Modelling and eal- ime op imisa ion o an
indus ial cooling-wa e ne wo k
Ma ´
ıa P. Ma cos
Sys ems Enginee ing and Au oma ic Con ol Depa men
Uni e sidad de Valladolid
Valladolid, Spain
[email p o ec ed]a.es
C´
esa de P ada
Ins i u e o Sus ainable P ocesses (IPS)
Sys ems Enginee ing and Au oma ic Con ol Depa men
Uni e sidad de Valladolid
Valladolid, Spain
[email p o ec ed]a.es
Jos´
e Luis Pi a ch
Sys ems Enginee ing and Au oma ic Con ol Depa men
Uni e sidad de Valladolid
Valladolid, Spain
[email p o ec ed]a.es
Ch is ian Jasch
Lenzing Ak iengesellscha
We ks aße 2, 4860
Lenzing, Aus ia
[email p o ec ed]
Abs ac —This wo k deals wi h he p oblem o dis ibu ion o
cooling wa e in an e apo a ion p ocess. The aim is o de elop a
Real-Time Op imisa ion (RTO) ool which imp o es he esou ce
e ficiency by supplying he op imal wa e dis ibu ion wi hin a
su ace-condense s ne wo k o a gi en p oduc ion demand. The
app oach includes expe imen al models and he au oma ic upda e
o ouling ac o . The p oblem is o mula ed and sol ed ia non-
linea p og amming. P oduc ion cons ain s and conce ns abou
he p ac ical implemen a ion a e also aken in o accoun in he
design o he RTO ool.
Index Te ms—modelling, op imisa ion, e apo a ion plan , op-
imal dis ibu ion, expe imen al models, RTO
I. INTRODUCTION
In he p ocess indus y, he e is an inc easing consensus on
he impo ance o how o manu ac u e he p oduc s in he bes
possible way. To do his, we mus ake in o accoun he eal-
ime p oduc ion si ua ion and he global ene gy and esou ce
e ficiency. Fu he mo e, we mus add ha he egula ion o
en i onmen al ma e s is inc easingly es ic i e. As a esul , i
an indus y wan s o keep being compe i i e in a global ma ke ,
i will ha e o pe o m op imisa ion a di e en le els: con ol
laye , Real-Time Op imisa ion (RTO), p oduc ion scheduling
and economic planning [1]. Imp o emen s on all his le els can
lead o huge sa ings in consump ion o ene gy and esou ces,
and consequen ly o he educ ion o p oduc ion cos s [2].
In o de o do ha , i is necessa y o p o ide compu e -
based ools which acili a e he decision-making p ocess o he
ope a o and plan manage s. These ools a e no mally model-
based so ha an impo an e o in adap ing heo e ical models
o he eal sys em [3] has o be done. In pa icula , RTO ools
will need eal- ime inpu s, so hey mus be in eg a ed wi h he
in o ma ion echnology (IT) in as uc u e o he plan s, e.g.,
This esea ch is unded by he Eu opean Union’s Ho izon 2020 esea ch
and inno a ion p og amme unde g an ag eemen No. 723575, and by he
Spanish Go e nmen (MINECO/FEDER DPI2015-70975-P).
ia a neu al deploymen pla o m ha connec s o di e en
IT sys ems [4].
This wo k deals wi h RTO in he cooling sys em o he
e apo a ion ne wo k in Lenzing A.G., one o he wo ld leading
ac o ies o human-made iscose fib e p oduc ion, which is
loca ed in Aus ia. Hence, in his pape he app oach and a
p o o ypical ool o he op imisa ion o he cooling wa e
dis ibu ion in he cooling sys em is desc ibed , wi h he goal
o minimizing he ade-o de e mined by he cos o he s eam
and he cooling wa e consump ion. The op imisa ion has been
p og ammed in CasADi [5] using MATLAB, and hen linked
o he PI Sys em in he plan .
The pape o ganizes as ollow. Nex sec ion b iefly desc ibes
he indus ial p ocess and he ne wo k which is going o be
op imised. Sec ion 3 shows he models ha ha e been ob ained
om expe imen al da a, and he conside a ions aken in o ac-
coun o de elop hose models. In Sec ion 4 he ma hema ical
o mula ion o he op imisa ion p oblem is exposed, i.e he
cooling wa e dis ibu ion and he conce ns abou he p ac ical
implemen a ion. Some esul s and a summa y o he wo k done
a e gi en in Sec ion 5. Finally, in Sec ion 6 he u u e wo k
is shown.
