Meta-model based optimization of hot rolling processes in the metal industry

CIplus Band 6/2017 Me a-model based op imiza ion o ho olling p ocesses in he me al indus y Ch is ian Jung, Ma in Zae e e , Thomas Ba z-Beiels ein, Gün e Rudolph The final publica ion is a ailable a Sp inge ia h p://dx.doi.o g/10.1007/s00170-016-9386-6 Noname manusc ip No. (will be inse ed by he edi o ) Me a-model based op imiza ion o ho olling p ocesses in he me al indus y Ch is ian Jung · Ma in Zae e e · Thomas Ba z- Beiels ein · G¨un e Rudolph he da e o eceip and accep ance should be inse ed la e Abs ac To maximize he h oughpu o a ho olling mill, he numbe o passes has o be educed. This can be achie ed by maximizing he hickness educ ion in each pass. Fo his pu pose, exac p edic ions o oll o ce and o que a e equi ed. Hence, he p edic i e models ha desc ibe he physical beha io o he p od- uc ha e o be accu a e and co e a wide ange o di - e en ma e ials. Due o ma ke equi emen s a lo o new ma e i- als a e es ed and olled. I hese ma e ials a e chosen o be olled mo e o en, a sui able low cu e has o be es ablished. I is no easonable o de e mine hose low cu es in labo a o y, because o cos s and ime. A s ong demand o quick pa ame e de e mina ion and he op imiza ion o low cu e pa ame e wi h minimum cos s is he logical consequence. The e o e pa ame e es ima ion and he op imiza ion wi h eal da a, which we e collec ed du ing p e ious uns, is a p omising idea. P oduce s bene i om his da a-d i en app oach and ecei e a huge gain in lexibili y when olling new ma- e ials, op imizing cu en p oduc ion, and inc easing quali y. This concep would also allow o op imize low cu e pa ame e s, which ha e al eady been ea ed by s anda d me hods. In his a icle, a new da a-d i en ap- Facul y o Compu e and Enginee ing Sciences Cologne Uni e si y o Applied Sciences, 51643 Gumme sbach, Ge many [email p o ec ed] Facul y o Compu a ional In elligence TU Do mund Uni e si y, 44227 Do mund, Ge many [email p o ec ed] p oach o p edic ing he physical beha io o he p od- uc and se ing impo an pa ame e s is p esen ed. We demons a e how he p edic ion quali y o he oll o ce and oll o que can be op imized sus ainably. This o e s he oppo uni y o con inuously inc ease he wo kload in each pass o he heo e ical maximum while p oduc quali y and p ocess s abili y can also be imp o ed. Keywo ds lowcu e ·K iging ·me a-model ·me al · ho olling 1 In oduc ion The complex p ocess o ho olling equi es e y accu- a e physical models. Se e al di e en physical models a e used in ho olling mills. These models include he slab o ingo hea ing a he u nace, whe e he he -5 mal beha io is modeled, as well as he olling p o- cess i sel . The ask o he di e en models is o p e- dic he he mal and physical beha io o he p oduc and se impo an pa ame e s o achie ing maximum p oduc h oughpu while inc easing he quali y o he10 inal p oduc . The e a e se e al p ocess and p oduc pa ame e s which play impo an oles. They a e inde- penden o he plan ype and applicable o s eel and aluminum ho mills. Some o hem, o example plan geome ies o d i e pa ame e s, emain cons an du ing15 olling. O he pa ame e s may a y du ing olling bu a e no domina ed by he ma e ial, e.g., he maximum possible hickness educ ion depends mainly on he ac- ual hickness and he wo k olls cu en ly ins alled. The maximum edcu ion is o cou se also dependen 20 om he ic ion be ween he olls and he ma e ial bu his e ec is no as huge as he geome ical limi a- ions. Pa ame e s, which depend on he ma e ial o he p oduc , a e usually ha d o op imize, because (i) he numbe o di e en ma e ials is s eadily inc easing due25 o ma ke equi emen s and (ii) measu emen s o he p ocess a e only indi ec ly co ela ed o he ma e ial. The mos impo an pa ame e s o he p edic ion o he oll o ce a e he low cu e pa ame e s o each ma e ial. A mo e e icien me hod o op imizing hese30 pa ame e s is necessa y o inc ease he lexibili y and educe he cos o he olling p ocess. A ecen app oach uses a i icial in elligen ech- niques o op imiza ion o shape olling sequences [17]. Especially in he ield o cold olling, se e