Developing Grey-Box Dynamic Process Models

DEVELOPING GREY-BOX DYNAMIC PROCESS MODELS C. de P ada *, D. Hose**, G. Gu ie ez*, J.L. Pi a ch*  * Depa men o Sys ems Enginee ing and Au oma ic Con ol, Uni e si y o Valladolid c/ Real de Bu gos s/n. Sede Me gelina EII, 47011, Valladolid, Spain (e-mails: p ada@au om.u a.es , glo[email p o ec ed].es , jose.pi a ch@au om.u a.es) ** Ins i u e o Enginee ing and Compu a ional Mechanic, Uni e si ä S u ga P a enwald ing 9, 70569 S u ga , Ge many (e-mail: dominik.hose@i m.uni-s u ga .de) Abs ac : This pape p esen s a me hodology o de eloping g ey models o p ocess sys ems, ha is, models ha , being based on undamen al p inciples and laws o na u e, combine hem wi h sub-models ob ained om expe imen al da a. The me hod ollows wo s eps: he i s one akes ad an age o wha is known, while he second uses he da a and mixed-in ege op imiza ion algo i hms o iden i y he s uc u e and pa ame e s o he emaining pa s o he model. The me hod is illus a ed in a challenging bio echno- logical p ocess: he Ace one-Bu anol-E hanol (ABE) e men a ion p ocess. Keywo ds: Modelling me hodology, g ey-box models, s uc u e, iden i ica ion, e men a ion p ocess.  1. INTRODUCTION Qui e o en people say ha modelling is an a , which su ely is ue. Ne e heless, e e y a equi es an associa ed ech- nique and de eloping dynamic models o indus ial p ocesses is no an excep ion. T adi ionally, wo main app oaches ha e been used: Fi s -p inciples and Da a-based me hodologies. The o me ies o de elop models o mula ing equa ions acco ding o physical laws ha a e pe inen o he p ocess conside ed. Typical o mula ions in he p ocess indus y use mass and ene gy balances, phase equilib ium, e c. This e- qui es knowledge o he p ocess and he conce ned laws and good judgemen in es ablishing hypo hesis ha suppo he alidi y o he model. Deciding which phenomena and equa- ions should be inco po a ed in o he model is no easy, as hey ep esen he comp omise be ween simplici y o use and ideli y in he ep esen a ion o eali y. The ime needed o de elop hese models should no be unde es ima ed, bu he use o a mode n simula ion en i onmen and a ce ain expe i- ence should acili a e he ask. As physic-chemical laws a e usually alid unde a wide ange o condi ions, one impo an ad an age when using hese models e e s o he con idence hey p o ide and he associa ed ex apola ion capabili ies. On he opposi e hand, da a based me hods y o disco e he model ha ela es se e al p ocess a iables by analyzing and co ela ing se s o expe imen al alues. Then, a model s uc- u e wi h adjus able pa ame e s is p oposed and he pa ame- e s can be es ima ed so ha he models ou pu adjus s as well as possible o he expe imen al da a. He e, a selec ion o he candida e model s uc u e and he pa ame e es ima ion algo- i hm a e he key componen s. These ypes o models a e, in p inciple, easie o o mula e and unde s and, and ha e he ad an age o being closely ela ed o eali y as hey a e ob- ained di ec ly om da a, hei main d awback being he limi ed ex apola ion capabili ies hey ha e ou side he egion co e ed by he expe imen al da a used in hei de elopmen . No ice ha , in p ac ice, he i s p inciple models always inco po a e unknown pa ame e s, hence, when a model o eali y is p oposed, de eloping i always in ol es a s age o da a collec ion and adjus men o he model pa ame e s so ha he model ou pu s i he expe imen al da a, as in (1): min , ()−(,,,)   s. .: 󰇗=  (,,),  (0)= =ℎ(,,) (1) whe e  is he ec o o known manipula ed a iables, x a e he s a e a iables wi h ini ial alue ,  a e he p ocess measu emen s o he ou pu a iables  o he model and  ep esen s he model pa ame e s. The dynamic model is de- sc ibed in e ms o he ec o unc ions (⋅),ℎ(⋅). In he same way, a p ope selec ion o he model s uc u e in da a based models implies ce ain knowledge