II. DESCRIPTION OF THE PROCESS
As ea lie was poin ed ou , his wo k is done in collab-
o a ion wi h Lenzing AG, a ac o y which p oduces iscose
fib es based on a enewable esou ce: wood. Once he wood
is sh edded, he cellulose pulp ha con ained he wood is
chemically ea ed and i becomes a iscose solu ion. The
key s age o p oduc ion is he spinning, i.e he con e sion
o his solu ion in o fib es by passing i h ough fine diame e
sie es unde p essu e, and in oducing i in o an acid ba h
(called spinba h he eina e ). In addi ion o he new solid
fib es, sodium sulpha e (Na2SO4) and wa e a e also p oduced
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as by-p oduc s. This causes he deg ada ion o he spinba h,
and, consequen ly, he quali y o he fib es dec eases.
The e o e, i is necessa y o egene a e he acidi y o
he spinba h h ough he con inuous ex ac ion o wa e and
Na2SO4. In o de o do ha , an e apo a ion ne wo k and a
c ys allisa ion sec ion, a ached o he p incipal p ocess a e
used.
The e apo a ion ne wo k is composed o fi een e apo a ion
plan s wi h di e en nominal capaci ies. This ne wo k should
be able o egene a e he acid ba hs ha come om spinning,
aking in o accoun ha he plan p oduces di e en fib es.
The e ficiency o each plan depends on di e en ac o s:
he e apo a ion load, he ope a ion condi ions (i.e spinba h
empe a u e and flow), he pe o mance o he cooling sys em
and he ouling s a e.
Fo a u he explana ion o he e apo a ion ne wo k and he
cha ac e is ics o each plan and i s con ol sys em, he eade
is e e ed o [6], [7].
Fig. 1. Simplified scheme o an e apo a ion plan
A. Cooling sys em ne wo k
The e apo a ion plan s could ha e wo di e en cooling
sys ems: one is based on Su ace Condense s (SC, Fig. 1)
and he o he one on cooling owe s (ou o he scope o
his wo k). The e a e fi een e apo a ion plan s which use
SC as cooling sys em. This E=15e apo a o s a e g ouped
in wo sub-ne wo ks depending on hei cooling wa e sou ce
(see Fig.2). The aim o his SC is o condense he s eam
apou ha comes om he e apo a ion plan s, o be used
la e in o he pa s o he ac o y.
Hence, he cooling wa e dis ibu ion p oblem is, he mo e
cooling wa e is p o ided o he SC, he mo e e ficien he
e apo a ion plan becomes, because less specific s eam apou
is needed. Ne e heless, he o al a ailable cooling wa e om
sou ces is limi ed and sha ed wi h o he depa men s o he
ac o y, he e o e i s use implies a cos .
Fig. 2. Cooling sys em ne wo k
III. MODELLING
Se e al expe imen s ha e been done in o de o s udy
he beha iou o SC in s eady s a e1in di e en ope a ing
condi ions. These expe imen s consis o unning he SCs in
di e en condi ions co e ing hei usual ange o ope a ion
and collec da a o he inle and ou le empe a u es o he
cooling wa e . I has o be aken in o accoun ha o all he
expe imen s, he e apo a ion capaci y (EC), i.e he flow o
e apo a ed wa e om he spinba h, is kep cons an . F om
collec ed da a, di e en s a ic models ha e been buil o
ep esen he cooling sys em o each plan .
A. Ou le empe a u e
Fi s , models p o iding he ou le empe a u e o cooling
wa e Tou wi h espec o he cooling wa e flow Fha e been
ob ained. I has o be emphasised ha he inle cooling wa e
empe a u e is assumed cons an in all he expe imen s. As
a esul , he ela ion be ween ou le empe a u e and cooling
flow can be fi ed o a polynomial cu e, di e en o each
e apo a o . (See an example in Fig. 3.)
Fig. 3. Expe imen al model empe a u e s. flow
1Dynamics o p essu e d op and hea ans e in he SCs a e neglec ed
because o hem being qui e as in compa ison o he ime he plan is going
o ope a e in he op imal s eady s a e.