al me hods35 o simula ion and op imiza ion we e published [18, 24]. The e we e also s udies, which we e based on ini e el- emen me hods [25]. In his pape we p opose me a- model based op imiza ion s a egies o he ask o ho olling mill low cu e pa ame e op imiza ion. Me a40 models, also e e ed o as su oga e models, simpli y 2 Ch is ian Jung e al. he simula ion op imiza ion as he un imes a e gene - ally much sho e han he o iginal unc ion e alua ions [1, 16] and a e a p o en s a egy o global op imiza ion [15]. The esul s a e compa ed o classical op imiza ion45 s a egies. Hence, his pape add esses he ollowing e- sea ch ques ions: (R-1) Can he olling p ocess bene i om me a-model based op imiza ion? (R-2) How ime- and cos e icien is his app oach in50 compa ison o he es ablished indus y p oce- du es? This pape is s uc u ed as ollows. Sec ion 2 in o- duces echnical e ms and undamen al p inciples o ho olling and he ela ed pa ame e s. I desc ibes he cu -55 en s a e-o - he-a app oach in indus y. P oblems e- la ed o he olling p ocess a e desc ibed in Sec. 3. Ou me hodology is de ailed and compa ed o exis ing ap- p oaches in Sec. 4. The expe imen al se up is desc ibed in Sec. 5. Resul s a e p esen ed in Sec. 6. The pape 60 concludes wi h a discussion in Sec. 7. 2 Ho Rolling 2.1 Fundamen als and Technical Te ms Be o e desc ibing some o he main aspec s, he unda- men al echnical e minology will be in oduced. Fig-65 u e 1 shows a common ho s ip mill o s eel. The p o- cess in gene al is e y simila o an aluminum mill. The wo k low o he p ocess is om he le side o he igh side. The main componen s a e a ehea ing u - nace, a e e sing oughing mill, a con inuous inishing70 mill, a cooling line, and a downcoile . The coilbox be- ween oughing mill and inishing mill is mo e o less op ional. I is used o achie e be e empe a u e p o- iles and allows a mo e compac olling mill. The p ocess s a s wi h he cha ging o he u nace.75 He e, slabs which a e usually a oom empe a u e a e cha ged and ehea ed o empe a u es a ound 1200 deg C o s eel mills and 500 deg C o aluminum mills. Fig. 1 Ho Rolling mill o s eel wi h coilbox. The oughing mill consis s o one ho izon al s and wi h ou olls, he so- called qua o s and and an op ional e ical olling s and wi h wo olls, he so-called edge (no shown in he igu e). The slab geome ies may a y. Usually, hey ha e an inpu hickness,hini , be ween 200 mm and 300 mm,80 wid h o 600 mm o 2500 mm and leng h be ween 3 m and 10 m o s eel p oduc ion. Fo aluminum, he hick- ness a e discha ge is commonly a ound 600 mm be- cause he empe a u e loss o aluminum du ing olling is much less han o s eel. A e discha ging, he i s ma-85 jo p ocess is o educe he hickness o he slab by 30 mm o 40 mm. This is done in a so-called oughing mill (RM). The oughing mill in a con en ional ho s ip mill consis s o one ho izon al s and wi h ou olls, which is hen called qua o s and and an op ional e ical olling90 s and wi h wo olls, he so-called edge (no shown in he igu e). The edge is op ional and has he ask o educe he wid h o he slab and o imp o e he shape o he slab especially a bo h ends. The educ ion om he ini ial hickness hini down95 o he a ge hickness h a ge is done in se e al de o - ma ion s eps which a e called passes. In each o hese de o ma ion s eps, he hickness o he slab will be con- inuously educed un il he a ge hickness is a ained. The educ ion has o be spli o se e al passes because100 he easible educ ion in one pass is limi ed. The de- o ma ion in he oughing mill can be done in bo h ope a ing di ec ions. Hence, each pass changes he di- ec ion o mo emen o he slab. The o al numbe o passes has o be odd, since he slab has o be mo ed o105 he nex p ocess s ep. A e olling in he oughing mill, he slab is ans e ed in he di ec ion o he inishing mill. I a coilbox is used, he ma e ial is coiled i s and hen di ec ly uncoiled o s a he olling in he inish- ing mill. He e, he p oduc is olled in se e al s ands o110 he inal hickness (speci ied by by he cus ome ) and is di ec ly cooled a e wa ds. The addi ional cooling line is only used o s eel mills. Finally, he p oduc