Full-Scale Digesters: Model Predictive Control with Online Kinetic Parameter Identification Strategy

Ci a ion: Co és, L.G.; Ba bancho, J.; La ios, D.F.; Ma in-Ba is a, J.D.; Mohedano, A.F.; Po illa, C.; de la Rubia, M.A. Full-Scale Diges e s: Model P edic i e Con ol wi h Online Kine ic Pa ame e Iden i ica ion S a egy. Ene gies 2022, 15, 8594. h ps://doi.o g/10.3390/ en15228594 Academic Edi o : Gio anni Esposi o Recei ed: 21 Augus 2022 Accep ed: 10 No embe 2022 Published: 16 No embe 2022 Publishe ’s No e: MDPI s ays neu al wi h ega d o ju isdic ional claims in published maps and ins i u ional a il- ia ions. Copy igh : © 2022 by he au ho s. Licensee MDPI, Basel, Swi ze land. This a icle is an open access a icle dis ibu ed unde he e ms and condi ions o he C ea i e Commons A ibu ion (CC BY) license (h ps:// c ea i ecommons.o g/licenses/by/ 4.0/). ene gies A icle Full-Scale Diges e s: Model P edic i e Con ol wi h Online Kine ic Pa ame e Iden i ica ion S a egy Luis G. Co és 1,*, J. Ba bancho 1, D. F. La ios 1, J. D. Ma in-Ba is a 2,3, A. F. Mohedano 3, C. Po illa 4 and M. A. de la Rubia 3 1Depa amen o de Tecnología Elec ónica, Escuela Poli écnica, Uni e sidad de Se illa, 41011 Se ille, Spain 2E uels Technologies L d., 42-44 Bishopga e, London EC2N 4AH, UK 3Depa amen o de Ingenie ía Química, Campus de Can oblanco, Uni e sidad Au onoma de Mad id, 28049 Mad id, Spain 4Facul ad de Minas, Uni e sidad Nacional de Colombia, Robledo, Medellín 050034, Colombia *Co espondence: luico [email p o ec ed] Abs ac : This wo k p esen s a nonlinea model p edic i e con ol scheme wi h a no el s uc u e o obse e s aiming o c ea e a me hodology ha allows easible implemen a ions in indus ial anae obic eac o s. In his way, a new s ep-by-s ep p ocedu e scheme has been p oposed and es ed by sol ing wo speci ic d awbacks epo ed in he li e a u e esponsible o he ine iciencies o hose sys ems in eal en i onmen s. Fi s ly, he implemen a ion o con ol s uc u es based on modeling depends on mic oo ganisms’ concen a ion measu emen s; he echnology ha achie es his is no cos -e ec i e no iable. Secondly, he eac ion a es canno be conside ed s a ic because, in he ex ended anae obic diges ion model (EAM2), he la ge luc ua ion o pa ame e s is una oidable. To ace hese wo d awbacks, he concen a ion o acidogens and me hanogens, and he alues o he wo eac ion a es conside ed ha e been es ima ed by a s uc u e o wo obse e s using da a collec ed by senso s. A e 90 days o ope a ion, he e o in con e gence was lowe han 5% o bo h obse e s. Fou model p edic i e con olle (MPC) con igu a ions a e used o es all he p e ious in o ma ion ying o maximize he olume o me hane and demons a e a sa is ac o y ope a ion in a wide ange o scena ios. The esul s demons a e an inc ease in e iciency, anging om 17.4% o 24.4%, using as a e e ence an open loop con igu a ion. Finally, he ope a ional obus ness o he MPC is compa ed wi h simula ions pe o med by adi ional al e na i es used in indus y, he p opo ional-in eg al- de i a i e (PID) con olle s, whe e some simple ope a ional scena ios o manage o an MPC a e longe su icien o dis up a no mal ope a ion in a PID con olle . Fo his con olle , he simula ion shows an e o close o he 100% o he e e ence alue. Keywo ds: anae obic diges ion; asymp o ic obse e ; homogeneous eac ion sys ems; kine ic pa ame e obse e ; model p edic i e con ol; s ep-ahead 1. In oduc ion The main a ibu e o indus ial p ocesses and manu ac u ing is he la ge-scale ans- o ma ion o mass and ene gy in o high-end aluable speci ic ma e ials and compounds [ 1 ]. In essence, chemical and biological p ocesses a e impe a i e o hose pu poses because i is p ima ily a his s age whe e he elemen s in a eac ion can be manipula ed o achie e he desi ed a ibu es [ 2 – 4 ]. All he abo e is possible i a con ol and moni o ing scheme is added o gua an ee sa e and s able ope a ion [ 5 ]. Speci ically, in anae obic sys ems, he ans o ma ion o subs a es in o biogas and diges a e has a ac i e ad an ages in compa ison o classical al e na i es such as ae obic diges ion o compos ing [ 6 , 7 ]; i e u ns li le sludges, has a posi i e o e all ene gy balance, and also has eno mous po en ial o educe challenging and concen a ed subs a es such as animal was es [ 8 , 9 ]. Howe e , his is an elabo a ed biochemical p ocess ha includes di e se mic oo ganisms, making he Ene gies 2022,15, 8594. h ps://doi.o g/10.3390/en15228594 h ps://www.mdpi.com/jou nal/ene gies Ene gies 2022,15, 8594 2 o 23 p ocess nonlinea [ 10 ]. In his a ea, i is g ea o