Sch i en eihe CIplus, Band 2/2014
He ausgebe : T. Ba z-Beiels ein, W. Konen, H. S enzel, B. Naujoks
Mul i- ideli y Modeling and
Op imiza ion o Biogas Plan s
Ma in Zae e e , Daniel Gaida, Thomas Ba z-Beiels ein
Mul i- ideli y Modeling and Op imiza ion
o Biogas Plan s
Ma in Zae e e ∗, Daniel Gaida, Thomas Ba z-Beiels ein
Facul y o Compu e and Enginee ing Sciences, Cologne Uni e si y o Applied Sciences, S einm¨ulle allee 1, 51643 Gumme sbach, Ge many
Abs ac
An essen ial ask o ope a ion and planning o biogas plan s is he op imiza ion o subs a e eed mix u es. Op imizing
he mone a y gain equi es he de e mina ion o he exac amoun s o maize, manu e, g ass silage, and o he subs a es.
Accu a e simula ion models a e manda o y o his op imiza ion, because he unde lying chemical p ocesses a e e y
slow. The simula ion models hemsel es may be ime-consuming o e alua e, hence we show how o use su oga e-
model-based app oaches o op imize biogas plan s e icien ly. In de ail, a K iging su oga e is employed. To imp o e
model quali y o his su oga e, we in eg a e cheaply a ailable da a in o he op imiza ion p ocess. Doing so, mul i-
ideli y modeling me hods like Co-K iging a e employed. Fu he mo e, a wo-laye ed modeling app oach is employed
o a oid de e io a ion o model quali y due o discon inui ies in he sea ch space. A he same ime, he cheaply
a ailable da a is shown o be e y use ul o ini ializa ion o he employed op imiza ion algo i hms.
O e all, we show how biogas plan s can be e icien ly modeled using da a-d i en me hods, a oiding discon inui ies
as well as including cheaply a ailable da a. The applica ion o he de i ed su oga e models o an op imiza ion p ocess
is shown o be e y di icul , ye success ul o a lowe p oblem dimension.
Keywo ds: Biogas Plan , Simula ion, Op imiza ion, Su oga e Models, Mul i- ideli y, Co-K iging
1. In oduc ion
Op imizing he ope a ion o biogas plan s is and will
be one o he main challenges in he ield o anae obic
diges ion (AD) in he nea u u e. Due o a s eady de-
c ease in unding and inc easing subs a e cos s only op-
imal ope a ing biogas plan s will be economically ad-
an ageous.
The ope a ion o biogas plan s is e y sensi i e o he
mix u e o he used subs a es. Hence, op imizing he
mix u e is an impo an ask o un o plan such plan s
e icien ly. Due o he e y slow p ocesses in ol ed,
op imizing he plan s in eal- ime would consume oo
much ime. Models like he Anae obic Diges ion Model
No. 1 (ADM1) allow o compu e a good p edic ion o a
biogas plan ’s p ocess a iables, based on he used sub-
s a es [4]. Thus, ADM1 can be used as a subs i u e in
he op imiza ion p ocess ins ead o a eal plan .
∗Co esponding au ho . Phone: +49 2261 8196 6327
Email add esses: [email p o ec ed] (Ma in
Zae e e ), [email p o ec ed] (Daniel Gaida),
[email p o ec ed] (Thomas
Ba z-Beiels ein)
While such models a e much cheape o e alua e han
hei eal-wo ld coun e -pa , hey do ake some ime o
e alua e. Hence, me hods ha use he smalles amoun
o e alua ions possible a e o in e es . This si ua ion
mo i a ed he cen al ques ion ha will be ackled in his
s udy:
(Q-1) How can he p ecision o simula ion models be
imp o ed wi hou inc easing he numbe o e alu-
a ions?
Su oga e modeling echniques a e he e o e a
p omising choice. Besides he expensi e in o ma ion
de i ed om ADM1, addi ional pe o mance in o ma-
ion is a ailable. A ough pe o mance es ima e can be
de e mined based on he biogas po en ial o he used
subs a es and hei associa ed cos s. This addi ional
knowledge can be in eg a ed in o he op imiza ion p o-
cess, by bols e ing he quali y o he chosen su oga e-
modeling echnique. This app oach o in eg a ing di -
e en le els o g anula i y o cos has p e iously been
called mul i- ideli y op imiza ion [14]. I is wo h in-
es iga ing whe he hese app oaches a e applicable o
eal-wo ld se ings. This can be o mula ed as he sec-
ond ques ion o be analyzed in his s udy:
P ep in submi ed o Applied So Compu ing Augus 26, 2014
(Q-2) Wha a e he bene i s and limi a ions o mul i-
ideli y modeling app oaches?