is coiled in he downcoile . One impo an quali y c i e ion o he inal p od-115 uc is he de ia ion o he ac ual hickness om he a ge hickness h a ge . The hickness de ia ion a he head o he p oduc is ypically a esul o he oll o ce p edic ion accu acy o he physical model. The head o he p oduc is he i s pa which encoun e s he120 de o ma ion. In inishing mills we will mos o en ind hickness gauges a e he las s and. These a e used o con ol he hickness once he head passes he gauge so he oll o ce de ia ion would mainly be esponsible o he head hickness. Addi ionally, he oll o ce is used125 as e e ence o he bending and o p o ile con ol and is he e o e also e y impo an o he p oduc p o ile. Inline con ol o he hickness is usually no ins alled in o he mill ypes such as pla e mills. This is because he o al leng h o he p oduc is much smalle han130 in mills wi h coil p oduc ion, whe e he o al leng h Me a-model based op imiza ion o ho olling p ocesses in he me al indus y 3 migh accumula e o mo e han 1000m. In ha case, a eedback o he measu ed hickness du ing olling can be used o adjus he oll gap and he e o e he inal hickness. Fo mill ypes wi hou inline con ol, i is135 c ucial o imp o e he oll o ce p edic ion in o de o minimize he hickness de ia ion. Ano he aspec is he h oughpu o he mill, which should always be maximized. To maximize he h ough- pu , he numbe o de o ma ion s eps, i.e. he numbe 140 o passes, should be minimized wi hou iola ing o he cons ain s. Two o he mos limi ing pa ame e s when ying o inc ease he educ ion in each pass a e (i) he maximum oll o ce capabili y and (ii) he maximum oll o que capabili y o he s and. The e o e, i is es-145 sen ial o ha e good models o he p edic ion o oll o ce and o que. As men ioned abo e, de ia ions in oll o ce p e- dic ion will also a ec he hickness and he e o e he quali y o he p oduc . Because cu en ma ke equi e-150 men s co e a e y wide ange o ma e ials and geome- ies i is impo an o inc ease he model quali y o he oll o ce p edic ion o all p oduc s which may be olled on hese mills. Ideally, he oll o ce p edic ion is comple ely independen o he geome y and o he 155 pa ame e s and will only depend on he quali y o he ma e ial pa ame e s. Each ma e ial is usually classi ied acco ding o i s chemis y. A ma e ial da abase s o es mechanical and he mophysical pa ame e s o he de- sc ip ion o he di e en p ope ies o each class. These160 pa ame e s a e used o he p edic ion o beha io du - ing he de o ma ion p ocess and a e he e o e o majo impo ance o he olling p ocess. 2.2 Flow Cu e and Roll Fo ce Model The low cu e pa ame e s a e mos ele an o he oll o ce model. The low cu e exp esses he ma e ial esis ance du ing plas ic de o ma ion in dependence on he chemis y, he empe a u e, he de o ma ion and he de o ma ion a e. The de o ma ion ϕ, also called e ec i e, loga i hmic de o ma ion o ue s ain is ex- p essed by: ϕ= ln h0 h1 . He e, h0is he inpu hickness and h1is he ou pu hickness o he pass. The i s low cu e o mulas we e de eloped by Geleji and Ekelund a ound 1950. These o mulas we e only linea ly dependen on he empe a- u e and only alid o s anda d low ca bon s eel [10]. A e wa ds se e al o he o mulas we e de eloped wi h polynomial componen s and exponen ial e ms which also ook in o accoun he de o ma ion and de o ma- ion a e. While he i s o mulas we e only alid o some low alloyed ca bon s eels, Hajduk de eloped o - mulas which we e also alid o some medium and high alloyed ca bon s eels [9],[8]. A good o e iew and de- sc ip ion o he di e en low cu es can be ound in [10, 12, 26]. These o mulas model he de o ma ion e- sis ance,k , in dependence on he de o ma ion ϕ, he de o ma ion a e, ˙ϕ, and he empe a u e,ϑ. The de- o ma ion esis ance o low s ess exp esses he s ess which is needed o sus ain a plas ical de o ma ion. In gene al, he de o ma ion a e can be o mula ed as: k =AKϕKϑK˙ϕ, whe e A∈R+is a cons an ac o and he e ms K(·) 165 ep esen unc ions o he co esponding a iables ϕ, ˙ϕ, and ϑ, espec i ely. The mos common model o hese o mulas was de eloped by Hensel and Spi el [10]. I was ex ended a he Uni e si y o F eibe