in e es o de i e models ha would be as sensi i e as possible o he lack o phenomenological knowledge [ 11 , 12 ]. Thus, an in- e es ing al e na i e o ep esen his phenomenon is he ex ended anae obic diges ion ma hema ical model 2 (EAM2) [ 13 ], designed o con ol and moni o ing pu poses. I ci cum en s he di icul ies in loca ing he biological lack o knowledge in dedica ed e ms, namely eac ion a es [ 14 ]. Addi ionally, compa ed wi h i s p edecesso , in he anae obic diges ion ma hema ical model 2 (AM2), wo addi ionally yield pa ame e s we e included o conside he in luence o p o eins and aminoacids on he dynamic o alkalini y and conside mo e subs a es o be modeled. In o de o alloca e a s a egy ha be e i s he challenge men ioned abo e, se e al echniques ha e been p oposed om di e en con ol a eas. Some o hem a e nonlinea p opo ional-in eg al-de i a i e (PID) con olle s, linea quad a ic egula ion (LQR) o nonlinea sys ems [ 15 ], nonlinea model p edic i e con ol (NMPC) [ 16 ], and eceding ho izon con olle s [ 17 , 18 ], o name he mos ele an cases epo ed in he li e a u e. Al- hough hese s a egies ha e p o en o be e ec i e, he emaining ine iciencies su ge om he ac ha some o he pa ame e s a e no a ailable o be measu ed online, and he concen- a ions o he mic oo ganisms canno be measu ed pe iodically because on-si e analy ical p ocedu es a e challenging o pe o m [ 5 ]. A nonlinea model-based con ol scheme has been p oposed o achie e a sa is ac o y ope a ion. A he same ime, he ep esen a ion o anae obic diges ion (AD) p ocesses is educed o a minimal numbe o assump ions in he model-building exe cise, main aining he pe o mance [ 18 , 19 ]. Among he al e na i es eg- is e ed in he li e a u e, he Model P edic i e Con ol (MPC) [ 17 , 18 ] o e s se e al bene i s compa ed wi h adi ional me hods [ 14 , 20 , 21 ]. This algo i hm uses a ma hema ical model o p edic and ollow he sys em’s e olu ion in ad ance. One o he main ad an ages is he explici use o physical and ope a ional cons ain s in he con olle , allowing i o s and a ound wo king unde easible ope a ional condi ions [22,23]. To achie e his objec i e, a necessa y condi ion mus be sa is ied o unlock he doo and enable he ope a ion o con ol schemes based on modeling. All he in o ma ion needed om he anae obic eac o mus be a ailable pe iodically, so a g oup o eliable senso s should be conside ed o measu e, in eal- ime, he e olu ion o eac ions in he homoge- neous eac ion sys em [ 24 ]. Some key a iables, he concen a ions o mic oo ganisms, as well as he g ow h a es o eac ions (conside ed a iable due o he la ge luc ua ion o pa ame e s in AM2 [ 10 ]), a e usually es ed using analy ical me hods h ough specialized labo a o ies on-si e. The cos and du a ion o hese analyses limi he equency o he sampling [ 25 ]. Addi ionally, he a ailable online senso s on he ma ke o biomass o me aboli es need o be adequa e o egula indus ial applica ions [ 14 ]. The p e ious assump ion se s a signi ican challenge because collec ing his in o ma ion is c i ical o he ope a ion o he con olle algo i hm. In gene al, he una ailabili y o specialized ins u- men s incen i e he design and cons uc ion o obse e s dedica ed o he eac ion p ocess. Hence, he design o so wa e senso s, based on he a ailable measu emen s [ 14 , 26 ], allows o gua an ee a s eamline o in o ma ion be ween he eac o and he con ol algo i hm [ 27 ]. S a e and pa ame e obse e s es ablished o e accu a e and s ic models a e he bes al e na i es o ind a way o ob ain he lack o he da a needed [ 28 – 30 ]. Fo anae - obic eac ion p ocesses, he nonlinea AM2 has been widely p o en o moni o ing and con ol pu poses [ 10 , 13 ]. This model, cons uc ed unde a mass-balance amewo k, can adap i s dynamics o a ious AD con igu a ions o eac o s and elude he d awback ela ed o he absence o phenomenological knowledge, concen a ing his in o ma ion in non-sensi i eness-dedica ed e ms named eac ion a es [ 10 , 23 ]. The complexi y o he biochemical p ocess conside s di e en ypes o mic oo ganisms, making he p ocess nonlinea and unp edic able, imp ac ical o he adi ional online es ima ion s a egies, especially i he ope a ion o he homogeneous sys em is condi ioned by cons ain s [ 5 ]. To deal wi h hese incon eniences, a sequen ial s uc u e in cascade has been p oposed [ 14 ]. The main idea is o es ima e he concen a ion o acidogens and me hanogens in he absence o eac ion a es. Then, he nex s ep is o use his in o ma ion o pe o m a new obse a ion Ene gies 2022,15, 8594 3 o 23 ha es ima es he eac ion a es. Acqui