o he in e ac- ions and phenomena aking place in he p ocess, so ha bo h me hodologies ha e some poin s in common. The main di - e ences a e ela ed o which elemen , knowledge o da a analysis, play he cen al ole. Choosing one o he o he app oach is dic a ed mainly by he inal aim and use. The model equi emen s a e di e en , o ins ance, o ope a o s’ aining han o con olle design. I he pu pose is using he model o ake decisions abou p o- cess ope a ion, hen p obably a dynamic op imiza ion p ob- lem simila o (2) is o be sol ed a egula ime in e als: min   () s. .: 󰇗=  (,,), (0)= =ℎ(,,), (,,)≤0 (2) whe e  is now he ec o o decision a iables,  a e he s a e a iables,  he model ou pu s and  ep esen s he model pa ame e s. The dynamic model is again desc ibed in e ms o he unc ions (⋅),ℎ(⋅), whe eas (⋅) deno es he p oblem speci ic cons ain s and () is he cos unc ion o be minimized. Sol ing (2) may in ol e a lo o compu a ion depending on he size and s uc u e o he model and he deg ees o eedom. This o mula ion assumes ha he s uc u e o he model is co ec and he pa ame e s a e being es ima ed in he igh way. No mally, a i s p inciples model is p e e able in deci- sion making, as i may p o ide mo e con idence and a wide ange o alidi y. Ne e heless, hese assump ions and expec a ions may ail due o: The di icul y o model ce ain ela ionships among a i- ables, due o unknown phenomena, complex ela ion- ships, impossibili y o measu e ce ain a iables equi ed o adjus he model pa ame e s, e c. The compu a ional load associa ed o e y de ailed mod- els ha makes sol ing (2) imp ac ical In hese cases, i may be sensible o combine wha is known wi h ce ain y abou he model, such as equa ions ep esen - ing mass o ene gy balances, wi h o he ypes o models ob ained om measu emen s, ep esen ing he mo e di icul o complex pa s o he p ocess. This esul s in a hyb id mod- el ha , being a mix u e o “whi e box” i s p inciples and “black box” sub-models is known as a “g ey box” one. The e a e many good e iews and publica ions on modelling, bo h co e ing i s p inciples (Cellie , 1991) and da a based ap- p oaches (Zou, Li and Zhang, 2017), bu he e is a lack o li e a u e o he sys ema ic de elopmen o g ey-box ones. The e o e, his pape p esen s a me hodology o de eloping g ey-box models, dealing in pa icula wi h he p oblem o s uc u e iden i ica ion o he black-box elemen s o he mod- els and discussing he so wa e ools a ailable. The me hod is illus a ed in a challenging applica ion: he Ace one-Bu anol- E hanol (ABE) e men a ion p ocess. The pape is o ganized as ollows: The modelling me hodol- ogy is explained in Sec ion 2 besides so wa e ools ha help in hese asks. The ABE p ocess and a p elimina y model a e desc ibed in Sec ion 3. Then, he g ey-box model o he ABE p ocess is de eloped in Sec ion 4 oge he wi h some alida ion esul s. Finally, he pape ends wi h some ema ks. 2. GREY BOX MODELLING When de eloping i s -p inciple models in he p ocess indus- y, one may ace si ua ions in which a pa ial se o equa ions (⋅),ℎ(⋅) is known, as in (2), bu a subse o a iables (,) is no (o oo complex o model). Howe e , hese ela ions be ween , and  a e equi ed o comple e he model. 󰇗=  (,,(,),), =ℎ(,,(,),) (3) 2.1 P oblem o mula ion Gi en he pa ial model (3), and assuming ha one expe i- men has been pe o med so ha alues o  and  a e a ailable om he p ocess ia any da a-collec ion sys em, he p oblem can be o mula ed as inding he sub-model (,) and pa ame e s  such ha he esponse o (3) i s he expe - imen al alues in he bes possible way. A ypical app oach o he p oblem is p oposing a unc ional s uc u e o (,), such as =(,,) and, as measu e- men s o  a ely a e accessible, adjus he pa ame e s  by sol ing he ollowing op imiza ion p oblem: min ,, ()−(,,,)   s. .: 󰇗=  (,,(,,),), (0)= =ℎ(,,(,,),) (4) As he ini ially p oposed s uc u e o (⋅) will likely no be co ec , he i is o be epea ed wi h a modi ied candida e s uc u e ollowing a ial and e o p ocedu e. This is a ime consuming p ocedu e wi h no gua an ee o success. 