g. Thus, hese ex ensions a e called F eibe ge App oach. The170 ex ended e sions o his low cu e model gi es a be - e app oxima ion o he low s ess wi hin high de o - ma ion g ades. Some o he a ailable low cu e mod- els, which a e ypically used in p ocess models o ho olling, a e p esen ed in Table 1. Thei co esponding175 equa ions ead as ollows. k =k ,0A0A1em1ϑA2ϕm2em4 ϕA3˙ϕm3(1) k =A0em1ϑϕm2em4 ϕ˙ϕm3(2) k =A0em1ϑϕm2em4 ϕ(1 + ϕ)m5ϑem7ϕ˙ϕm8ϑ(3) k =k ,0A1em1ϑA2ϕm2A3˙ϕm3(4) The mul iplie s Ai(i= 0,1,2,3) can be educed o one pa ame e , A. The pa ame e s mj(j= 1,2,...,8) a e de ining he exponen ial beha io o he ma e ials in dependence o he empe a u e ϑ, he de o ma ion180 ϕ, and he de o ma ion a e ˙ϕ. The pa ame e s ϑ,ϕ, and ˙ϕa e de ining he wo king poin in each de o ma- ion. The alue k ,0used in he equa ions (1) and (4) is he basic de o ma ion and is calcula ed by empi ical o mula ions based on he chemis y. Each ma e ial is185 classi ied acco ding o i s chemis y and ge s i s own pa ame e se Mwi h pa ame e alues m1 o m8and A0 o A3, espec i ely. Usually, he e is one pa ame e - se M o each ma e ial, which is hen alid o a spe- ci ic equa ion only. Besides he pa ame e alues o 190 he models also he alid egion o hese pa ame e s is s o ed. Summa izing, he pa ame e s k , ϕ, ˙ϕ, and ϑ, he mul iplica o s Ai, mj, (i= 0,1,2,3; j= 1,...,8), and ela ed unc ions k ,0, Kϕ, K ˙ϕ, Kϑ, a e used. Nowadays, hund eds o di e en ma e ials a e known.195 The pa ame e k is almos linea ly co ela ed wi h he 4 Ch is ian Jung e al. Table 1 O e iew o ypical equa ions o he eg ession o he ma e ial low s ess. The mul iplie s Aia e educed o one pa ame e , A. Pa ame e s mja e de ining he exponen ial beha io o he ma e ials in dependence o he empe a u e ϑ, he de o ma ion ϕo he de o ma ion a e ˙ϕwhich de ines he wo king poin . Pa ame e k ,0is de ined by a simple equa ion based on he chemis y. En ies in he column ”Equa ion” e e s o he equa ions de ined on p. 3. Eq. Name #Pa ams Pa ame e Lis (1) F eibe g 1 5 A, m1, m2, m3, m4 (2) F eibe g 4 5 A, m1, m2, m3, m4 (3) F eibe g 8 7 A, m1, m2, m4, m5, m7, m8 (4) Hensel Spi el 4A, m1, m2, m3 oll o ce and oll o que. Thus, he model p edic ion quali y and he ewi h he p ocess s abili y and p od- uc quali y a e co ela ed o he pa ame e -se M. I is he e o e impo an o op imize hose pa ame e s in o -200 de o inc ease he model quali y and o ensu e a s able p ocess wi h maximum h oughpu and p oduc qual- i y. A s anda d p ocedu e o ob aining hese pa ame- e s is he measu emen o he de o ma ion esis ance in a labo a o y. Those measu emen s can be used o a205 eg ession on o one o he o mulas shown in Equa ions (1) o (4). O cou se, o he o mulas exis , and migh be used o eg ession. Especially when ying o model he de o ma ion o mic o alloyed o high alloyed s eel o when complex ma e ials wi h phase ans o ma ions210 should be desc ibed hese o he models migh be mo e sui able. 2.3 Desc ip ion o he analy ical model The analy ical model used o he calcula ion o he oll o ce is based on he elemen a y olling heo y [12, 27].215 Some o he limi a ions o ha heo y a e compensa ed wi h co ec ion unc ions. Fo example, one o he e- qui emen s o he elemen a y heo y is ha du ing each pass he e is a comple e plas ic de o ma ion o he whole ma e ial. Fo p oduc s wi h hickness abo e 500220 mm his is clea ly no gi en. The e o e, a compensa ion cu e, which is empi ically de e mined, is applied. Fo he calcula ion o he oll o ce in each pass he de o - ma ion zone is di ided in o single s ipe-like elemen s and he o ce balance o each s ipe is calcula ed. The225 solu ion yields o he basic di e en ial equa ion o he plas ically de o ma ion heo y which was de eloped by Ka man in 1925. When calcula ing he oll o ce o one pass, he low esis ance has o be conside ed. This low esis ance k is ma e ial dependen and is in lu-230 enced by he pa ame e alues o ϕ, ˙ϕand ϑ. Fu he - mo e, he o ces induce empe a u e in o he ma e ial so he calcula ion o he nex pass depends on he p e- ious passes. Op imizing he low cu e by analyzing he olling esul s is no su icien . I he pa ame e o 235 he low cu e