ing in-dep h knowledge o he e olu ion o eac ion a es ep esen s a challenge since i e eals he deg ee o comple ion o bo h eac ions du ing he p ocess [5,31]. This pape is o ganized as ollows. Sec ion 2explains he ex ended AM2 ma hema ical model used o con ol pu poses, whe e addi ional e ms a e included due o he in luence o p o eins and aminoacids on he dynamics o alkalini y. The modi ica ion o he AM2 model [ 10 ] unlocks he doo o add a b oad spec um o subs a es allowed o be used by he con ol algo i hm. Then, Sec ion 3p esen s he pa ame e iden i ica ion p ocedu e based on op imiza ion ha uses a no el s ep-ahead algo i hm o conside ably imp o e he e iciency o adjus ing he dynamics o he model o he da a. The asymp o ic s a e obse e ha ope a es wi hou in o ma ion on he eac ion a es is p esen ed o es ima e he concen a ions o acidogens and me hanogens. Nex , using he p e ious in o ma ion, a second obse e es ima es he e olu ion o he wo eac ion a es conside ed. In Sec ion 4, he in o ma ion aken om obse e s and he eac o ’s measu emen s is used o enable he ope a ion o he MPC. Finally, Sec ion 5shows he imp o emen achie ed by using he MPC con olle , measu ing he inc ease in he olume o me hane p oduced. The obus ness o he MPC p oposed is compa ed wi h he PID con olle . 2. Op imal Pa ame e Iden i ica ion Algo i hm The model EAM2, ini ially p oposed by Be na d e al. [ 10 ] as AM2 in 2002 and modi ied by Co és e al. [ 13 ] in 2022, was selec ed o suppo he new moni o ing and con ol s uc u e p esen ed in his pape . The new model sol es wo issues documen ed in he li e a u e as he main d awbacks in implemen ing au oma ed decision suppo sys ems on anae obic eac o s [ 11 ]. Fi s , he necessi y o cha ac e ize he inle subs a e [ 10 ]. EAM2 conside s a wide ange o subs a es han AM2 because i akes in o accoun he e ec o ammonium p oduced om p o eins and aminoacids ha a ec he dynamics o alkalini y. Second, he concen a ions o acidogens and me hanogens a e es ima ed using a speci ic s a e obse e . The abo e-men ioned solu ions allow balancing he absence o eliable senso s o analy ical me hods o ob ain he needed in o ma ion pe iodically [10]. 2.1. P oposed Mass-Balance Ma hema ical Model o Anae obic Reac o s The ex ended ma hema ical model used in his pape is based on he AD ep esen ed by wo main g oups o mic oo ganisms. Two consecu i e p ocess desc ibe he AD phe- nomenon on eac o s, he acidogenic bac e ia ( ep esen ed by X 1 ) ha consume he o ganic subs a e ( ep esen ed by S 1 ) and p oduce mainly ola ile a y acids (mainly acid ace ic, ep esen ed by S 2 ). Thus, he second g oup o mic oo ganisms, he me hanogenic a chaea ( ep esen ed by X2), consumes S2and p oduces CO2and CH4. Two yield pa ame e s we e added o ep esen he gene a ion o ammonium due o he e men a ion o aminoacids o he consump ion by he g ow h o mic oo ganisms. The e ms added a e K Z,1 and K Z,2 , espec i ely. The model p esen ed in Co és e al. [ 13 ] conside s he ollowing dynamical equa ions: dX1 d =X1(µ1−αD)(1) dX2 d =X2(µ2−αD)(2) dS1 d =D(S1in −S1)−k1ψ1S1 KS1+S1(3) dS2 d =D(S2in −S2) + k2ψ1S1 KS1+S1−k3ψ2S2 KS2+S2(4) Ene gies 2022,15, 8594 4 o 23 dZ0 d =D(Zin −Z) + kZ,1ψ1S1 KS1+S1+kZ,2ψ2S2 KS2+S2(5) dC d =D(Cin −C)−qC+k4µ1X1+k5µ2X2(6) whe e X 1 is he concen a ion o acidogens (kg/m 3 ), X 2 is he concen a ion o me hanogens (kg/m 3 ), S 1 is he concen a ion o o ganic subs a e cha ac e ized by chemical oxygen demand (COD) (kg/m 3 ), S 2 is he concen a ion o ola ile a y acids (VFA) (g ace ic acid/L), Zis he o al alkalini y (kg/m 3 ), and Cis he concen a ion o o al ino ganic ca bon (kg/m 3 ). The a iables S 1in ,S 2in ,C in , and Zin a e, espec i ely, he in luen concen a ions o S 1 ,S 2 ,C, and Z. The a iable α is ela ed o he eac o design, wi h alues anging be ween 0 and 1. The alue 0 co esponds o an ideal ixed-bed eac o , and α= 1, o an ideal con inuously s i ed ank eac o (CSTR). Dis he dilu ion a e (d −1 ) and is he in e se o he hyd aulic e en ion ime, HRT = 1/D, ψ1 is he maximum acidogenic bac e ia g owing a e (d −1 ), ψ2 is he maximum me hanogenic a chaea g owing a e (d −1 ), K S1 is he hal -sa u a ion cons an o S 1 (kg/m 3 ), and K S2 is he hal -sa u a ion cons an o S 2 (g ace ic acid/L). The yield pa ame e k 1 is a cons an o subs a e deg ada ion, k 2 is a cons an o VFA p oduc ion (mmol/kg), k 3 is a cons an o VFA consump ion (mmol/kg), k 4 is a cons an o CO 2 p oduc ion (mmol/kg), k5 is a cons an o CO 2 p oduc ion (mmol/kg), and k 6 is a cons an o CH 4 p oduc ion (mmol/kg). The Monod- ype Equa ions (7) and (8) a e cha ac e ized by he ollowing wo eac ion a es: µ1=ψ1S1 KS1+S1(7) and, µ2=ψ2S2 KS2+S2(8) Bo hMonod- ypekine icsdesc ibe heg ow h o acidogenic bac e ia ψ1 andme hanogenic a chaea ψ2 because, du ing he e men a ion p ocess, he biomass does no egis e possible VFA accumula ion and consequen ly inhibi ion. Finally, he me hane low a e p oduced, qM, is p opo ional o he eac ion a e o me hanogenesis, as shown in Equa ion (9): qM=k6ψ2S2 KS2+S2(9) 2.2. Expe imen al Da a Resul s The da a conside ed o modeling pu poses [ 32 ] we e used p e iously in Co és e al. [ 13 ] o cha ac e ize a CSTR pilo he mophilic diges e (150 L) using he AM2 ex ended ma he- ma ical model. The o al ex ension o he expe imen was 338 days, whe e he e ec o solid e en ion ime (SRT) and he dynamics o he sys em in a wide ange o ope a ional condi- ions we e s udied. Howe e , only a speci ic ange o da a was used, whe e he acidogens and me hanogens ope a ed unde s able and homogeneous condi ions disca ding uns able scena ios un a o able o modeling pu poses (see he condi ions a gued o disca d he da a). Finally, he da a conside ed ep esen a o al o 207 days. The alues o COD a he inle and VFA and COD a he e luen a e shown in Co és e al. [13]. 3. Op imal Pa ame e Iden i ica ion and Online Measu emen s 3.1. Pa ame e Iden i ica ion Using Pa e n Sea ch by S ep-Ahead E e y sampling ime, he pa ame e iden i ica ion algo i hm s ep-ahead uses he EAM2 o calcula e he ollowing alue o he sys em in ad ance. The p ocedu e es ablished ha on he ollowing s ep, he p edic ed alue is compa ed wi h he same a iable measu ed. The di e ence in magni ude o bo h ep esen s he cumula i e e o . The p ocedu e is pe o med du ing all he expe imen . The ollowing ec o Es o es all his in o ma ion. Ene gies 2022,15, 8594 5 o 23 E= [e o 1,e o 2, . . . , e o n](10) n ep esen s he o al numbe o days e alua ed. Finally, an op imiza ion p oblem is sol ed aiming o minimize all he e o s calcula ed along he simula ion. min p(k)E subjec o: nm(k+1) = (nm(k),u(k)), (11) nnm(k+1) = (nnm(k),u(k)), 0⩽p(k)⩽pmax,∀k=1, . . . , nm ep esen s all he s a es alues measu ed S 1 ,S 2 ,C, and Z. nnm ep esen s all he non-measu able s a es X 1 and X 2 . The pa ame e s conside ed o be iden i ied a e p(k) e{µ1max , µ2max , KS1 , KS2 , k1 , k2 , k3 , k4 , k5 , k6 , KZ1 , KZ2 , Zin} . pmax ep esen s he maximum alue a o dable by each pa ame e . This algo i hm has al eady been used in a p e ious wo k wi h he same elemen s, eal da a, and phenomenological models [13]. 3.2. Asymp o ic Es ima o When Reac ion Ra es A e Unknown The absence o in o ma ion due o he lack o eliable senso s o analysis esul s opens up an oppo uni y o subs i u e he unce ain y by he design o so wa e senso s. In anae - obic eac o s, his echnology es ima es he concen a ion o acidogens and me hanogens wi hou he in o ma ion o he eac ion a es [ 11 ]. I esul s in a pa icula ca ego y o obse e -denomina ed asymp o ics, because wo condi ions suppo i s ope a ion; he sys em is s ill no exponen ially obse able, and he eac ion kine ics a e unknown [ 14 ]. The de ailed in o ma ion o he design is p esen ed in Co és e al. [13]. In he ollowing, we p esen he condi ions ha a ec he design o he algo i hm: he in o ma ion o he ma ix φ is unknown, he yield coe icien s om K a e ully known, and he numbe o q s a e , he numbe o measu ed s a e a iables, is he same o highe han he ank o he ma ix K( ha is, q s a e = dim( ξ1)≥ ank(K)). Hence, based on he p e ious in o ma ion, he gene al equa ion o homogeneous eac ion sys ems desc ibed by a gene al nonlinea s a e space model is p esen ed: dξ d =Kφ(ξ, )−Dξ−Q(ξ) + F(12) whe e dim( ξ ) = dim(F) = dim(Q) = N, dim( φ ) = M and dim(K) = N × M. Thus, he gene al nonlinea model Equa ion (12) can be di ided as dξa d =Kaφ(ξa,ξb)−Dξa−Qa+Fa(13) dξb d =Kbφ(ξa,ξb)−Dξb−Qb+Fb(14) whe e he ank o Kis p. The subma ix Ka esul s om a sec ion o Kwi h p × M. The subma ix Kb has he emaining in o ma ion o K. Finally, he ma ices (ξa , ξb) , ( Qa , Qb ), and (F a ,F b) a e he co esponding pa s o ξ ,Q, and Fcaused by he in luence o K a and K b . The p e ious o mula ion has he ollowing ea u e: Zob =A0ξa+ξb(15) The e exis s a ans o ma ion ha conside s Zob as a linea combina ion o ξa and ξb as seen in Equa ion (15). Finally, a e p ocessing Equa ions (13) – (15) , he new s a e space model is equi alen o: Ene gies 2022,15, 8594 6 o 23 dξa d =Kaφ(ξa,ξb)−Dξa−Qa+Fa(16) dZob d =−DZob +A0(Fa−Qa) + (Fb−Qb)(17) The exp ession Fa−Qa = 0 means he pa i ion made by Equa ions (13) and (14) a e