2.2 P oposed modelling me hodology Ins ead, he ollowing wo-s age app oach can be used. 1. Es ima ion. Va iables  a e conside ed as independen and included in he da a- i ing p oblem (5) as new deci- sion a iables wi h an app op ia e pa ame e iza ion : min ,, ()−(,,,)   s. .: 󰇗=(,,,), (0)= =ℎ(,,,), (,,,)≤0 (5) 2. Reg ession. Simula e he model 󰇗=(,,,) in (5) o gene a e alues o , using  and he es ima ed  as inpu s. Find co ela ions o  wi h any  and/o , and o mula e eg ession cons ain s (,). In his way, all alues a e consis en wi h he emaining model. Finally, he unc ional (,) is added o he model in o de o ge he inal exp ession (3). No e ha addi ional cons ain s (,,,)≤0 ha e been included in S age 1 o gua an ee ha he alues o  o e ime con o m o physically admissible ones, such as being posi- i e, la ge han o he a iables, e c. The pa ame e iza ion ,=1,…, can be simple (cons an alues o e a se o disc e e- ime in e als), o mo e complex ones based on colloca ion poin s, acco ding o he p oblem na u e and he expec ed ime e olu ion o hese a iables. Addi ionally, he conside a ion o  as independen a iables adds ex a deg ees o eedom ha acili a e he model i o he expe imen al alues. No e ha (5) could be o mula ed al e na i ely as a dynamic da a econcilia ion p oblem wi h obus es ima o s (Hube , 2014) ins ead o he quad a ic cos , i enough measu emen s we e a ailable o achie e enough edundancy. Sol ing (5) p o ides a se o poin s  cohe en wi h bo h he expe imen al da a and he model. The esolu ion can be done ei he ia sequen ial o simul aneous app oaches: Depending on he p oblem s uc u e, a combina ion o a dynamic simula- o and an op imiza ion algo i hm ( SQP like SNOPT o an e olu iona y one) can be a good choice, bu mode n op imi- za ion en i onmen s like CasADi (Ande sson e al., 2012) o Pyomo, (Ha e al., 2012) o e excellen ea u es, including au oma ic disc e iza ion by o hogonal colloca ion and au o- ma ic di e en ia ion, ha acili a e he use o e icien in e i- o poin codes such as IPOPT in a simul aneous app oach. The e a e di e en ways o app oach he p oblem in S age 2. Among hem, an op ion is o pos ula e a lexible gene al s uc u e wi h, o ins ance, a neu al ne wo k adjus ing i s pa ame e s la e on o i he ,  and  alues. Ne e heless, his app oach has some d awbacks: on he one hand, i does no ake ad an age o he pa ial knowledge ha one may ha e abou  and, on he o he hand, i does no gua an ee a easible ex apola ion when  akes alues ou side he es ima ed ange, unless ex a condi ions a e imposed. A good al e na i e is o use mixed-in ege op imiza ion and global me hods o selec among a combina ion o use -p o ided po en ial basis unc ions, hose ha p o ide he bes i aking in o accoun possible ex a cons ain s o gua an ee physical cohe ence. Algeb aic modelling en i onmen s such as ALAMO (Cozad, Sahinidis and Mille 2014; 2015) o e e y good suppo o he i ing ask using global MINLP sol e s like BARON and adap i e-sampling p ocedu es. In he nex sec ion, his me hodology is applied o a challenging bio- echnological p ocess. 3. THE ABE PROCESS The Ace one-Bu anol-E hanol (ABE) e men a ion p ocess has expe ienced a ise in popula i y due o i s possibili ies in he p oduc ion o bio-bu anol which is being used in a lo o p oduc s such as bio uels (Mayank e al. 2013). In ou case, he ABE ins alla ion is a ba ch p ocess (Hose e al. 2016), i.e. i is ca ied ou in a closed e men e wi hou any inpu o ou pu low, he cells consume he subs a e and gene a e he use ul p oduc s. Only he empe a u e and he pH a e con- olled a ound a ce ain alue. Howe e , o he o ms such as he con inuous e men a ion a e possible. The