Ne e heless, hese models depend on he inle wa e em-
pe a u e, so ha we p opose emo ing he e e ence inle wa e
empe a u e Tin eco ded a he ime he es s we e ca ied ou .
Hence, we ge an inc emen al model (1), so ha , gi en a eal-
ime measu emen o he inle empe a u e, deno ed by ˆ
Tin,
he ou le empe a u e can be compu ed by (2).
ΔT= (F)−Tin (1)
¯
Tou =ΔT+ˆ
Tin (2)
Whe e (·)is a non-linea unc ion ha ep esen s he expe -
imen al model, e.g. he polynomial cu e depic ed in Fig. 3.
In addi ion, he SC su e s om ypical ouling e ec s as
in o he simila indus ies, p o oking ha he hea ans e
dec eases wi h ime and, consequen ly, he ou le empe a u e
o he cooling wa e will also dec ease. Thus, assuming he
es s we e ca ied ou wi h he SC ully clean he eal ime
ou le empe a u e will be lowe han he p edic ed as i is
shown in Fig. 4.
Fig. 4. Model adap a ion o cu en ouling s a e
To o e come his issue, he idea is o add a bias pa ame e
K o he abo e base model which adjus s he cu e o he
eal- ime measu emen :
Tou =ΔT+ˆ
Tin −K (3)
In his way, he cu en s a e o ouling in he SC is aken in o
accoun o u he op imisa ion. The bias K can be easily
upda ed wi h eal- ime measu emen s o he ou le empe a u e
(ˆ
Tou ) by:
K =¯
Tou −ˆ
Tou (4)
No e ha his app oach allows o isola e he ouling e ec s
in he SC sys em om he ones in he spinba h hea ing
line (Fig. 1), which also a ec s he o e all specific s eam
consump ion. Fouling in he hea ing line is ou o his wo k,
o de ails on how o deal wi h i see [8].
B. Specific S eam Consump ion
On he o he hand, om he collec ed da a o inle and
ou le empe a u e and olume ic flow o he cooling wa e ,
he ac ual cooling capaci y in he SC can be compu ed by:
Cpow =4.18
3600F(Tou −Tin)(5)
Mo eo e , by eco ding he li e s eam consump ion o he
e apo a ion plan in he es s, we can depic he specific s eam
consump ion (SSC) e sus he a ailable cooling powe in
he SC sys em and fi a model o i . Howe e , analogous
o he empe a u e model, o emo e he dependency on he
ope a ing poin (load) om he base model is needed. To do
so, he simples idea is o compu e he bes specific s eam
consump ion (BSSC), i.e he minimum SSC ob ained in he
expe imen s, and o build an inc emen al model:
ΔSSC =g(Cpow)−BSSC (6)
Whe e g(·)is ano he non-linea unc ion ha ep esen s he
expe imen al model, e.g. he polynomial cu e depic ed in
Fig. 5.
Fig. 5. Specific s eam consump ion s. cooling powe
Fo ha o be ue, wo assump ions a e made:
•The es s we e ca ied ou wi h clean SC.
•The model o ΔSSC (i.e., he shape o he cu e in
Fig. 5 o ins ance) does no a y significan ly om one
ope a ion poin o ano he (plan e apo a ion loads).
C. Modelling me hodology
The polynomial unc ions (·)and g(·)ha e been ob ained
fi ing he cu es o he expe imen al da a by leas squa es
me hod independen ly o each e apo a o .
Based on he da a, polynomials wi h deg ades no g ea e
han h ee a e enough o ge a good fi . These models ep esen
sa is ac o ily he sys em in he ope a ion ange and a e sui able
o he op imisa ion. Ne e heless, his me hodology does no
ake in o accoun possible ou -laye poin s in he expe imen al
da a due o dis u bances o noise. Consequen ly, hese fi s
migh no be he bes o ep esen he eal beha iou o he
sys em. Thus, a mo e sophis ica ed modelling ou ine o ob ain
hese cu es, as i is p oposed in [9], could be aken in o
accoun .