app op ia e since he new dynamics on Zob a e independen om K and φ . In Equa ion (16) , he independence o ˙ ξa om φ is s a ed. To ex end o de ailed in o ma ion, check he comple e demons a ion in Co és e al. [13]. Obse e Design Using he nonlinea EAM2, he ollowing equa ions desc ibe he decoupled subsys em conduc ed by he s a e a iables X1,X2,S1, and S2: ξ=    X1 X2 S1 S2    ,F=    0 0 DS1in DS2in    ,Q=    0 0 0 0    ,K=    1 0 0 1 −k10 k2−k3    ,φ=µ1X1 µ2X2(18) Conside ing he p e ious subsys em, he subsequen s a e equa ion is s uc u ed as ollows: d d     X1 X2 S1 S2    =    1 0 0 1 −k10 k2−k3    φ1 φ2−D    X1 X2 S1 S2    +    0 0 DS1in DS2in    (19) whe e ξa and ξb g oup he measu able and non-measu able s a es, espec i ely. Thus, using he equa ions desc ibed in Co és e al. [13], he exp ession o Zob is as ollows: dZob d =−DZob +A0(Fa −Qa) + (Fb−Qb)(20) whe e he ma ices ela ed o he equa ion a e: ξb=X1 X2,Zob =Zob1 Zob2,A0="1 k10 k2 k1k3 1 k3#,ξa=S1 S2(21) Fa=DS1in DS2in,Fb=0 0(22) wi h: Zob =Zob1 Zob2=A0ξa+ξb="1 k10 k2 k1k3 1 k3#S1 S2+X1 X2(23) 3.3. Kine ic Pa ame e Reac ion Es ima o This sec ion p esen s an addi ional ool ha complemen s he lack o measu emen s in he eac o . The goal is o es ima e he kine ic eac ions coming om he eac ion sys em. Figu e 1shows he s uc u e p oposed o ob ain all he in o ma ion needed o eed he ma hema ical model con ained in he con ol scheme s a egy. Ene gies 2022,15, 8594 7 o 23 Figu e 1. The con olle and he obse e s uc u e p oposed o he anae obic eac o . The discon inuous line ep esen s he in o ma ion gene a ed by he obse e s. A he end, he measu ed a iables nm and he es ima ed ones con e ge o eed he con olle algo i hm. In o de o design he kine ic es ima o algo i hm, he ollowing nonlinea equa ion ha ep esen s he sys em mus be conside ed: ˙ ξ=Kφ−Dφ−Q+F(24) Equa ion (24) assumes ha he coe icien s om K a e known, while he dilu ion a e D , he inpu lows F , and he gas ou pu lows Q a e measu ed in eal- ime. Addi ionally, i is assumed ha ξ is ully-known because he non-measu ed dynamics a e econs uc ed by he p e ious asymp o ic obse e . The ec o φ is pa ially known and is di ided as ollows: φ=Hρ(25) The ma ix H con ains he in o ma ion om he known kine ic eac ions, and ρ con- ains he emaining in o ma ion, he unknown kine ic eac ions. Thus, using Equa ion (25) in Equa ion (24) esul s in Equa ion (26). ˙ ξ=KHρ−Dφ−Q+F(26) The es ima ion o eac ion kine ics, 1 and 2 , is equi alen o es ima ing he en i e ec o ρ . Hence, ollowing he s uc u e p oposed by Bas in e al. [ 14 ], a new dynamic o he sys em is p esen ed as ollows: ˙ ˆ ξ=KH ˆ ρ−Dξ−Q+F−Ω(ξ−ˆ ξ), (27) ˙ ˆ ρ=(KH)TΓ(ξ−ˆ ξ)(28) whe e ˆ ρ ep esen s he eal- ime es ima ion o ρ . When ξ−ˆ ξ∼ = 0 occu s in Equa ion (28) , i means ha ˙ ˆ ρ∼ = 0 esul s in a pe ec endency p ocess o con e gence o desi able eac ion kine ics alues ˆ ρ∼ =ρ . The e is a manda o y condi ion. Equa ion (27) ha ep esen s he kine ic eac ion es ima o has an equi alen s uc u e o he Luenbe ge obse e used in homogeneous eac ion sys ems [14]. ˙ ˆ ξ=Aˆ ξ+Bu | {z } 1∗ +Ω(ξ−ˆ ξ)(29) Ene gies 2022,15, 8594 8 o 23 The unde lined e m 1 ∗ in Equa ion (29) ,A ˆ ξ +Bu, is equi alen o he e m KH ˆ ρ− D ξ− Q +Fin Equa ion (27). Thus, he abo e-men ioned compa ison esul s in Equa ion (30): ˙ ˆ ξ=KH ˆ ρ−Dξ−Q+F | {z } 1∗ −Ω(ξ−ˆ ξ)(30) In his way, Equa ion (30) ep esen s he nonlinea sys em, while Equa ion (29) is he equi alen linea ized sys em. Obse e Design To design he kine ic eac ion es ima o , only he measu ed and es ima ed s a es on ˙ ξ a e needed, while he e m ξ−ˆ ξ is used as a e e ence o moni o and ollow he es ima o ’s pe o mance. ˆ ξ should con e ge o ξ as soon as possible ( ξ≈ˆ ξ ). The dynamics o ˆ ˙ ξ depends on he e olu ion o he dynamics o he e o ˙ e = (A −Ω L)e. Thus, he e e ences ( he known alues) ha assis in uning he kine ic eac ion es ima o a e ξ , ˆ ξ , K , and H . Fo be e unde s anding, he ollowing de ini ions a e p oposed: he e m e= ξ−ˆ ξ means he obse a ion e o , and he e m ˜ ρ=ρ−ˆ ρ means he acking e o . Hence, using Equa ions (31) and (32): ˙ ˆ ξ=KH ˆ ρ−Dξ−Q+F−Ω(ξ−ˆ ξ)(31) ˙ ξ=Kφ−Dξ−Q+F(32) De i ing on bo h sides o e and ˜ ρ , i esul s in ˙ e=˙ ξ−˙ ˆ ξ and ˙ ˜ ρ=˙ ρ−˙ ˆ ρ . Taking Equa ions (31) and (32) and eplacing hem by ˙ e esul s in: ˙ e=Kφ−Dξ−Q+F−KH ˆ ρ+Dξ+Q−F+Ω(ξ−ˆ ξ)(33) ˙ e=Kφ−KH ˆ ρ+Ωe(34) ˙ e=KHρ−KH ˆ ρ+Ωe(35) ˙ e=KH(ρ−ˆ ρ) + Ωe(36) ˙ e=KH ˜ ρ+Ωe(37) In he same way, he dynamic o he acking e o is as ollows: ˙ ˜ ρ=−(KH)TΓe+dρ d (38) Me ging Equa ions (37) and (38) esul s in he ollowing