changes, he whole p ocess has o be sim- ula ed again and hen he calcula ed oll o ces based on he new low cu e can be compa ed wi h he o ig- inal eedback, i.e., measu emen s o oll o ce, o que, empe a u e, and speed. Addi ionally, i is also no su -240 icien o op imize he esul o a single p oduc be- cause he pa ame e s ϕ, ˙ϕand ϑmay no a y enough o achie e s able esul s. The e o e i is p e e able o conside a campaign wi h a wide a ia ion o p oduc geome ies, empe a u e anges, and de o ma ions.245 2.4 S anda ds in Indus y Cu en ly, he a ailable concep s o he op imiza ion o low cu e pa ame e s a e mos ly dealing wi h de e mi- na ion o hose pa ame e s in labo a o y a he han op imizing hose pa ame e s wi h eal p ocess da a.250 T adi ionally, he pa ame e s a e measu ed wi h small samples o one piece in labo a o y and a e hen gene - alized o e e y ma e ial which is close o he sample in e ms o ma e ial composi ion. Some companies a e modi ying he low cu e pa ame e s wi h linea mod-255 els. Tha is, hey a e de e mining he p edic ion accu- acy o hei model and a e a ying some o he in lu- ence pa ame e s. Mos o he esea ch in his a ea is on he de elopmen o sui able low cu e equa ions espe- cially o high and mic o alloyed s eel [11, 19, 28] a he 260 han using a da a d i en app oach o he op imiza ion o hose pa ame e s. 3 P oblem Desc ip ion Acco ding o ou bes knowledge, low cu e pa ame e de e mina ion in labo a o y as desc ibed in Sec. 2 will265 equi e se e al weeks and cos s se e al housand Eu o o he equi ed ma e ials. Some imes, his is no a - o dable and he e o e no a sui able way o de e mine hose pa ame e s. Hence, we a e looking o a cheape app oach o pa ame e es ima ion.270 Due o hei highly nonlinea beha io , he low cu e equa ions canno be sol ed di ec ly. Fu he mo e, a di e en oll o ce would esul in a di e en empe - a u e balance o he p oduc and hus he empe a u e in he nex pass di e s om he o iginal calcula ion.275 Because hese aspec s canno be neglec ed, we ha e o simula e a whole scena io when es ing new pa ame e se s o he low cu e o a speci ied ma e ial. The cal- cula ions o he oll o ces and oll o ques wi hin his simula ion a e a e wa ds compa ed wi h he measu e-280 men s o ge a quali y c i e ion o he new pa ame e Me a-model based op imiza ion o ho olling p ocesses in he me al indus y 5 se . The de ailed desc ip ion o he simula ion scena io is p esen ed in Chap e 4. I is impo an ha he simula ion scena io beha es in he same way as he online p ocess. The e o e, he285 simula ion uses eedback o he measu ed speed, he educ ion, and empe a u e o calcula e he new se - ings. This enables he simula ion o achie e he same wo king poin as in he online p ocess. Ano he p ob- lem can be he amoun o da a. The simula ion o a290 whole ba ch whe e only one ma e ial g oup was olled consis s o housands o di e en de o ma ion s eps and will he e o e be highly expensi e in e ms o simula ion ime. Se e al op imiza ion algo i hms equi e bound cons ain s o he op imized pa ame e s. In ou case,295 pa ame e s Aand mia e dependen on each o he . The only limi a ion which can be se is a plausible egion o he esul ing alue k o he basic de o ma ion. In ho mills, he maximum basic de o ma ion alue o k is usually below 300 N mm2, bu always posi i e. Then, o a300 gi en maximum wo king ange o he de o ma ion, de- o ma ion a e, and empe a u e, he easibili y o he pa ame e se can be es ed. Due o he ac ha e e y company has usually i s own classi ica ion sys em i migh be ha ma e ials305 which a e g ouped oge he in one company a e sepa- a ed in o he companies. In his case, he op imiza ion, which has been done in he i s company canno be di- ec ly used o o he companies and has o be enewed e e y ime.310 Summa izing, i is desi able o op imize he p ocess in o de o – eliably es ima e alid low cu es, – educe lab cos s, –sa e ime,315 –de e mine pa ame e s in hei wo king en i onmen s, and –make he p ocess mo e lexible and adap o new (ma e ial) changes quickly. 