Once bo h models, (·)and g(·), ha e been fixed by
iden ifica ion, hey can be used o p edic ion, ecei ing he
wa e flows Fand he inle empe a u es ˆ
Tin as inpu s and
p o iding he wa e ou le empe a u es Tou and he inc ease
o specific s eam consump ion ΔSSC. No e ha al hough he
measu ed ou le empe a u e ˆ
Tou is also used o upda e he
pa ame e K , i is no conside ed as a model inpu in he
p edic ion s a e.
IV. NETWORK OPTIMISATION
As men ioned abo e, he e apo a ion ne wo k ha uses SC
as cooling sys em can be g ouped in wo sub-ne wo ks: he
fi s one (SN1) is composed o ESN1=4plan s and he second
one (SN2) includes he o he s ESN2=11. I should be aken
in o accoun ha all he SCs a e connec ed in pa allel and
ha SN2 can ecei e wa e om SN1 bu no he o he way
a ound.
Fig. 6. Simplified scheme o he cooling sys em ne wo k
The op imal dis ibu ion s ongly depends on he e ficiency
o each plan , i.e he nominal specific-s eam consump ion SSC,
de e mined by he assigned e apo a ion capaci y EC and he
ouling s a e K .
Thus, he p oblem objec i e is minimizing he ade-o
be ween he cos o li e s eam and wa e usage, which is
gi en by he absolu e s eam consump ion (ASC) imes i s p ice
(Ps eam) plus he wa e flow (F) imes i s p ice (Pwa e ).
The Ps eam is gi en by he ene gy depa men , meanwhile,
he Pwa e is calcula ed by nego ia ion be ween all in ol ed
depa men s.
A. Ma hema ical o mula ion
Fo he wo se s o SC e apo a o s, he p oblem cons ain s
a e as ollow:
•The o al flow in each subne (SN1, SN2) has o be lowe
han he maximum limi (FS1,FS2 espec i ely).
•Exceeding wa e can go om SN1 o SN2 bu no
backwa ds.
•Uppe and lowe flow limi s defined o each SC (Fe,Fe),
i.e sui able ope a ion ange in o de o a oid p oblems o
en ainmen 2in o he SC.
•The ou le wa e empe a u e pe plan has o be lowe
han he maximum allowed (Tmax). This cons ain is
because he ou le cooling wa e goes o he i e and
i has o ulfil he cu en en i onmen al cons ain s.
2En ainmen he e is unde s ood as he p esence o he acid ba h in he SC
pipes. This e ec is pa icula ly ha m ul o he equipmen , so i needs o be
a oided.
Hence, he op imisa ion p oblem is:
min
Fe,FN12∈RE+1 J=
E

e=1
(ASCe·Ps eam +Fe·Pwa e )(7a)
s. .:
ESN1

e=1
Fe+FN12 ≤FS1 (7b)
ESN2

e=1
Fe−FN12 ≤FS2 (7c)
FN12 ≥0(7d)
Fe≤Fe≤Fe∀e∈E (7e)
Tou e≤Tmax ∀e∈E (7 )
Taking in o accoun ha FN12 s a es o he wa e om SN1
o SN2 and ASCecan be ob ained as o he specific s eam
consump ion and he assigned load capaci y o each plan , as
shown in:
ASCe=ΔSSCe·ECe.(8)
Whe e Tou has been calcula ed by he expe imen al model o
each plan as s a ed in sec ion III, o mula (3), and ΔSSC
can be ob ained combining (5) and (6) as ollows:
ΔSSCe=g4.18
3600Fe(Tou e−Tine)−BSSCe(9)
Finally, as he expe imen al models a e C1non-linea unc-
ions, usually quad a ic polynomials, he op imisa ion p ob-
lem (7) is easily handled ia non-linea p og amming (NLP).
B. Implemen a ion
Once he p oblem is o mula ed, we coded i in CasADi-
Ma lab using an in e io poin op imise [10]. Ne e heless,
some conside a ions ha e o be aken in o accoun .
Fi s , as he expe imen al models ha e been buil o an
specific flow ange, i he eal- ime cooling wa e flows ˆ
Fe
a e ou o ange, he op imisa ion should no be execu ed, as
he es ima ion o he ouling pa ame e K in (4) may be w ong
due o plan -model misma ch. The e o e, a wa ning message
should appea o in o m o such si ua ion.