dynamic sys em: ˙ e ˙ ˜ ρ="KH ˜ ρ+Ωe −(KH)TΓe+dρ d #(39) Thus, a e o ganizing he p e ious equa ion, i esul s in he ollowing sys em: ˙ e ˙ ˜ ρ=ΩKH −HTKTΓ0e ˜ ρ+"0 dρ d #(40) Replacing he ma ices: A=ΩKH −HTKTΓ0(41) V="0 dρ d #(42) This esul s in: ˙ e ˙ ˜ ρ=Ae ˜ ρ+V(43) Ene gies 2022,15, 8594 9 o 23 Acco ding o he in o ma ion desc ibed, he es ima ion p ocedu e is pe o med as ollows: Ka=−k10 k2−k3,Kb=1 0 0 1,Fa=DS1in DS2in,Fb=0 0(44) The nex s ep is o calcula e he ma ix A0; hus, i means ha : A0=−KbK−1 a(45) In he ollowing equa ions, he sys em is di ided in o he dynamic s a es o be es i- ma ed, using he da a om he asymp o ic obse e , Z 1 and Z 2 , and he measu ed da a S 1 and S2. Sol ing ξb om Equa ion (15), we ob ain: ξb=Zob −A0ξa(46) Equa ion (46) allows o calcula e he ollowing ma ices: K=Kb Ka,H=1 0 0 1,ξ=  ξb S1 S2 ,F=Fb Fa(47) Finally, all he in o ma ion needed o supply Equa ions (27) and (28) a e ob ained abo e. 4. Nonlinea Model P edic i e Con olle (Mpc) As i is shown in Figu e 1, he obse e s uc u e calcula es he non-measu able dynamic s a es, he concen a ions o X 1 and X 2 (wi h he asymp o ic obse e ), and he kine ic eac ion a es 1 and 2 (wi h he eac ion a e obse e ). Once he in o ma ion om measu emen s and i ual senso s has been deli e ed o he con olle , he algo i hm is eady o s a calcula ing he con ol ac ions a e ecei ing he es o he in o ma ion: he ope a ional and physical ins uc ions, as well as he con ol objec i es. 4.1. Con olle Design The subsequen condi ions eme ge as he equi ed ules o ake in o conside a ion o design he con olle . • The eac o has o be balanced o eac agains pe u ba ions, always wo king wi hin he physical and ope a ional bounda ies. • Me hane p oduc ion needs o be maximized all he ime. • The en i onmen al egula ions and he capaci y o he eac o o educe he concen- a ion o subs a es a he inle , S 1 and S 2 , condi ioned he p og amming. The e o e, he ule S 1 ( ) + S 2 ( ) ≤ K d , whe e K d deno es he maximum e luen concen a ion o he bo h subs a es conside ed, has o be ollowed. • The eac o has o be p o ec ed agains ailu es due o unexpec ed high a ia ions on VFA, and in consequence, inhibi ions on he me abolism o mic oo ganisms. Thus, an al e na i e is o use alkalini y as a base o neu alize he le el o acids. S2( ) Z( )=λ(48) Equa ion (48) ies o p ese e a ela ion be ween VFA and Zdu ing he ope a ion o he eac o . Acco ding o he li e a u e, λ has o be wi hin 0.1 and 0.3, o main ain he mic oo ganisms in a zone o com o in ega d o he le el o acids [11]. 4.2. S uc u e o he Con olle Finally, he s uc u e o he con olle is shown in Figu e 2. The dilu ion a e D con ols he amoun o subs a e a he inle . I esul s in u C , he a iables used by he MPC o manipula e he dynamics o he eac o . u i ep esen s he non-con olle a iables conside ed as inle : S 1in ,S 2in ,C in , and Z in .u pe holds he in o ma ion abou he inpu s conside ed as dis u bances. y con ains he in o ma ion o he a iables a ailable o be measu ed. n m a e he s a e dynamics ha can be measu ed: S 1 ,S 2 ,Z, and C. The alue o pH Ene gies 2022,15, 8594 16 o 23 5.4. Mpc Con olle wi hou Res ic ions Figu e 7shows he esul s o maximizing he me hane p oduced by he eac o using an MPC wi hou he ope a ional and physical es ic ions shown in Equa ion (49). (a) (b) (c) (d) (e) ( ) Figu e 7. MPC con olle wi hou es ic ions, maximizing he olume o CH 4 . Time cou se o ( a ) ol- ume o CH 4 ; ( b ) dilu ion a e D; ( c ) alue o pH; ( d ) dynamic o VFA; ( e ) alues o λ ; ( ) dynamics o S2and Z. Figu e 7a shows a subs an ial inc ease in he olume o me hane p oduced (see discon- inuous ed line), whe e he black line ac s as he e e ence ( olume o me hane p oduced due o p o ile inpu s used in pa ame ic op imiza ion), o measu e he imp o emen s on e - iciency. The co esponden con ol ac ions calcula ed a e shown in Figu e 7b. The absence o es ic ions in he algo i hm allows he con olle o calcula e con ol ac ions, neglec ing s ong changes in ope a ion ha could a ec he s abili y o he eac o . As can be seen, maximizing he objec i e u ns he alues o D ex eme; showing oscilla ions om one Ene gies 2022,15, 8594 17 o 23 ex eme o he o he . Se e al imes he alue o D dec eases o ze o and suddenly ise o he maximum. In he same way, pH had se e e changes e en beyond he ope a ional limi s (see Figu e 7c). The ela ionship be ween pH and VFA is clea in Figu e 7c,d showing a co ela ion along he simula ion. This algo i hm exposes high ex e nal isks in ope a ion because he e exis s a lack o in o ma ion o e he sys em’s ope a