Multi-fidelity Modeling and Optimization of Biogas Plants

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, . . . , θkTde 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 Xcand 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 =gX,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