Howe e , i his happens because he plan s a e in
main enance, i.e hey a e no p ocessing any spinba h, he
flow o his e apo a o mus no be op imised, bu he
es o he ne wo k mus . To do ha , i he load o an
e apo a o (EC) is less han 1, he flow o ha condense
is se o he one ha is measu ed a ha momen , i.e we
p opose eplacing cons ain (7e) by he ollowing exp essions:
Fe≤Fe≤Fe∀e∈{e|ECe>1}(10)
Fe=ˆ
Fe∀e/∈{e|ECe>1}(11)
Finally, i may be he case ha , due o he s a e o ouling
and/o he wa e inle empe a u e o he condense s, he
op imum cooling capaci y is ou o he ange whe e he expe -
imen al models (6) we e buil . In his case, he op imisa ion
should no un and a wa ning should be displayed, in o de o
in o m he ope a o s o his si ua ion.
V. RESULTS AND DISCUSSION
The op imisa ion o mula ed in he sec ion abo e has been
es ed o fline wi h eal sample da a, eco ded in a pa icula
ime ins an . The alues o he pa ame e s3used o sol e he
p oblem a e:
•Maximum allowed empe a u e, Tmax =31
◦C
•Cooling wa e a ailable om sou ce 1, FS1 = 864 m3/h
•Cooling wa e a ailable om sou ce 2, FS2 = 943 m3/h
In Figs. 7-9 he ob ained esul s a e shown along wi h he
eal- ime measu ed da a o he ime ha he pa ame e s we e
eco ded.
Fig. 7. Cooling wa e dis ibu ion
Compa ing he op imised flows wi h he measu ed ones
(Fig.7), i is obse ed ha mos e apo a o s mus inc ease
hei cooling wa e consump ion. No e ha , his can look
inconsis en a p io i as he cooling wa e has a cos . Howe e ,
as shown in he Fig.8, he consump ion o s eam has dec eased
because i s p ice is 10 imes highe han he wa e one, so when
adding bo h cos s in he objec i e unc ion he esul is ha
benefi s ha e been ob ained.
Fig. 8. Inc emen al Specific S eam Consump ion
I should be no ed ha he ΔSSC o e apo a o s 18 and 33
is ze o. This is because he expe imen al models o ob ain he
SSC a e no consis en , so we assumed ha he measu ed SSC
3The alues o he Tin, EC, Fe,Fe,Ps eam and Pwa e a e omi ed due o
confiden iali y easons wi h Lenzing AG.
is he bes SSC ha hese e apo a o s can achie e. Doing ha
he flow o hese SCs will fi wi hin he ope a ion flow ange,
wi h he unique cons ain o complying wi h he maximum
ou le wa e empe a u e. In con as , o e apo a o s 35 and
36, he e is a eal specific s eam consump ion bu he op i-
misa ion adjus s he flow in o de o make ha op imal SSC
ma ches wi h he BSSC in bo h e apo a o s.
Analysing Fig.9 we can obse e ha he eal- ime measu e-
men o he ou le empe a u e goes beyond he maximum al-
lowed in se e al cases. In con as o ha , he op imal solu ion
chooses he flows in o de o fi he ou le empe a u e below
he maximum, hus ulfilling he en i onmen al es ic ions.
Fig. 9. Ou le cooling wa e empe a u e
Fu he mo e, flow om he subne 1 o subne 2, FN12,
is 19 m3/h, so we can sum up ha he o al cooling wa e
a ailable in subne 2 is no enough o ope a e he SCs in hei
op imal poin .
Table I shows he cos s due o he cooling sys em be o e
and a e he op imisa ion, as well as he alue o he sa ings.
No e ha hese alues only ep esen a snapsho in a pa icula
ime ins an , bu he annual po en ial benefi ha would be
ob ained by applying his op imisa ion ool in daily ope a ion
looks p omising. The e o e, he ool is now deployed on si e,
cu en ly unde es ing pe iod.
TABLE I
RESULTS OF COSTS AND SAVINGS
Cos be o e op imisa ion 71.39e/h 625376.40e/yea
Cos a e op imisa ion 19.88e/h 174148.80e/yea
Sa ings 51.51e/h 451227.60e/yea
Summa y and conclusions
In his wo k we add essed a p oblem on esou ce e ficiency
in a cooling sys em o a eal indus ial e apo a ion ne wo k.