ac ual e men - e used in his s udy can be seen in Figu e 1. Figu e 1. ABE e men e wi h con ol equipmen . Ini ially, he subs a e, in his case glucose, is gi en in o he eac o along wi h he so-called inoculum, he mic obiologi- cal s a ing cul u e, in his case Clos idium ace obu ylicum. A e a sho lag phase in which he bac e ia adjus hem- sel es o he new en i onmen , he acid p oduc ion phase o acidogenesis s a s in which mainly ace a e, bu y a e and lac a e a e p oduced. This phase is ypically indica ed by a apid g ow h o cells un il he pH has d opped om abou 7 o 4.5. In a second s ep, he sol en ogenesis, he cell numbe s alls o e en dec eases and he p oduc ion o he sol en s s a s, ha is ace one, bu anol and e hanol, which is he main aim o he p ocess. The empe a u e is usually main ained cons an a i s op imum which lies be ween 30 and 40ºC h oughou he whole p ocess. Fo u he insigh in o he biochemis y o ABE e men a ion one can e e o Hube , Ande sch and Go schalk (1982). 3.1 P ocess model The impo an ea u es o be modelled ha e been iden i ied as: cell g ow h and dea h dynamics, subs a e u iliza ion o he acid, sol en p oduc ion and cell main enance, as well as inhibi ion mechanisms due o an excess o bo h subs a e and sol en s in he b o h. A mac oscopic model is employed o cap u e he quan i a i e p ocess dynamics wi h he pu pose o de e mining he bes ope a ing condi ions la e on. The use o mic oscopic models based on me abolic pa hways has no been conside ed as hey a e no adequa e o he inal use o he models in economic op imiza ion o he p ocess ope a- ion. The nomencla u e o he concen a ions (in g/L) is gi en by X: Biomass (C. ace obu ylicum), S: Subs a e (Glucose), Pa: Sol en (Ace one), Pb: Sol en (Bu anol) and Pe: Sol en (E hanol). Conside ing ha he olume is sensibly cons an , one possible model ha cap u es he ea u es explained abo e and should, he e o e, be able o ep oduce he gene al ajec- o ies o expe imen al da a is (6): XYPXYPXYP mXXYSXXX xeexbbxaa xs        , , , ,(6) Basically he model is composed o mass balances o cells, subs a e and p oduc s, ha we know mus be sa is ied. The accumula ion o cells pe ime uni is equal o he di e ence be ween in low and ou low o cells (which a e ze o as he p ocess ope a es in ba ch mode), he a e o g ow h and he a e o cell dea h (bo h assumed p opo ional o he numbe o cells). Simila a gumen s a e used o de i e he o he equa- ions. The model employs a g ow h e m µ, dea h and main enance coe icien s,  and , as well as he a es Yxs, Yxa, Yxb and Yxe indica ing how subs a e is con e ed in o cells, and p oduc s a e gene a ed as a esul o he cell ac i i- y. Ne e heless, i is also well known ha he g ow h e m µ is no cons an , bu depends on se e al ac o s and he ela- ion among hem, µ = (X, S, Pa, Pb, Pe), is no well known. Se e al models o µ ha e been p oposed in he li e a u e, some o which a e shown in Table 1 (Heijnen and Romain, 1995; Yang and Tsao, 1994). No e, ha he o iginal model by Monod is by a he mos popula due o i s simplici y and he ac ha i o en su ices o ep oduce he gene al beha io o cell g ow h acco ding o he Michaelis-Men en kine ics. The o he models a e o en ex ensions o he one by Monod. 3.2 Di ec pa ame e es ima ion Fo iden i ica ion and alida ion pu poses wo da a se s we e p o ided. The co esponding expe imen s we e ca ied ou in he ba ch e men e o Figu e 1 wi h glucose as he subs a e and he bac e ia C. ace obu ylicum as biomass. Measu e- men s o concen a ions o biomass, subs a e and he h ee p oduc s we e aken o e he ba ch cycle. Un o una ely, no in o ma ion abou he empe a u e o pH du ing he expe i- men s we e p o ided making i impossible o include hese a iables in he models. The adi ional app oach o pa ame e es ima ion chooses a model s uc u e o µ and hen sol es (4) o es ima e he model pa ame e s. In ou