sil a - Pé ez e al., 2014) eaching
di e ences o up o 9% o si es wi h a simila annual DNI (Chha ba and Meye , 2011).
The e o e, i is in e es ing o compa e he measu ed DNI agains he DNI se s gene a ed wi h
he o iginal model, wi h M5 and wi h I3, o iden i y he syn he ic da ase ha be e esembles
he s a is ical cha ac e is ics o he measu ed DNI. Figu e 9 shows he PDF´s and CDF’s o he
add essed da ase s depending on he sky condi ion.
In compa ison o he equency dis ibu ion o he o iginal (measu ed) da ase , he cubic
in e pola ion o he hou ly alues a a 10-min scalese (I3) exhibi s a comple ely di e en
equency dis ibu ion, especially o pa ly cloudy sky condi ions whe e he g ea es
luc ua ions ake place. The CDF o he imp o ed model (M5) is much close and imp o es he
pe o mance o he o iginal model.
Fo quan i ica ion o his s a emen , we ha e calcula ed he Finkels ein-Scha e (FS) s a is ic
(Finkels ein and Scha e , 1971) o each da ase . This s a is ic akes in o accoun he di e ences
be ween he CDF o he measu ed and syn he ic da ase s, and pe mi s he compa ison wi h he
esul s o o he models ega dless o he ime esolu ion and analysis pe iod.
𝐹𝑆=1𝑛
⁄∑𝛿𝑖
𝑛
𝑖=1 (12)
Whe e, δ is he absolu e di e ence be ween he measu ed and syn he ic CDF a each poin , i,
and n ep esen s he numbe o eadings.
Table 6 p esen s he FS o he o iginal model, imp o ed model, and cubic in e pola ion o he
hou ly alues a a 10-min scale. The imp o ed model signi ican ly educes he FS in compa ison
wi h he o iginal and I3 model. The syn he ic da ase gene a ed wi h he imp o ed model is
simila o he measu ed da ase ega dless o he sky condi ion, as shown in Figu e 10.
The obus ness o he model, ained wi h da a om Se ille only, can be obse ed when applying
i o o he si es wi h di e en clima ic condi ions and no p e ious adap a ion ( he pa ame e s
o he be a dis ibu ion used o gene a e he s ochas ic componen a e hose ob ained o
Se ille). We es ed i on i e si es in Spain a di e en la i udes in he Medi e anean, A lan ic,
and Con inen al clima es. The esul s a e summa ized in Table 7.
The imp o ed model p o ides he bes esul s when applied o Se ille and sligh ly educes i s
pe o mance o he o he i e selec ed si es, bu s ill he esul s a e signi ican ly be e han
hose o he o iginal model. The declining pe o mance o he model in he loca ions selec ed
o he alida ion is mainly caused because he s ochas ic componen o he model has been
ained wi h da a om Se ille.
7 Conclusions
In his pape , we p opose imp o emen s o he model o Polo e al. (2011) o he gene a ion o
syn he ic DNI da a in a 10-min esolu ion aking hou ly mean alues as inpu . The main
weaknesses o he o iginal model we e ound while compa ing he di e ences be ween he
hou ly and daily sums o he gene a ed and measu ed da ase due o he s ochas ic na u e o
he me hodology. The applica ion o he p oposed imp o emen s, based on he knowledge o
he dis inc i e ea u es o he DNI, kep he daily cumula i e alues and consis ency o he
luc ua ions o he syn he ic DNI wi h he obse ed sky condi ion o educe bo h he e o s in
mo e han 40% RMSD b inging hem o 1.3% and 7.9% in daily and hou ly sums, espec i ely.
The use o kb ins ead o k’ esul ed in a signi ican educ ion o he RMSD while he ou come o
he es o he imp o emen s showed a be e pe o mance wi h espec o he equency
dis ibu ion o he syn he ic da a. In addi ion, he imp o ed model has been applied wi hou any
adap a ion o o he loca ions wi h di e se clima ic condi ions, achie ing excellen esul s
compa ed o he o iginal model, and demons a ing he obus ness o he p oposed
me hodology.
Fig 7. Va ia ion o he daily RMSD wi h he applica ion o he p oposed imp o emen s.
Fig 8. Va ia ion o he hou ly RMSD wi h he applica ion o he p oposed imp o emen s.
0
1
2
3
4
5
6
7
8
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
O iginal O iginal_kb M1 M2 M3 M4 M5 I3
kWh/m2
%
RMSDDAILY
0
2
4
6
8
10
12
14
16
18
20
0
10
20
30
40
50
60
70
80
90
O iginal O iginal_kb M1 M2 M3 M4 M5 I3
W/m2
%
RMSDHOURLY

Fig 9. PDF (le ) and CDF ( igh ) analysis o he syn he ic and measu ed da ase s.
