An improved model for the synthetic generation of high temporal resolution direct normal irradiation time series

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