In his pape , se e al mul i- ideli y modeling ap-
p oaches a e compa ed, and he bes a e es ed o hei
pe o mance in an op imiza ion p ocess. Sec ion 2 gi es
an o e iew o ele an p e ious wo k. The speci ic
p oblem o be sol ed is in oduced in Sec ion 3. In
Sec ion 4, me hods ha we e used in his s udy a e de-
sc ibed. Sec ion 5 p esen s expe imen s, in which a i-
ous mul i- ideli y app oaches a e es ed o hei model-
ing quali y, whe eas Sec ion 6 es s he bes o hese o
hei success in sol ing he ac ual op imiza ion p oblem.
A concluding summa y o indings as well as an ou look
on u u e esea ch is gi en in Sec ion 7.
2. Fo me Resea ch
2.1. Biogas Plan Simula ion
Islam e al. [19] analyze he impac o di e en ac-
o s on p oduc ion o biogas in di e en biogas plan s
o Bangladesh. The da a was collec ed om 18 poul y
a ms. Thei analysis is based on collec ed da a om
su ey, In e ne , and o he sou ces. To ob ain u he in-
sigh in he beha io o biogas plan s, simula ion models
such as he ADM1 can be used. ADM1 is e y popula
and he nowadays mos complex ma hema ical model
used o simula e he anae obic diges ion p ocess ( o a
e iew see [3]). In se e al publica ions i is u ilized o
dynamically model ull-scale ag icul u al and indus ial
biogas plan s [5, 23, 29]. ADM1 is a s uc u ed model
inco po a ing disin eg a ion and hyd olysis, acidogen-
esis, ace ogenesis, and me hanogenenesis s eps. The
ADM1 is implemen ed as a s i di e en ial equa ion
sys em in a MATLABR
oolbox o biogas plan model-
ing, op imiza ion and con ol published by Gaida e al.
[16]. In his oolbox, a model o a ull-scale ag icul u al
biogas plan is de eloped ha is used in he empi ical
pa o his publica ion. The simula ion model o he
biogas plan includes he ADM1 and u he mo e mod-
els o elec ical and he mal ene gy sinks and sou ces
as well as models o pe o mance and s abili y c i e ia.
Typical c i e ia include cos s. bene i (wi h espec o
he Renewable Ene gy Sou ces Ac (EEG 2009) in Ge -
many [6]), s abili y o subs a e deg ada ion p ocesses
and ope a ing cons ain s such as uppe and lowe pH
limi s, maximum VFA/TA [33] alue, maximum o al
solids con en in he diges e , and minimum me hane
concen a ion o he biogas.
2.2. Biogas Subs a e Feed Op imiza ion
Biogas plan subs a e eed mix u es ha e p e iously
been op imized wi h a Gene ic Algo i hm and Pa icle
Swa m Op imiza ion by Wol e al. [35]. Mo e ecen ly
Ziegenhi e al. [39] used s a e o he a e olu ion
s a egies like Co a iance Ma ix Adap ion E olu ion
S a egy (CMAES) [18, 17] o Di e en ial E olu ion
(DE) [34] o educe he numbe o needed simula ions.
They also used he Sequen ial Pa ame e Op imiza ion
Toolbox (SPOT) [2] o une he employed algo i hms.
In ou wo k, we di ec ly use SPOT on he subs a e eed
op imiza ion p oblem. Tha is, we suppo he op imiza-
ion p ocedu e wi h su oga e-models.
Bo h p e ious s udies used a biogas plan model
based on he MATLABR
SimulinkR
Toolbox SIMBA,
de eloped by i ak sys em GmbH1. The he ein p esen ed
esea ch on he o he hand is based on he MATLABR
Toolbox o Biogas Plan Simula ion [16]. In con as
o ea lie wo ks by Wol e al. [35] and Ziegenhi e
al. [39] ou app oach is no limi ed o he ADM1. A
simple es ima e o a subs a e mix u es quali y is de-
i ed om he biogas po en ial o each ing edien .
2.3. Su oga e Modeling in Op imiza ion
Especially when he e alua ion o a ge unc ions is
expensi e, i is a well es ablished app oach o exploi
su oga e models o he a ge unc ion o sa e expen-
si e unc ion e alua ions.
A me hodical amewo k o su oga e model based
op imiza ion o noisy and de e minis ic p oblems is Se-
quen ial Pa ame e Op imiza ion (SPO) in oduced by
Ba z-Beiels ein e al. [2]. SPO has been de eloped o
sol ing expensi e algo i hm uning p oblems bu can
be di ec ly employed o sol ing eal wo ld enginee ing
p oblems as well.
One o he mos o en used su oga e-models is K ig-
ing, which is an especially p omising model o con inu-
ous, smoo h p oblem landscapes. Besides i s p edic ion
pe o mance, i is o en employed because i p o ides
an es ima o o he local ce ain y o he model, which
can be used o calcula e he Expec ed Imp o emen (EI)
o a new sample o e he bes known sample. Jones
e al. [22] in oduced his concep o balance exploi a-
ion and explo a ion in expensi e op imiza ion, e ming
i E icien Global Op imiza ion (EGO).