ion. Figu e 7e shows he co esponding alues o λ along he simula ion; i appea s ha he alues go beyond he limi s epea edly. As i seems in Figu e 7 , he alue o alkalini y Z exceeds he alue o S2 whe e a his poin , he alue o λ goes beyond 1, meaning he eac ion sys em uns inadequa ely. Gi en he p e ious esul s in Figu e 7, he maximiza ion o me hane wi hou es ic- ions leads he sys em o ope a e beyond non- easible limi s. The e o e, i is necessa y o include ex a in o ma ion abou he p ocess o a oid inhibi o s along he simula ion. Figu e 8shows he esul s ob ained due o he ope a ion o he MPC wi hou he es ic ion o bu e ing capaci y ( λ ) and wi hou he mul is a unc ion. Figu e 8a shows he olume o me hane p oduced by bo h he e e ence (black line), and he MPC ( ed discon inuous line). A signi ican inc ease in he olume o me hane p oduced compa ed o he e e ence is obse ed. Figu e 8b shows he con ol ac ions calcula ed by he MPC. As i is shown, he oscilla ion is se e e, howe e , he esul s a e eliable. Figu e 8c,d shows he esul s o he pH and S2 . As expec ed, he oscilla ion o alues is in ense a ound he e e ence (black line) bu no a ac i e o con ol pu poses. Finally, Figu e 8e shows he e olu ion o he non- es ic ed pa ame e λ . As expec ed, due o he absence o es ic ions on p og amming, he maximum alue o λhas been exceeded once. (a) (b) (c) (d) Figu e 8. Con . Ene gies 2022,15, 8594 18 o 23 (e) ( ) Figu e 8. MPC con olle wi hou es ic ions. ( a ) Volume o CH 4 ; ( b ) dilu ion a e D; ( c ) alue o pH; (d) dynamic o VFA; (e) alues o λ; ( ) dynamics o S2and Z. 5.5. Model P edic i e Con olle wi h Res ic ions The in o ma ion abou ope a ional and physical es ic ions is added o he MPC aiming o main ain he mic oo ganisms unde easible ope a ional condi ions. As shown in Figu e 9a, he discon inuous ed line shows he olume o me hane p oduced. The e iciency is main ained despi e he es ic ions; he e o e, his op ion be e i s he sys em unde s able condi ions. The co esponden con ol ac ions calcula ed by he op imize (see Figu e 9b) shows a sligh a enua ion compa ed wi h p e ious simula ions. The esul s o pH in Figu e 9c show ha he ange o alues becomes close o he e e ence han he p e ious esul s. Compa ed wi h he p e ious esul s, he acidi ica ion le el is much lowe (see Figu e 9d). In Figu e 9e, he discon inuous blue line ep esen s he limi s a ed o a egula ope a ion. As i is shown, he discon inuous ed line ne e su passes he limi s beyond λ = 0.8; in some cases, i ge s close . Howe e , i is because he eac o is p oducing a high amoun o me hane close o he ope a ional limi s. Figu e 9 shows he e olu ion o he balance be ween VFA and Z. I is clea ha he bu e capaci y is always unde con ol. Al hough he algo i hm is es ic ed, he amoun o me hane CH 4 p oduced is highe han he e e ence. Finally, Figu e 10 shows simila esul s o hose shown in Figu e 8bu p og ammed wi hou he mul is a unc ion. Despi e he dec ease in e iciency, he esul s show an adequa e ope a ion. (a) (b) Figu e 9. Con . Ene gies 2022,15, 8594 19 o 23 (c) (d) (e) ( ) Figu e 9. MPC con olle wi h es ic ions, maximizing he olume o CH 4 . ( a ) Volume o CH 4 ; (b) dilu ion a e D; (c) alue o pH; (d) dynamic o VFA; (e) alues o λ; ( ) dynamics o S2and Z. (a) (b) (c) (d) Figu e 10. Con . Ene gies 2022,15, 8594 20 o 23 (e) ( ) Figu e 10. MPC con olle wi h es ic ions. ( a ) CH 4 e olu ion; ( b ) D alues calcula ed by op imize ; (c) pH alues; (d) VFA alues; (e)λo e he p ocess; ( ) e olu ion o S2and Z. Table 1shows a condensed ision o he p e ious esul s. Two same al e na i es we e simula ed wi hou he mul is a unc ion. As expec ed, he e is a educ ion in e iciency because he algo i hm chose a andom ini ial poin o s a he p ocess. The black line is used as a e e ence o compa e easily he esul s om he me hane p oduced by he MPC [ 13 ]. In he case o he MPC con olle ha ope a es wi hou ope a ional es ic ions, he imp o emen achie ed was 17.4% and 24.4%, espec i ely, o he algo i hm wi h no mul is a and he mul is a unc ion. The eac o ha ope a es wi h ope a ional es ic ions shows an imp o emen o 18.8% and 20.8% o bo h MPC schemes, wi hou and wi h he mul is a unc ion, espec i ely. Al hough he pe o mance is educed signi ican ly because o he inse ion o es ic ions, he con inuous ope a ion o he eac o is no b oken and can un pe manen ly. Table 1. Imp o emen in he me hane p oduced by each MPC con ol scheme. Schemes Wi hou Res ic ions Wi h Res ic ions No Mul is a Mul is a No Mul is a Mul is a Inc ease on e iciency 17.4 % 24.4% 18.8% 20.9% 6. Conclusions