case, as an example, we ha e cho- sen he Hinshelwood model om Table 1, including an inhib- i o y e ec by bu anol, as in (7): = (1−) (7) The ini ial alue o he concen a ions is assumed o be known om measu emen s. The coe icien s  and , as well as a es Yxs, Yxa , Yxb and Yxe and pa ame e s , Kµ, Kb, a e unknown decision a iables o (4). As he concen a ions and pa ame e s should be posi i e, he cons ain s y = [X, S, Pa, Pb, Pe] ≥ 0, p = [  , , Yxs, Yxa , Yxb, Yxe , µ0, Kµ ,Kb] ≥ 0 a e imposed o he dynamic op imiza ion, as well as no maliza- ion ac o s in he objec i e unc ion. A sequen ial app oach, connec ing he dynamic simula ion o (6)-(7) wi h a nonlinea op imiza ion (NLP) algo i hm is o en used o his ype o pa ame e es ima ion (Boada e al., 2016). He e, he gene ic algo i hm p o ided in MATLAB was used o sol e he op imal i ing o he expe imen al da a. No e ha he p oblem is highly non-linea and non-con ex, which jus i ies he use o he e olu iona y algo i hm o a oid local minima in spi e o he highe compu a ion imes. The esul s compa ing he model esponse (line) and he expe i- men al da a (do s) can be seen in Figu e 2a. The cell dynam- ics a e app oxima ed easonably well and, al hough he sub- s a e u iliza ion yields some e o , his i ing could be con- side ed adequa e. No e ha he aining se on he le p e- sen s some w ong da a, because he concen a ions o bu anol and e hanol canno dec ease o e ime. Un o una ely, he model alida ion depic ed in Figu e 2b demons a es how di icul inding uni e sal models ha apply in he gene al case a e, as hey do no i a all in di e - en condi ions. a) Response agains he iden i ica ion da ase . b) Response agains he alida ion da ase . Figu e 2. Model esponse (line) and expe imen al da a (do s). The unde lying p oblem wi h his echnique is ha a model o he cellula g ow h µ has o be chosen in ad ance which may no apply in his pa icula case. Howe e , because all models p o ided in Table 1 a e heu is ic, hey canno gene - ally be conside ed applicable and ha e o be chosen ca e ully. 4 GREY MODEL OF THE ABE PROCESS Nex , he me hodology p esen ed in Sec ion 2 will be used o de i e a g ey model o he ABE p ocess. O he success ul applica ions a e epo ed in, o ins ance, Pi a ch e al. (2017). 4.1 Es ima ion wi h ee g ow h a e In he i s s age, he g ow h a e µ is pa ame e ized o e ime and i s alues a e conside ed as independen coe icien s o be es ima ed besides he o he model pa ame e s using a o mula ion simila o (5). De ining 1 disc e iza ion poin s , =1,…,, a common pa ame e iza ion is ()=,∈󰇟,󰇠 as in Figu e 3. Table 1. Se e al models o µ. In he model by Yang, Paa deno es ace a e, Pba deno es bu y a e and Pl deno es lac a e. Model  Monod     Teissie   1−    Haldane         Hinshelwood     1−     ∈(,,,,) Yang    󰇧1−       −      −      −        ⋅      −        ⋅      −  5.6− 1.6 󰇨 Figu e 3. A ze o-o de pa ame e iza ion o (). The es ima ion p oblem can hen be o mula ed as ollows:  0][ ,0],,,,[ ],,[,)( , , , , , :s. . )()()()(min 1 1 2 1 ,          xexbxaxs eba iiixee xbbxaa xs N i i N i imi T imi p , Y , Y, Yλ , m,Yp PPPSXy XYP XYPXYP mXXYSXXX y yW y y i          (8) Wi h Δ deno ing changes o e consecu i e ime ins an s. No e ha , in addi ion o he posi i e cons ain s on he con- cen a ions and pa ame e s, a Tikhono  egula iza ion e m has been added o he quad a ic objec i e, penalizing he changes o e ime o , so ha inc easing o dec easing he weigh ing ac o  we can a ou mo e smoo h e olu ions o gi e mo e eedom o i he expe imen al da a ym. In he objec i e unc ion W is a no maliza ion ma ix ha can also be used o balance he quali y o he adjus men among he componen s o he ec o y. No e also ha , in spi e o he la ge numbe o pa ame e s o be es ima ed, he s uc u e o (8) is now a mo e simple, and he p oblem can be ecas as a quad a ic p og amming