0
1000
2000
3000
4000
5000
6000
7000
F equency
W/m2
Yea 2013 Se ille
0< kb ≤ 0.2
DNI_Measu ed
DNI_Syn h_M5
DNI_Syn h_O iginal Model
DNI_Syn h_I3
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Cummula i e p obabili y
W/m2
Yea 2013 Se ille
0< kb ≤ 0.2
DNI_Measu ed
DNI_Syn h_M5
DNI_Syn h_O iginal Model
DNI_Syn h_I3
0
200
400
600
800
1000
1200
F equency
W/m2
Yea 2013 Se ille
0.2< kb ≤ 0.4 DNI_Measu ed
DNI_Syn h_M5
DNI_Syn h_O iginal Model
DNI_Syn h_I3
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Cummula i e p obabili y
W/m2
Yea 2013 Se ille
0.2< kb ≤ 0.4
DNI_Measu ed
DNI_Syn h_M5
DNI_Syn h_O iginal Model
DNI_Syn h_I3
0
100
200
300
400
500
600
700
800
F equency
W/m2
Yea 2013 Se ille
0.4< kb ≤ 0.6
DNI_Measu ed
DNI_Syn h_M5
DNI_Syn h_O iginal Model
DNI_Syn h_I3
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Cummula i e p obabili y
W/m2
Yea 2013 Se ille
0.4< kb ≤ 0.6
DNI_Measu ed
DNI_Syn h_M5
DNI_Syn h_O iginal Model
DNI_Syn h_I3
0
500
1000
1500
2000
2500
3000
3500
4000
4500
F equency
W/m2
Yea 2013 Se ille
kb ≥ 0.6
DNI_Measu ed
DNI_Syn h_M5
DNI_Syn h_O iginal Model
DNI_Syn h_I3
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Cummula i e p obabili y
W/m2
Yea 2013 Se ille
kb > 0.6
DNI_Measu ed
DNI_Syn h_M5
DNI_Syn h_O iginal Model
DNI_Syn h_I3
Fig 10. Daily examples
TABLES
Table 1. Selec ed loca ions o he model alida ion.
La i ude
(°N)
Longi ude
(°W)
Al i ude
(m)
Clima e
Yea
G anada
37.1
−3.0
1100
Con inen al
Medi e anean
2013
Badajoz
38.9
−6.4
200
Con inen al
Medi e anean
2011
Alican e
38.6
−0.8
500
Con inen al
2009
Pamplona
42.8
−1.6
450
A lan ic
2010
Alme ia
37.1
−2.3
500
Medi e anean
2012
Se ille
( ained)
37.4
−6.0
10
Medi e anean
2013
Table 2. Sky condi ion o each kb in e al.
Sky condi ion
Type o day
Kb≤ 0.1
To ally co e ed
Thick clouds, no luc ua ions
0.1<kb≤ 0.2
To ally co e ed
Thick clouds, some luc ua ions
0.2<kb≤ 0.3
Mos ly co e ed
Al e na i e clouds and clea s, some luc ua ions
0.3<kb≤ 0.4
Mos ly co e ed
Al e na i e clouds and clea s, g ea luc ua ions
0.4<kb≤ 0.5
Pa ly co e ed
Thin clouds, g ea e luc ua ions
0.5<kb≤ 0.6
Pa ly co e ed
Thin clouds, some luc ua ions (and hazy days)
0.6<kb≤ 0.67
Mos ly clea
No clouds bu ce ain u bidi y
kb>0.67
To ally clea
No clouds
Table 3. Maximum de ia ion o he 10-min di ec no mal i adiance wi h espec o he hou ly
mean o each kb in e al.
Maximum s anda d de ia ion analysis
kb
[0, 0.1]
[0.1, 0.2]
[0.2, 0.3]
[0.3, 0.4]
[0.4, 0.5]
[0.5, 0.6]
[0.6, 0.67]
𝑚𝑎𝑥
𝑖𝑚𝑝𝑟𝑜𝑣Wm
215
345
401
431
442
435
430
𝑚𝑎𝑥
𝑜𝑟𝑖𝑔Wm
442
442
442
442
442
442
442
Di (W/m2)
227
97
41
11
0
7
12
Table 4. E ec o he imp o emen s in he daily ime scale.
Daily ime scale
O iginal
O iginal_kb
M1
M2
M3
M4
M5
I3
RMSD (kWh/m2)
0.42
0.16
0.16
0.12
0.12
0.12
0.05
0.05
RMSD (%)
7.4
2.9
2.8
2.8
2.2
2.1
0.9
0.8
Table 5. E ec o he imp o emen s in he hou ly ime scale.
Hou ly ime scale
O iginal
O iginal_kb
M1
M2
M3
M4
M5
I3
RMSD (W/m2)
77.8
41.0
40.7
39.1
34.8
34.0
31.9
18.1
RMSD (%)
17.2
9.1
9.0
8.7
7.8
7.6
7.1
4.0
Table 6. Finkels ein-Scha e (FS) analysis
FS
O iginal model
0.0233
I3
0.0160
M5 model
0.0061
Table 7. Compa ison o RMSD, NRMSD, and FS when using he M5 in di e en loca ions.
RMSEdaily
(%)
NRMSEdaily
(%)
RMSEhou ly
(%)
NRMSEhou ly
(%)
NRMSE10-min
(%)
FS
G anada
1.5
0.8
10.4
5.5
12.3
0.0090
Badajoz
1.3
0.6
6.3
3.1
11.8
0.0092
Alican e
1.5
0.7
7.2
3.4
12.5
0.0101
Pamplona
1.9
0.7
9.6
3.6
13.0
0.0099
Alme ia
0.7
0.4
8.4
4.2
10.5
0.0100
Se ille
0.8
0.5
5.7
3.2
9.8
0.0061