O he models include A i icial Neu al Ne wo ks
(ANN) o Suppo Vec o Reg ession (SVR) [11]. Non-
con inuous p oblem landscapes, o p oblems which a e
no ha expensi e, may be ackled wi h app oaches like
Random Fo es (RF) [8] o Mul i a ia e Adap i e Re-
g ession Splines (MARS) [15].
1www.i ak-sys em.com
2
A comp ehensi e o e iew o su oga e model as-
sis ed op imiza ion was p o ided by Jin [20], ocusing
on single objec i e p oblems.
Ex ensions o he abo e concep s o mul i-objec i e
p oblems a e a ailable (e.g., mul i objec i e EGO [25,
30, 12] and SPO [37, 38]). Since mul i-objec i e p ob-
lems a e no in he ocus o his pape , we e e o he
o e iew by Knowles and Nakayama [26] o u he
in o ma ion.
2.4. Mul i- ideli y Op imiza ion
Mul i- ideli y op imiza ion [14] deals wi h p oblems
whe e he a ge unc ion can be e alua ed a di e en
le els o ideli y. Tha is, he ac ual a ge unc ion ep-
esen s he highes le el o ideli y, yielding he mos
accu a e bu also mos expensi e i ness es ima e. A
he same ime, one o se e al cheape , less accu a e es i-
ma es can ep esen he lowe ideli y le els. The ac ual,
expensi e a ge unc ion will be e e ed o as he ine
unc ion, whe eas he cheape and less accu a e unc ion
will be e e ed o as coa se unc ion, espec i ely.
Such si ua ions o en a ise, especially in enginee -
ing p oblems. The e, he e alua ion o he ac ual p ob-
lem may be an expensi e eal-wo ld e alua ion mea-
su emen , o a ime consuming Compu a ional Fluid
Dynamics (CFD) simula ion. In hese cases, a simpli-
ied physics-based model may yield an inexpensi e bu
less accu a e quali y es ima e. Fo some models, i-
deli y may e en be scalable. Fo ins ance, simpli ied
meshes wi h less densi y can be employed wi h CFD,
o i a ailable p e-con e ged simula ion esul s may be
ha nessed. To exploi in o ma ion om di e en i-
deli y le els in model-based op imiza ion, se e al me h-
ods exis , including Co-K iging. Fo es e e al. [14]
show how his can be applied o enginee ing p oblems.
Co-K iging exploi s co ela ion be ween coa se and ine
unc ion o gene a e a be e su oga e model o he ine
unc ion.
Se e al mo e simple su oga e modeling app oaches,
e.g. whe e he su oga e ies o model he e o o he
coa se unc ion, hus co ec ing i , a e possible, some o
which a e used in his wo k. The de ails o he applied
me hods will be desc ibed in Sec. 4.
3. P oblem Desc ip ion
In his pape we deal wi h a p oblem whe e wo i-
deli y le els a e a ailable. The op imiza ion objec i e
as well as i s wo ideli y le els a e desc ibed in his
sec ion.
3.1. The Objec i e
The objec i e o he op imiza ion is o maximize he
mone a y gain (in Eu os pe day) o a biogas plan . The
decision space is spanned by he amoun o each sub-
s a e in he mix u e which is ed in o he plan . The
objec i e is composed o se e al ela ed con ibu ions:
• e enue om selling elec ical ene gy p oduced in
combined hea and powe plan s
• e enue om selling he mal ene gy p oduced in
combined hea and powe plan s
•cos o ene gy used in plan ope a ion, e.g., s i ing
he diges e con en , subs a e anspo a ion, hea -
ing he diges e s, e c.
•cos o subs a es.
All hese a e mo e o less dependen on he speci ic
subs a e mix u e, and o cou se dependen on plan -
speci ic pa ame e s which can be assumed o be con-
s an , e.g., size o e men e s o ou side empe a u e.
The maximal possible gain is limi ed by he maximum
load o he combined hea and powe plan , as well as
he limi s o he anae obic diges ion p ocess i sel .
3.2. The Fine Func ion
He e, he ine objec i e unc ion is he comple e bio-
gas plan simula ion, based on he ADM1. The mod-
eled biogas plan con ains wo diges e s and p oduces
an elec ical powe o 500 kW. As men ioned abo e,
he used implemen a ion is he MATLABR
oolbox de-
eloped by Gaida e al. [16].