A s uc u e o wo obse e s demons a es o be e ec i ely sol ing wo speci ic d aw- backs esponsible o he ine iciencies implemen ing con ol schemes based on modeling. The Asymp o ic obse e ha ope a es in he absence o he eac ion a es demons a es an e o o con e gence less han 5% es ima ing he concen a ions o acidogens and me hanogens. Then, he p e ious in o ma ion enables an ups eam ope a ion eeding he kine ic obse e ha es ima es he eac ion a es wi h an e o o con e gence lowe han 5%. In he ollowing, he es o he pa ame e s, conside ed as s a ic due o hei co ela ion wi h he speci ic biochemical cha ac e is ics o he o ganic ma e , we e calcula ed using algo i hms based on op imiza ion. All he abo e-men ioned aspec s ep esen he key ha makes i possible o ope a e MPCs in he indus y. Fou MPC con igu a ions we e used o es he obse e and iden i ica ion s uc u e p oposed. The objec i e was o maximize he olume o me hane. While he con olle wi hou he ope a ional es ic ions showed a conside able inc ease in e iciency, a e ope a ional limi s we e added o he algo i hm, he e iciency o he con olle sligh ly dec eased. Howe e , i s educ ion on e iciency is a o dable because he algo i hm helped he sys em o s ay in ope a ion con inuously. Wi hou ope a ional es ic ions, he e iciency eached 24.4%, while he es ic ed algo i hm showed an e iciency o 20.9%. The exhaus i eness in he me hodology p oposed is wo h using because he added in o ma ion gua an ees obus ness in ope a ion and an inc ease in e iciency compa ed wi h adi ional con ol al e na i es. Ene gies 2022,15, 8594 21 o 23 Au ho Con ibu ions: Concep ualiza ion, L.G.C., J.B., D.F.L., J.D.M.-B., C.P. and M.A.d.l.R.; Da a cu a ion, L.G.C., C.P. and M.A.d.l.R.; Fo mal analysis, D.F.L., J.D.M.-B., C.P. and M.A.d.l.R.; In es iga- ion, L.G.C., J.B., D.F.L., J.D.M.-B., A.F.M., C.P. and M.A.d.l.R.; Me hodology, L.G.C., J.B. and C.P.; P ojec adminis a ion, J.B. and C.P.; Resou ces, J.D.M.-B. and A.F.M.; So wa e, L.G.C.; Supe ision, D.F.L., J.D.M.-B. and M.A.d.l.R.; Valida ion, D.F.L. and M.A.d.l.R.; Visualiza ion, L.G.C.; W i ing— o iginal d a , L.G.C.; W i ing— e iew & edi ing, J.B., D.F.L. and C.P. All au ho s ha e ead and ag eed o he published e sion o he manusc ip . Funding: The au ho s wish o exp ess hei g a i ude o Fundación FIDETIA (G91045419), Uni e si- dad de Se illa, Spain, o unding his wo k. The au ho Luis G. Co és exp ess special acknowledge- men s o Fundación Cen o de Es udios In e disciplina ios Básicos y Aplicados (CEIBA), Colombia, o unding. Labo a o io de In es igaciones en Ca álisis y nue os ma e iales LICATUC (Uni e sidad de Ca agena). Da a A ailabili y S a emen : The ini ial expe imen al esul s de eloped in his s udy a e in he nex link: h ps://doi.o g/10.1016/j.p ocbio.2005.03.073 (accessed on 20 Augus 2022). Con lic s o In e es : The au ho s decla e ha hey ha e no known compe ing inancial in e es s o pe sonal ela ionships ha could ha e appea ed o in luence he wo k epo ed in his pape . Nomencla u e C,Cin To al ino ganic ca bon concen a ion (kg/m3) DDilu ion a e (d−1) k1Yield subs a e deg ada ion k2Yield o VFA p oduc ion k3Yield o VFA consump ion k4Yield o CO2p oduc ion k5Yield o CO2p oduc ion k6Yield o CH4p oduc ion (m3) KS1Hal -sa u a ion cons an (kg/m3) KS2Hal -sa u a ion cons an (kg/m3) KZ,1 Yield o ammonium p oduc ion (kg/m3) KZ,2 Yield o ammonium p oduc ion (kg/m3) qCCa bon dioxide low a e (m3/d) qMMe hane low a e (m3/d) 1, 2Reac ion a es (d−1) S1,S1in O ganic subs a e concen a ion (kg acid ace ic/m3) S2,S2in Vola ile a y acids concen a ion (kg/m3) X1Concen a ion o acidogenic bac e ia (kg/m3) X2Concen a ion o me hanogenic bac e ia (kg/m3) Z,Zin To al alkalini y (kg/m3) αF ac ion o bac e ia in he liquid phase µ1Speci ic g ow h a e o acidogenic bac e ia (d−1) µ2Speci ic g ow h a e o me hanogenic bac e ia (d−1) ψ1Maximum acidogenic bac e ia g ow h a e (d−1) ψ2Maximum me hanogenic bac e ia g ow h a e (d−1) nmVec o o measu ed s a e a iables nnm Vec o o non-measu able s a e a iables p,pmax Vec o o pa ame e s u,umin,umax Con ol ac ion (d−1) ξVec o o s a e a iables KMa ix wi h he kine ics o he biochemical φMic obiological eac ions in ol ed on sys em QGaseous a e o mass ou low om eac o FMass eed a e due o ex e nal subs a es ρUnknown unc ions o all s a es HKnown unc ions o all he s a es ΓGain ma ix o he upda ing law ΩSqua e gain con e gence ma ix Ene gies 2022,15, 8594 22 o 23 λBu e capaci y NpP edic ion ho izon (d) NuCon ol ho izon (d) Re e ences 1. 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