4 Me hodology320 4.1 Simula ion en i onmen The en i onmen o he online p ocess is shown in Fig. 2. Fi s o all he model ge s in o ma ion abou he p od- uc which includes ini ial geome y da a, discha ging empe a u e da a, and in o ma ion abou he chemis y325 o he p oduc . The discha ge empe a u e is an ini- ial empe a u e ield o he p oduc . One pa o he olling model calcula es he empe a u es losses du ing he whole p ocess. Finally, he impo an pa ame e ϑ is a esul o he empe a u e losses om discha ge o330 ha poin o he p ocess. Fu he mo e, a ge da a is Fig. 2 Model en i onmen in he eal-wo ld p ocess o each pass: The p oduc and cus ome da a such as ma e ial de- sc ip ion, ini ial and a ge geome y (1) a e combined wi h ope a o da a (2) and a e ed o he model which calcula es all equi ed se ings (3) o he nex olling pass (se up o nex pass). A e olling o his pass he model ge s eedback (4) o he jus olled pass and combines his in o ma ion o he ecalcula ion o he p e ious pass and o he nex calcu- la ions (2-4). Addi ionally, changes om he ope a o o he nex pass a e send o he model. The p oduc and cus ome da a a e only p oduc and no pass dependen and may only be send once. To enable a simula ion o his p ocess e e y in- and ou pu o he model is s o ed in a da abase. Fig. 3 Model en i onmen o he o line simula ion. The da a which has been collec ed in he eal wo d p ocess is send o he model which calcula es a new se up o he nex pass. This se up may be di e en om he o iginal one. Bu because o he ac ha we also s o e he eedback om he d i es and gauges he model will ecei e also he o iginal se - ing and ecalcula es he pass as i eally has been olled. also coming om he cus ome . Bo h da a can be seen as cons an and a e deno ed wi h Ncons , i.e., he ope a- o canno change hem as hey a e pa o he p oduc- ion planning sys em which handles he o de s o he335 mill owne . A e wa ds, he Ncons is used o calcula e he i s se ings o he mill and his esul is shown o he ope a o . The se ing consis s o oll gap se ings, speed se ings, geome y, empe a u e and ime calcu- la ions. Wi h his da a he i s planned se ings NSe ,340 which also include he pa ame e ϕ, ϑ and ˙ϕ, a e calcu- la ed and a e shown o he ope a o . The ope a o can in e e e and modi y he way his p oduc is olled. This is e e ed o as olling s a egy. This olling s a egy de ines how a p oduc is olled which includes olling345 speed, numbe o passes, de o ma ions, speed se ings, 6 Ch is ian Jung e al. possible olling b eaks and much mo e. These s a egies may also be speci ic cons ain s like absolu e educ ion, de o ma ion ϕ, o ce, o que, bu also o he es ic ions o he p ocess like d i e limi a ions.350 I he ope a o is sa is ied wi h he se ings NSe calcula ed by he model, he se ings a e sen o he plan whe e he i s pass is olled. A e his pass, all measu emen s collec ed du ing olling which include o ces, o ques, speeds, empe a u es, gap se ings, de-355 lay imes, and se e al mo e da a, a e sen o he model and he da abase bu a e also shown o he ope a o . Now, he ope a o and also he model can adap he se - ings o he nex passes and eac o any unexpec ed beha io o he mill. Usually, no big changes a e made360 by he ope a o and he olling o he u he passes is s a ed di ec ly. Again, o each pass he se ings a e sen o he plan and he eedback o measu emen s is ecei ed om he plan . Fo analysis and o line simu- la ions all inpu s and ou pu s a e s o ed in a da abase.365 This da abase is he basis o he o line simula ion shown in Figu e 3. He e, we eed he same model which was used in he online p ocess wi h he da a s o ed du - ing he eal-wo ld p ocess. The e o e we can gua an ee ha he model eac s in he same way as i would eac 370 in he olling p ocess. The da a, which was coming om he ope a o and he geome y da a a e aken om he da abase. Hence, he model will no ecognize i i is used o an o line calcula ion o o an online scena io. Al hough i may calcula e di e en se ings o he p o-375 cess, i will ecei e he o iginal eedback om he plan and calcula e e e y hing based on he o iginal se ings. Fo a p oduc wi h 19 passes he model is igge ed 20 imes. The i s igge c ea es he ini ial se up and all o he igge s a e eedbacks o he 19 passes wi h380 which he o iginal p oduc s we e olled. The