This oolbox is able o yield in o ma ion abou all el-
e an p ocess a iables, as well as calcula es he mon-
e a y gain o a gi en se ing. Depending on he exac
se up, he model will ake a leas 30 seconds, wi h an
a e age o abou 1 minu e o compu e he daily mone-
a y gain o a ce ain subs a e mix u e in equilib ium
s a e. Simula ions may ail, o a e s opped i hey do
no yield a esul a e 10 minu es. These cases ha e o
be deal wi h du ing op imiza ion, as discussed in Sec-
ion 4.3.
The olume o each a ailable subs a e in he in- eed
mix u e is a ied du ing he op imiza ion p ocess. Tha
means, he dimension o he decision space depends
on he numbe o a ailable subs a e ypes. The e m
”a ailable” can e e o physical a ailabili y o a sub-
s a e a he plan , o he a ailabili y o calib a ion da a
o ha subs a e. Only subs a es wi h known pa ame-
e s can be ep esen ed by he ADM1.
In his s udy, wo cases will be es ed.
3
•Two-dimensional case: I is assumed ha only he
subs a es maize and pig manu e a e a ailable. The
esul ing op imiza ion p oblem is ha o inding
he bes mix u e o bo h. The low dimensionali y
allows o isual analysis hus p o iding an in u-
i i e unde s anding o he p oblem.
•Fi e-dimensional case: He e, h ee addi ional
subs a es a e a ailable, namely cow manu e, g ass
and co n-cob-mix. This is a ealis ic scena io o
many plan s.
The exac limi s o he op imized pa ame e s a e sum-
ma ized in Table 1.
Table 1: Lowe and uppe bounda ies o he op imized pa ame e s,
ha is he amoun o subs a es in he mix u e.
Subs a es Lowe [m3
d] Uppe [m3
d]
Maize 5.00 40.00
Pig Manu e 5.00 60.00
G ass Silage 0.00 20.00
Co n-Cob-Mix 0.00 10.00
Cow Manu e 0.00 10.00
3.3. The Coa se Func ion
The coa se, mo e simple objec i e unc ion is mos ly
based on he biome hane po en ial o each subs a e,
hence called biome hane po en ial (BMP) model. The
BMP can be calcula ed o each subs a e using he
Buswell equa ion [9].
Thus, he BMP model es ima es ha he amoun o
p oduced gas ises linea ly wi h he amoun o each sub-
s a e eed in o he plan . The es ima e o p oduced en-
e gy is limi ed by he maximum load o he block hea
and powe plan . One e alua ion akes wo hund ed mi-
c oseconds o less. Hence, en housands o coa se unc-
ion e alua ions could be made du ing one e alua ion o
he ine objec i e unc ion.
The BMP model is able o yield basic in o ma ion
like amoun o me hane gas p oduced o daily mone a y
gain. Howe e , i can no yield he comple e se o p o-
cess a iables ha a e a ailable wi h he ADM1 and is
less accu a e.
3.4. Ad an ages o he Coa se Func ion
In he case o his applica ion, he op imiza ion p o-
cess can p o i in wo di e en ways om he da a a ail-
able in o m o he coa se unc ion. Fi s , he low i-
deli y models op imum can be used o enhance he ini-
ial expe imen al design c ea ed by SPOT, o used as
a s a ing guess o non-se -based app oaches. Sec-
ond, he su oga e model o he global landscape can
be enhanced by he low- ideli y model, e.g., using Co-
K iging o simila me hods.
4. Me hods
4.1. Modeling
4.1.1. Co-K iging
K iging is a me hod o in e pola ion and eg ession
based on Gaussian p ocess modeling. The ollowing no-
a ion is adop ed om Fo es e e al. [13]. Gi en a se
o nsolu ions X={x(i)}i=1...nin a k-dimensional con-
inuous sea ch space wi h obse a ions y={y(i)}i=1...n,
K iging is a me hod o ind an exp ession o a p e-
dic ed alue a an unknown poin by in e p e ing he
obse ed esponses yas i hey a e ealiza ions o a
s ochas ic p ocess. The ollowing se o andom ec-
o s Y={Y(x(i))}i=1...nis used o de ine his s ochas ic
p ocess. The co ela ion o he andom a iables Y(·) is
modeled as ollows [13]:
co hY(x(i)),Y(x(l))i=exp −
k
X
j=1
θj|x(i)
j−x(l)
j|pj.(1)
The ma ix ha collec s co ela ions o all pai s {(i,l)}is
called he co ela ion ma ix Ψ. I is used in he K iging
p edic o
ˆy(x)=ˆµ+ψTΨ−1(y−1ˆµ),(2)
whe e ˆy(x) is he p edic ed unc ion alue o a new sam-
ple x, ˆµis he maximum likelihood es ima e o he mean
and ψis he ec o o co ela ions be ween aining sam-
ples Xand he new sample x. The wid h pa ame e
θ=θ1, . . . , θj, . . . , θkTde e mines how a he in lu-
ence o each sample poin xsp eads. The pa ame e pj
is usually ixed a pj=2, and de ines he shape o he
co ela ion unc ion.