The modelling, op imisa ion and isualiza ion concep s p e-
sen ed in his pape suppo he ope a o s when i comes o
aking be e decisions in eal ime o imp o e he ne wo k
ope a ion. The models ob ained allow o execu es an au oma ic
upda e o he SCs ouling s a e based on eal- ime measu e-
men s. Ne e heless, he models de eloped a e e y sensi i e

o dis u bances and noise e o o he da a used o ob ained
he polynomial unc ions, so a modelling ou ine o calcula e
hese cu es is ecommended.
The esul ing models a e inco po a ed in he RTO scheme
ha sol es an NLP p oblem acco ding o he cu en p oduc-
ion cons ain s and he plan s ouling s a es. Ne e heless, he
ins an aneous RTO does no ake in o accoun any p edic ion
o he ouling e ec , so he p oposed con ol ac ions may be
subop imal in he long e m. In despi e o ha , significan
s eam consump ion sa ings ha e been ob ained in he es s.
The de eloped RTO ools a e cu en ly unde e alua ion
a Lenzing AG: he implemen a ion in exis ing sys ems and
ope a ional policies is pe o med s ep by s ep o ge expe ience
in li e es ing and o ensu e accep abili y om he plan
pe sonnel. Abou a yea o no mal ope a ion is equi ed o
assess he impac , bu p elimina y es s wi h his o ical da a
p edic sa ings a ound 400000 e/yea .
Howe e , he op imisa ion canno un au oma ically in he
cu en o m due o exis en ou -o - ange si ua ions. In ad-
di ion, he a e some e apo a o s which miss consis en da a
se s. Fo his eason u he es s a e planned on si e in o de
o imp o e he SCs models, he e o e he ool eliabili y.
VI. FURTHER STEPS
The comple e e apo a ion sys em in Lenzing A.G. can be
mainly desc ibed by wo ne wo ks o equipmen . The fi s
one conce ns he e apo a ion plan s whe e decisions on load
alloca ion need o be aken [8]. The second one e e s o
he cooling sys ems a ached o each plan , eed by a wa e
dis ibu ion ne wo k, whe e decisions on wa e flows o plan s
need o be aken, i.e he p oblem add essed in his wo k.
Cu en ly, independen op imisa ion se ups a e al eady de-
eloped o each ne wo k. Howe e , bo h sys ems a e coupled
by he spinba h loads ECeand he ou le wa e empe a u es
Tou ,e in he cooling sys ems, so ha he decisions on he
load alloca ion influence he cooling-wa e dis ibu ion and
ice e sa (see he o mula ion o he op imisa ion in [8] and
compa e wi h he one in Sec ion IV-A).
Fig. 10. Rela ion be ween ne wo ks
I bo h p oblems a e sol ed independen ly, by ea ing hese
sha ed a iables as “a p io i” fixed da a o each p oblem, he
o e all op imisa ion becomes an i e a i e p ocedu e wi h no
global op imali y gua an ees. On he con a y, i bo h o mu-
la ions a e me ged in o a cen alized p oblem, he op imisa ion
becomes an MINLP one, as he wa e dis ibu ion is an NLP
p oblem and he load alloca ion in ol es disc e e decisions
(alloca ion o plan s o p oduc s). The e o e, he complexi y
o he hypo he ical cen alized p oblem is much highe han
he one o he wo sepa a e p oblems abo e, guessed o be
unsui able o eal- ime applica ion.
To o e come he abo e issues, we p opose as u u e wo k,
o add ess he p oblem in a dis ibu ed ashion ia Lag angean
decomposi ion [11] and p ice-coo dina ion schemes [12], i.e.,
adding he sha ed cons ain s ( a iables in his case) as a
penal y in he objec i e unc ion Jo each indi idual p oblem.
The modified objec i es o each p oblem will be in he o m:
JM=J+
e
pe(Re−ˆ
Re)(12)
Whe e Rea e he sha ed a iables ha should be equal
( o ˆ
Re) in bo h p oblems and pea e he associa ed Lag ange
mul iplie , also e e ed as “ esou ce p ices” in he li e a u e.
Then, a sma ule o upda ing ˆ
Reand pein each i e a ion is
equi ed. Resea ch on his will be conduc ed in o de o speed
up he esolu ion.
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