only di e - ence a e he pa ame e s used o he calcula ion o he oll o ce. The e o e, we ha e a calcula ed and measu ed alue o he o ce and o que o each p oduc and pass o e e y un. A e each simula ion un, he pa ame-385 e s may be changed and esul s o di e en pa ame e s may be compa ed. This enables he op imiza ion o he low cu e pa ame e s. Minimiza ion o he Roo Mean Squa e E o (RMSE) o he p edic ed oll o ce is he op imiza ion objec i e.390 4.2 Su oga e Modeling I he simula ion uns o he o iginal p oblem in gene al a e e y expensi e in e ms o e alua ion i would be e y ime consuming o pe o m pa ame e op imiza- ion on hose o iginal scena ios. The e o e, we use a395 su oga e-model based op imiza ion app oach. Su o- ga e models a e supposed o eplace he o iginal, expen- si e simula ion model and a e expec ed o be cheape o calcula e. The analy ical models, as in oduced in Sec. 2.2,400 a e cheape o e alua e in compa ison o he ini e ele- men me hod models. In his pape , he e m su oga e model is used o desc ibe da a d i en models, which a e buil om an analy ical model. The e o e, he simula- ion uns on he analy ical model wi h a speci ied se o 405 p oduc s is he expensi e model. The p oduc s which a e simula ed usually belong o he same ma e ial g oup and ha e p e iously been olled in a se ies on a eal olling mill. Each p oduc is calcula ed as i would be done du ing olling. The e o e, ime delays which oc-410 cu ed du ing he eal p oduc ion a e also aken in o accoun . Ou da a is sou ced om a e e sing aluminum ho mill, which has ypically a ound 19 passes. Hence, we ha e mo e han 20 calcula ions o each p oduc be-415 cause he model is igge ed a e inishing each pass. The da a o each pass is sen o he model, which may eac on unp edic ed ci cums ances. In gene al, da a-d i en su oga e models can be any kind o models, e.g., a i icial neu al ne wo ks, linea 420 models, K iging, andom o es and o he s. A de ailed o e iew o su oga e model based nume ical op imiza- ion is p esen ed by Jin [13] and Jones [15]. One amewo k o su oga e-model based op imiza- ion is sequen ial pa ame e op imiza ion (SPO) [3].425 SPO combines me hods om classical DoE and mode n Design and Analysis o Compu e Expe imen s (DACE) [2, 4] based on K iging models. Algo i hm 1 p esen s he pseudo code o SPO, as adap ed o he applica ion o ho mill pa ame e op i-430 miza ion. No e, ha we will use he no a ion x(i), y(i) o he da a om he i- h pass which is passed o he su oga e model. Du ing he i s s age o expe imen a- ion, SPO explo es he sea ch space o he op imiza ion p oblem A, which is ea ed as a black box. A se o in-435 pu design poin s xis passed o A. Usually hese a e c ea ed by a space illing design, e.g. La in hype cube sampling. Each call o he objec i e unc ion p oduces some ou pu y ega ding i s pe o mance. SPO now ies o de e mine a unc ional ela ion-440 ship be ween xand y.SPO hus uses a model Y(x) as su oga e o he ho mill simula ion model A. As men- ioned abo e, he chosen model ype is K iging K iging is equen ly used o su oga e-model based op imiza- ion, because i p o ides a powe ul and lexible p edic-445 o . I also p o ides an es ima e o he a iance o e o o each p edic ion. The obse a ions a e in e p e ed as ealiza ions o a s ochas ic p ocess. A gaussian ke nel is used o model he co ela ion be ween obse a ions [23]. Me a-model based op imiza ion o ho olling p ocesses in he me al indus y 7 Algo i hm 1: SPO-based ho mill simula ion uning. // phase 1, collec ini ial knowledge abou he op imized p ocess: 1le Abe he ho mill simula ion model we wan o une; 2gene a e an ini ial design DES = {x(1),...,x(n)}o n pa ame e ec o s; 3le k=k0be he ini ial numbe o eplica ions o de e mining es ima ed esponses; 4 o each x ∈DES do 5 un Awi h x o de e mine he es ima ed esponse yo x; // phase 2, building, using and imp o ing a su oga e model: 6while s op c i e ia no eached do 7build su oga e model Y(x) based on DES and {y(1),...,y(|DES|)}; 8op imize he model w. . some cos unc ion and cons ain s, hus p oduce a se DES’ o dnew pa ame e ec o s ; 9 un Awi h each x∈DES’ o de e mine he esponse; 10 ex end