As an ex ension o K iging, Co-K iging may in-
clude in o ma ion o a coa se unc ion in o he model.
To ha end, Co-K iging exploi s co ela ion be ween
he di e en ideli y le els. Acco ding o Fo es e e
al. [14], Co-K iging can be unde s ood o eg ess he
coa se unc ion while coinciding wi h he ine unc-
ion. We now ha e wo ec o s wi h n samples om
he ine unc ion and ncsamples om he coa se unc-
ion, i.e., X ={x(j)
}j=1...n and Xc={x(i)
c}i=1...ncin a k-
dimensional con inuous sea ch space wi h obse a ions
yc ={y(i)
c,y(j)
}i=1...nc,j=1...n . Acco dingly, he s ochas ic
p ocess can now be de ined by he se o andom ec-
o s Yc ={Yc(x(i)
c),Y (x(j)
)}i=1...nc,j=1...n . Then, we ge
4
he co a iance ma ix C
C= σ2
cΨc(Xc,Xc)ρσ2
cΨc(Xc,X )
ρσ2
cΨc(X ,Xc)ρ2σ2
cΨc(X ,X )+σ2
dΨd(X ,X )!
(3)
whe e we ha e he same co ela ion unc ion, bu wi h
wo se s o model pa ame e s o Ψdand Ψc espec-
i ely. An addi ional pa ame e ρis in oduced as a
cons an scaling ac o . While Ψcdoes ep esen he
co ela ion s uc u e in he coa se unc ion, Ψdcap u es
he di e ence be ween he Gaussian p ocess ep esen -
ing he cheap unc ion (scaled by ρ) and he unknown
Gaussian p ocess ep esen ing he ine unc ion. The
Co-K iging p edic o o he ine unc ion is
ˆy (x)=ˆµ+cTC−1(y−1ˆµ),(4)
whe e he cis he ec o o co a iances be ween he
known solu ions ( ine and coa se) Xc =X
Xcand he
solu ion o be p edic ed xand 1deno es a ec o o ones.
We e e o Fo es e e al. [14] o u he in o ma ion.
The model we use in his wo k is a e-implemen a ion
in R based on MATLAB R
code o Fo es e e al. [13].
Two expe imen al designs a e e alua ed o Co-
K iging, one is a la ge expe imen al design which co -
e s he design space o he coa se unc ion. Hence, all
poin s in his design a e e alua ed wi h he coa se unc-
ion. The second design is he smalle se o poin s
e alua ed on he ine unc ion. I is nes ed in he
la ge design, ha means, each poin e alua ed on he
ine unc ion is also e alua ed on he coa se unc ion.
Bo h designs should op imize some c i e ion o space-
illingness. In his wo k, we c ea e designs ha maxi-
mize he minimum dis ance be ween he samples.
4.1.2. Al e na i e mul i- ideli y models
Se e al simpli ied al e na i es o Co-K iging can be
used o in eg a e in o ma ion om bo h coa se and ine
unc ion in o he modeling p ocess. The ollowing
me hods a e all compa ed o Co-K iging in a p elimi-
na y in es iga ion o model quali y.
Di . Model A e y in ui i e idea is o assume ha he
coa se unc ion is able o model he gene al s uc-
u e co ec ly. The emaining e o can hen sim-
ply be co ec ed by modeling he di e ence be-
ween coa se and ine unc ion ( c, ). The esid-
uals o he coa se unc ion a e used as aining da a
o a da a-d i en model. Tha is, he su oga e
model c
Mis build wi h design Xand obse a ions
(X)=y , c(X)=ycand
c
Mdi =gX,y −yc.(5)
A new p edic ion is hen always based on he esul
o he model, as well as he esul o he coa se
unc ion:
ˆy =ˆ
(x)=ˆ
di (x)+ c(x),(6)
whe e ˆy is he p edic ed ine unc ion alue and
ˆ
di ep esen s he p edic ion o he su oga e
model c
Mdi . This model will be e e ed o as
he Di e ence Model (Di . Model).
Ra io Model The Ra io Model wo ks in a e y simila
way. Ins ead o di e ences, i.e., esiduals, he a io
be ween ine and coa se unc ion is modeled.
c
M a io =g X,y
yc!(7)
The p edic ion o he Ra io Model c
M a io is gi en
as
ˆy =ˆ
(x)=ˆ
a io(x) c(x) (8)
Inpu Model The inpu model akes a sligh ly di e en
app oach. The esponse o he coa se a ge unc-
ion is used as an addi ional inpu pa ame e o he
model, e.g., K iging.
c
Minpu =g({X,yc},y ),(9)
wi h p edic ion:
ˆy =ˆ
(x)=ˆ
inpu ({x, c(x)}).(10)
These h ee simple app oaches can all be applied o
a bi a y models, e.g., Neu al Ne wo ks o Suppo Vec-
o Machines. They equi e he coa se unc ion du ing
p edic ion. O cou se, i he coa se unc ion i sel is
somewha cos ly, al hough cheape han he ine unc-
ion, i can again be eplaced by a sepa a e su oga e
model.
