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].
Pa ame e : seq.p edic ionModel. unc.505
–The ini ial design consis s o 40 candida e solu ions,
which a e c ea ed by La in Hype cube Sampling
(LHS).
Pa ame e : ini .design.size.
Table 2 Uppe and lowe bounds o he pa ame e se M
in oduced in sec ion 2.2 which was used du ing he op imiza-
ion. All pa ame e s a e o ype FLOAT.
Fac o Low High
A0 2
m1-0.01 0
m2-0.3 0.4
m30 0.2
m4-0.1 0.1
14 Ch is ian Jung e al.
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