he design by DES = DES ∪DES’; // phase 3, inal exploi a ion and ine uning: 11 use local op imize o he bes ppa ame e se s x1...p ∈DES, wi hou cons ain s In he sequen ial imp o emen loop SPO op imizes450 he su oga e model Y(x) o e he conside ed space o inpu a iables by means o a cos unc ion. Once he new se o design poin s DES’ has been selec ed, he equi ed e alua ions o DES’ a e pe o med. Based on DES’, he su oga e model Y(x) is upda ed.455 In s ep 8 o Algo i hm 1, a sea ch on he su o- ga e model is pe o med. He e, he cons ain s o he mill pa ame e iza ion p oblem ha e o be conside ed. As he cons ain s a e no expensi e o e alua e, hey a e e alua ed oge he wi h he su oga e model i sel .460 Fo he inequali y cons ained op imiza ion, we use he popula me hod de eloped by Powell [21, 22], which does no equi e any de i a i es o he objec i e unc- ion o be a ailable. Du ing his op imiza ion s ep 8, he nex poin x o e alua e in he sequen ial loop o 465 SPO is de e mined. Fo expensi e, global, black-box op imiza ion Jones [14] in oduced e icien global op i- miza ion (EGO). EGO exploi s he in o ma ion gi en om a K iging model, i.e., he p edic ed mean and a i- ance, o compu e he expec ed imp o emen (EI) o a470 gi en solu ion. EI can hence be used as a cos unc ion du ing s ep 8, as an al e na i e o he p edic ed alue o he K iging model. In s ep 11, he well known downhill simplex algo- i hm in oduced by Nelde and Mead [20] is used o im-475 p o e he bes ound esul s by a local op imiza ion p o- cedu e. We choose he downhill simplex implemen a ion in he nlop R package. Du ing local e inemen , con- s ain s a e dis ega ded because hey no longe play a ole in he egion o good solu ions.480 5 Expe imen al Se up In ou case, he pa ame e op imiza ion was based on Equa ion (1). The easible ange was 0 ≤k ≤300. Tha is, solu ions ha esul in o nega i e k alues o k alues la ge han 300 a e conside ed o be in easible. The usual wo king poin o ou es da a was in he ollowing ange: 0≤ϕ≤0.5, 0 ≤˙ϕ≤600, 500 ≤ϑ≤600. Wi h ha said, he op imiza ion p oblem o be sol ed in his s udy is de ined as ollows: –Pa ame e s o be changed a e he low cu e pa am- e e ec o mand he consolida ed pa ame e Ao 485 he low cu e. –The de ia ion o simula ed oll o ce om he mea- su ed oll o ce is minimized. –Compu a ional cons ain s: The e alua ions o he objec i e unc ion is expensi e. (see Sec ion 4.2)490 The pa ame e s which ep esen he sea ch space and we e subjec o op imiza ion in his s udy a e summa- ized in Table 2. To e alua e he success o he op imiza ion, he e- sul ing pa ame e se is compa ed o a well-known, es-495 ablished pa ame e se used in p ac ice so a . This old pa ame e se has been de e mined by expe s ac- co ding o bes knowledge om li e a u e on simila ma e ials. We ha e chosen he SPO oolbox (SPOT) o con-500 duc he expe imen s [5]. SPOT i sel has pa ame e s as well ha a e se acco ding o he au ho s expe ience: –The chosen su oga e model is K iging, based on code by Fo es e e al. [7]. 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Meachanics Resea ch Communica ions (2008) Kon ak /Imp essum Diese Ve ö en lichungen e scheinen im Rahmen de Sch i en eihe "CIplus". Alle Ve ö - en lichungen diese Reihe können un e h ps://cos.bibl. h-koeln.de/home abge u en we den. Die Ve an wo ung ü den Inhal diese Ve ö en lichung lieg beim Au o . Da um de Ve ö en lichung: 14.11.2017 He ausgebe / Edi o ship P o . D . Thomas Ba z-Beiels ein, P o . D . Wol gang Konen, P o . D . Bo is Naujoks, P o . D . Ho s S enzel Ins i u e o Compu e Science, Facul y o Compu e Science and Enginee ing Science, TH Köln, S einmülle allee 1, 51643 Gumme sbach u l: www.ciplus- esea ch.de Sch i lei ung und Ansp echpa ne / Con ac edi o ’s office P o . D . Thomas Ba z-Beiels ein, Ins i u e o Compu e Science, Facul y o Compu e Science and Enginee ing Science, TH Köln, S einmülle allee 1, 51643 Gumme sbach phone: +49 2261 8196 6391 u l: h p://www.spo se en.de eMail: homas.ba z-beiels ein@ h-koeln.de ISSN (online) 2194-2870 This wo k has been pa ially suppo ed by he MIWF NRW unde g an ag eemen FH-STRUKTUR 2014/10 (ISAFAN).