4.1.3. Two-laye Modeling
One p oblem in su oga e modeling o biogas plan s
is ha he modeled landscape is no con inuous, as illus-
a ed in Fig. 1. In his example, he ac ual gain unc ion
has a sal us a x=30%. In ac , he op imum is o en
in he icini y o a sal us in decision space, which can
also be seen in Fig. 2. This beha io is caused by he
so called manu e bonus, which is a ixed bonus paid o
biogas p oduce s. This bonus is paid, i mo e han 30%
o he subs a e con ains speci ic manu es [6].
Models like K iging a e bes sui ed o con inuous
landscapes. To some ex end, hey a e able o deal wi h
5
Manu e
[%]
Gain
[€/d]
ac ual gain
heo . gain wi h manu e bonus
heo . gain wi hou manu e bonus
20 30 40
800
700
600
Figu e 1: Illus a ion o he wo di e en modeling laye s in he Two-
laye su oga e model. He e, only he pe cen age o manu e o a ixed
amoun o o he subs a es is assumed o a y. The discon inui y in he
cu e a ises a exac ly 30 pe cen manu e.
discon inui ies, a leas globally. S ill, a K iging model
will always de e io a e in egions close o he discon i-
nui ies. This is especially p oblema ic due o he ac
ha he op imum may o en be close o he 30 pe cen
bonus limi . The model quali y would he e o e be de-
e io a ed in he icini y o he op imum. To a oid his
p oblem, wo app oaches a e eligible o his applica-
ion.
1. While he modeled landscape is ha o he ac ual
mone a y gain, he simula ion does p o ide addi-
ional in o ma ion. This could be exploi ed by
modeling he exac amoun o p oduced gas (e.g.,
wi h Co-K iging), and calcula ing he mone a y
gain on- he- ly du ing p edic ion. The amoun o
p oduced gas would be con inuous o e he whole
design space, hus yielding a easonable su oga e
model. The d awback is, ha a leas wo models
would need o be ained: he i s , which models
he amoun o p oduced me hane gas, and he sec-
ond, which models u he esul s om he ADM1
simula ion which a ec he gain o he plan . Also,
he on- he- ly calcula ion o he mone a y gain
would be added on op o he e o o he p edic-
ion du ing he su oga e-op imiza ion p ocess.
2. The al e na i e is, o c ea e wo K iging models
o he mone a y gain. One ep esen s he mone-
a y gain wi hou manu e bonus, one wi h he ma-
nu e bonus. Du ing su oga e-op imiza ion he op-
imize will swi ch be ween he wo models, de-
pending on whe he he 30 pe cen bonus limi is
eached. The wo laye s a e illus a ed in Fig. 1.
The o me app oach would be mo e ime consuming,
since i needs o calcula e he mone a y gain o each
p edic ed sample. Also, each model would ha e o p e-
dic each sample, because bo h alues a e needed o he
mone a y gain calcula ion. The la e app oach would
only equi e o ake he 30 pe cen bonus limi in o ac-
coun , o swi ch be ween models, wi hou any u he
calcula ions. The d awback would be he loss o in o -
ma ion, since he o me app oach is able o gi e an es-
ima e o he p oduced amoun o gas o he in e es ed
use . In his s udy, i was decided o ake he less in o -
ma i e bu mo e e icien app oach wo. We e e o his
as he Two-laye app oach, due o he wo di e en mod-
els o he mone a y gain. The di e ence in model qual-
i y will be in es iga ed in a p elimina y s udy in Sec. 5.
Please no e, ha he ea lie desc ibed bump in he de-
cision space is no a cons an o se . The manu e bonus
which causes his discon inui y a ec s he e enue om
sold gas, hus ha ing a mul iplica i e in luence on a sin-
gle pa o he objec i e alue calcula ion. The impac
o gas p icing on he o e all gain a ies signi ican ly in
he gi en sea ch space.