one. Figu e 4. Compa ison o he model esponses and expe i- men al da a (do s) o he biomass, subs a e and p oduc s. Uppe le co ne in do s: es ima ed alues o µi -  . The solu ion o a ac o  = 1000 can be seen in Table 2 and Figu e 4, which also shows he ime e olu ion o he model ou pu besides he expe imen al da a o he biomass, sub- s a e and p oduc s. The i is as good as o be e han he one displayed in Figu e 2a. In addi ion, he igu e shows in do s he ime e olu ion o he es ima ed alues o he e m µ( ) -  , deno ed by x. Table 2: Values o he es ima ed model pa ame e s. Pa ame e Value Pa ame e Value 0.008 Y xa 2.625 0.106 Y xb 6.869 Y xs 19.224 Y xe 0.458 4.1 Es ima ion o he g ow h a e wi h ALAMO In o de o comple e he model, a unc ional ela ion = (,,,,) has o be ound. This can be done using he es ima ed alues o µ, µi, oge he wi h he ones o he o he a iables ob ained by simula ion. Fo ins ance, he ones de- pic ed in Figu e 4, which a e consis en wi h he whi e model. Fo his ask, ALAMO (Au oma ic Lea ning o Algeb aic MOdels) has p o en o be a use ul ool o ge algeb aic su o- ga e models om gi en da a se s, (Cozad, Sahinidis and Mille , 2014). Fo his pu pose, i p o ides some s anda d basis unc ions, such as monomials, loga i hms o exponen- ials, which can be combined. Ne e heless, in his con ex , i s s eng hs lie in he possibili y o including use de ined basis unc ions as well as cons ain s gua an eeing physical sense o he es ima ed unc ional, e en ou side he ange o he expe imen al da a. ALAMO au oma ically picks he mo e sui able basis unc ions h ough a mixed-in ege op imiza- ion, e ining he solu ion wi h adap i e sampling in he e- gions whe e he model p esen s la ge e o s. In o de o selec he basis unc ions, no ice ha we can d op he dependency on X, as he model (6) al eady includes he e m µX. A he same ime, i su ices o le µ depend on S and only one p oduc , e.g. Pb, as he o he ones a e p opo - ional o i . Then, in addi ion o he s anda d basis unc ions o ALAMO, wo use -de ined ones a e included in he lis : bbs KP S KS S /1 , /1  (9) The op imiza ion p oblem o be sol ed is gi en by (10), whe e he combina ion o basis unc ion i ha bes i s he alues µi is selec ed by he bina y a iables wj.  s ands o an allowed ange o he  pa ame e s and l is he maximum numbe o basis unc ions allowed in he solu ion. ]1,0[ ,...1 :s. . ),(min 1 1 2 1 ,             j M j j j up jj low N i M j biijji w wlw Mjww PS jj    (10) The p oblem is sol ed using he BARON code and he solu- ion ob ained is gi en below. 042.14/1 0095.0460 300 7.3 b b P S SP S S      (11) Top le pic u e in Figu e 5a depic s he esul s o he i compa ing he alues o µi wi h he app oxima ion (11). This exp ession is hen inco po a ed in o (6) o ge he inal g ey- box model below. a) Fi o he aining da ase . Top le : i o he g ow h a e. b) Response agains he alida ion da ase . Figu e 5. Fi o he g ey-box model o he expe imen al da a. 042.14/1 0095.0460 300 7.3 458.0869.6625.2 106.0224.19 008.0 b b eba P S SP S S XPXPXP XXS XXX               (12) The model ou pu is compa ed o expe imen al aining and alida ion da ase s in Figu e 5, showing a good ag eemen . I becomes e iden ha he eliabili y o he simula ion is high- ly dependen on he µ-model accu acy, since a small e o in he app oxima ion by ALAMO s ill leads o some e o . Compa ing Figu es 2b wi h 5b, he p oposed app oach yields much be e (al hough no pe ec ) i s. No e ha , as was al eady men ioned, some expe imen al da a a e no eliable. 5 CONCLUSIONS A me hodology o he de elopmen o g ey-box models, combining i s p inciples and da a d i en models, has been p esen ed and es ed success ully. I s s eng hs, compa ed o adi ional app oaches, a e ha ewe assump ions abou he model ha e o be made, lea ing addi ional deg ees o ee- dom. 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