4.2. E o Measu e
Two e o measu es will be used in he model qual-
i y expe imen s. The Mean Squa ed E o (MSE) o
he ec o o nobse a ions y=(y1,...,yi,...,yn)T
and he ec o o co esponding p edic ions ˆ
y=
(ˆy1,...,ˆyi,...,ˆyn)T
MSE(y,ˆ
y)=1
n
n
X
i=1
(yi−ˆyi)2(11)
The second e o measu e is he Scaled MSE (SMSE)
as in oduced by Keijze [24]. Keijze [24] de ines he
SMSE as ollows:
SMSE(y,ˆ
y)=MSE(y,1a+bˆ
y)=
(11)
=1
n
n
X
i=1
(yi−(a+bˆyi))2
whe e b=co (y,ˆ
y)
a (ˆ
y)and a=¯
ˆ
y−b¯
y
(12)
He e, ¯
ˆ
yand ¯
yindica e he espec i e mean alues. The
SMSE can be unde s ood o e alua e di e ences be-
ween wo ec o s a e scaling hem o a common
ange. Tha allows o igno e e o s ha a e i ele an
o he op imiza ion p ocedu e. A simple example would
be a p edic ion ˆ
y ha di e s om he obse a ions y
only by a cons an o se . While his p edic ion would
ecei e a compa a i ely la ge MSE alue, he loca ion
o he op imum would be pe ec ly accu a e. SMSE is
6
conside ed as he adequa e e o measu e in his case.
Sec ion 5 will u he mo i a e his choice o he biogas
applica ion, showing p elimina y esul s whe e SMSE
is clea ly he mo e easonable indica o .
4.3. Handling E alua ion Failu es
Simula ions o he expensi e, ine a ge unc ion
may ail and lead o un easonable o missing esul s.
Such poin s can no simply be igno ed. Igno ing hem
would lead o a si ua ion we e he op imiza ion p o-
cess would epea edly sugges an ins able con igu a ion.
This would possibly p e en he op imiza ion p og ess.
Ins ead, only ailed simula ions in he ini ial design a e
emo ed. Failed simula ions in he op imiza ion p o-
cess i sel a e eplaced by he impu a ion me hod sug-
ges ed by Fo es e e al. [13]. Tha means, ins ead o
he un easonable esul he p edic ed mean is impu ed in
such cases, penalized by adding he p edic ed a iance
o he K iging (o Co-K iging) model. Na u ally, such
a me hod is only possible whe e a a iance es ima e o
he model is a ailable. All o he models, o non-model-
suppo ed app oaches will use ixed, la ge penal y al-
ues.
4.4. Op imiza ion Algo i hms
Th ee di e en op imiza ion app oaches a e em-
ployed in his s udy, ei he o op imize he ine/coa se
unc ion di ec ly, o o op imize he co esponding su -
oga e models.
•Downhill Simplex Me hod (Simplex) is a clas-
sic, de i a i e ee, local op imiza ion me hod
de eloped by Nelde and Mead [28]. Fo he
expe imen s in his pape a bound cons ained
Simplex [7] implemen a ion om he NLop li-
b a y [21] is used, in e aced by he R-package
nlop [36].
•Di e en ial E olu ion (DE) [31, 32] is a s a e-o -
he-a de i a i e ee op imiza ion me hod based
on he p inciples o e olu ion. Due o being
popula ion-based and due o i s s ochas ic na u e i
has he capabili y (al hough no gua an ee) o lea e
local op ima. The R-implemen a ion in he pack-
age DEop im [1, 27] was used in he expe imen s.
•La in Hype cube Sampling (LHS) is a e y sim-
ple op imiza ion s a egy, whe e a space illing
La in Hype cube design o expe imen is e alua ed
in he decision space o he op imiza ion p oblem.
The bes ound solu ion in his design is he es i-
ma ed op imum.
5. P elimina y S udy: Model Quali y
A i s s udy was pe o med o analyze how well he
di e en su oga e-models ep esen he p oblem land-
scape, i.e., es ing o modeling e o . To ge a simple
and unde s andable example, we es ic his P e-S udy
o he wo dimensional case. Only he subs a es pig
manu e and maize a e assumed o be a ailable.
Th ee ques ions a e o in e es :
1. Wha e o measu e should be used?
2. Does he Two-laye modeling app oach imp o e
model quali y?
3. Which (mul i- ideli y-)modeling me hod, e.g..,
K iging, Co-K iging, di o a io model, wo ks
bes ?
Please no e, ha app oxima ely i e pe cen o an-
domly chosen subs a e-mix u es may yield simula ion
ailu es due o nume ical ins abili y. In his p elimina y
s udy, such poin s a e igno ed, i.e., manually emo ed.
A mo e complica ed impu a ion as desc ibed in Sec. 4.3
is only employed du ing he la e op imiza ion expe i-
men s.
5.1. Expe imen al Se up
We pe o m ou se s o expe imen s, whe e each ep-
esen s a di e en size o he expe imen al design. In
each se , wo design ypes a e c ea ed and e alua ed.
Type one is a smalle La in Hype cube Design (LHD)
(size 5, 10, 15 and 20 poin s) e alua ed wi h he ine
unc ion . Type wo is a e y la ge LHD (always 100
poin s) e alua ed wi h he coa se unc ion c. The de-
i ed in o ma ion is used o build su oga e models o
he ine unc ion, i.e., ˆ
.
To e alua e pe o mance, we will look a he ea lie
in oduced e o measu es (MSE, SMSE). The consid-
e ed su oga e models a e s anda d K iging, Co-K iging
and a Neu al Ne wo k app oach (Quan ile Reg ession
Neu al Ne wo k QRNN [10]). QRNN and K iging a e
also es ed wi h he ea lie in oduced simple mul i-
ideli y app oaches, ha is, Inpu , Ra io and Di e ence
modeling (see Sec ion 4.1.2). QRNN is in oduced as an
al e na i e app oach o de e mine whe he ce ain obse -
a ions a e ac ually linked o he Mul i- ideli y model-
ing app oach, o a he o he employed model ype.
I has o be no ed ha Co-K iging is he mos ime-
consuming me hod. Since he model building akes less
han a second in any case, his is no signi ican in com-
pa ison o he cos o e alua ion . Howe e , in he
la e op imiza ion expe imen s un ime dese es mo e
7
−1000
−500
0
500
1000
5 10 15 20 25 30 35 40
10
20
30
40
50
60
BMP
maize [ m3d ]
pig−manu e [ m3d ]
0
500
1000
1500
5 10 15 20 25 30 35 40
10
20
30
40
50
60
ADM1
maize [ m3d ]
pig−manu e [ m3d ]
Figu e 2: Con ou plo s o ADM1 and BMP model, based on 400
samples om a LHD. The wo con ou plo s a e bo h in e pola ed wi h
Two-laye K iging, using da a om 400 samples gene a ed wi h LHS.
The depic ed alues a e mone a y daily gain in Eu o, hus la ge al-
ues a e be e .
a en ion, since he ime-consump ion will sum up o e
all sequen ial op imiza ion s eps. Highe sea ch space
dimension will also play an impo an ole.
Fo each combina ion o e o measu e, design size
and chosen su oga e model, 20 epea s a e pe o med.
The modeling e o is es ima ed based on da a om a
la ge La in Hype cube Design, consis ing o 400 de-
sign poin s. This da a se is no a ailable du ing model
aining.
5.2. Resul s and Discussion
As a i s esul , Fig. 2 shows illed con ou plo s ep-
esen ing he ADM1 ( ) and BMP ( c) a ge unc ions,
espec i ely. He e, bo h a e in e pola ed wi h Two-laye
K iging, using da a om 400 samples c ea ed wi h LHS.
MSE: K iging
MSE: Coa se Func ion
SMSE: K iging
SMSE: Coa se Func ion
0.1 0.2 0.3 0.4
Figu e 3: Depic ed a e SMSE and MSE o a K iging model o he
ine unc ion as well as he BMP model (coa se unc ion) e alua ed
by wo di e en e o measu es. The K iging model was build based
on a LHD o size 5. The p ocess o c ea ing he LHD design was
epea ed 20 imes. Smalle alues a e be e .
Nai e K iging
Two−laye K iging
Coa se Func ion
0.02 0.04 0.06 0.08 0.10
Design size: 5
SMSE
● ●●
●
Nai e K iging
Two−laye K iging
Coa se Func ion
0.00 0.02 0.04 0.06 0.08 0.10
Design size: 10
SMSE
●● ●
●
Nai e K iging
Two−laye K iging
Coa se Func ion
0.00 0.01 0.02 0.03 0.04 0.05
Design size: 15
SMSE
● ● ●
Nai e K iging
Two−laye K iging
Coa se Func ion
0.005 0.010 0.015 0.020
Design size: 20
Figu e 4: Again, all plo s a e based on LHD designs, epea ed 20
imes. Design sizes a e in each heade . The wo-laye app oach mod-
els alues wi h and wi hou manu e bonus sepa a ely. Smalle alues
a e be e .
In he gi en egion o in e es , bo h show simila beha -
io . While he BMP has only a sligh ly di e en shape,
a s ong o se can be obse ed. S ill, he op ima o bo h
unc ions a e no o a om each o he . They a e also
close o he discon inui y, hence he need o he Two-
laye app oach.
The di e en scale indica es ha SMSE should be
p e e ed o e MSE. An unscaled e o measu e would
no be a ai compa ison, as he op imum o he coa se
unc ion is e y close o he one o he ine one. Fig. 3
shows how his choice a ec s he es ima ion o quali y,
compa ing MSE and SMSE o he coa se unc ion o
a K iging model o he ine unc ion. SMSE is be e
sui ed o e alua e he use ulness o he model o op i-
miza ion pu poses. As shown in Fig. 2, bad MSE alues
a e caused by he sal us, al hough he loca ion o he op-
imum is e y well app oxima ed.
Fig. 4 shows how SMSE esul s a y depending on
whe he o no he Two-laye app oach is used. As ex-
pec ed, he model p o i s om using he Two-laye ap-
p oach, as i a oids he discon inui y in oduced by he
8
