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Bioreactor Scaling Enhances Feedback Control Of Concentration, Rates, and Yields.
12th 12th IFAC IFAC Symposium Symposium on on Dynamics Dynamics and and Control Control of of 12th IFACSystems, Symposium on Dynamics and Control of Process Systems, including Biosystems Process including Biosystems Available online at www.sciencedirect.com Process Biosystems 12th IFACSystems, Symposium on Dynamics Control of Florianópolis - SC,including Brazil, April 23-26,and 2019 Florianópolis - SC, Brazil, April 23-26, 2019 Florianópolis - SC,including Brazil, April 23-26, 2019 Process Systems, Biosystems Florianópolis - SC, Brazil, April 23-26, 2019
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IFAC PapersOnLine 52-1 (2019) 237–242
Bioreactor Scaling Enhances Feedback Bioreactor Bioreactor Scaling Scaling Enhances Enhances Feedback Feedback Control Of Concentration, Rates, and Bioreactor Scaling Enhances Feedback Concentration, Rates, Control Of Control Of Concentration, Rates, and and Yields. Control Of Concentration, Rates, and Yields. Yields. Yields. Pedro A. Lira Parada, Even Pettersen, Nadav Bar ∗∗
Pedro A. Lira Parada, Even Pettersen, Nadav Bar ∗∗ Pedro A. Lira Parada, Even Pettersen, Nadav Bar ∗ ∗ Pedro A. Lira Parada, Even Pettersen, Nadav Bar of Chemical Engineering, Norwegian University of ∗ Department Department of Chemical Engineering, Norwegian University of ∗ ∗ Department of Chemical N-7491 Engineering, Norwegian University of Science and Technology, Trondheim, Norway (e-mail: ∗ Science and Technology, N-7491 Trondheim, Norway (e-mail: Department of Chemical Engineering, Norwegian University of Science and Technology, N-7491 Trondheim, Norway (e-mail: [email protected], [email protected], [email protected]) [email protected], Science and Technology,[email protected], N-7491 Trondheim, [email protected]) Norway (e-mail: [email protected], [email protected], [email protected]) [email protected], [email protected], [email protected]) Abstract: Fermentation Fermentation processes processes are are biologically biologically complex, complex, multiple-input, multiple-input, multiple-output multiple-output Abstract: Abstract: processes biologically multiple-output systems, and andFermentation as such, such, selection selection of an anare adequate controlcomplex, strategymultiple-input, is often often an an elaborate elaborate task. The The systems, as of adequate control strategy is task. Abstract: processes are biologically multiple-input, multiple-output systems, andFermentation as such, selection of anare adequate controlcomplex, strategy is the often an elaborate task. The process and disturbance dynamics easily oversimplified, and controller design in such process and dynamics easily oversimplified, and controller design in systems, anddisturbance as such, selection of anare control strategy is the often an elaborate task. The process and disturbance dynamics areadequate easily oversimplified, and the controller design in such such cases seldom seldom yield adequate performance. In the the present study, we developed a non-structured, non-structured, cases yield adequate performance. In present study, we developed a process and disturbance dynamics are easily oversimplified, and the controller design in such cases seldom yield adequate performance. In the present study, we developed a non-structured, dimensionless model model of of microbial microbial fermentation fermentation production. production. Analysis Analysis and and simulations simulations show show that that dimensionless cases yield adequate performance. In theproduction. present study, we developed a non-structured, dimensionless model ofmodel microbial fermentation Analysis simulations show that scalingseldom of aa bioreactor bioreactor facilitates parameter estimation, yield aand straightforward objective scaling of model facilitates parameter estimation, yield a straightforward objective dimensionless modeland ofmodel microbial fermentation production. Analysis simulations show that scaling a bioreactor facilitates parameter estimation, yield aand straightforward objective functionofdefinition, definition, promote control decisions with better performance. The results results suggest function and promote control decisions with better performance. The suggest scaling of a bioreactor model facilitates parameter estimation, yield a straightforward objective function definition, and promote control decisions with better performance. The results suggest that scaling scaling of of a bioreactor bioreactor model model enhances enhances the the performance performance of of feedback control, control, producing producing that function definition, andrates, promote control decisions with better performance. The results suggest that scaling of a a bioreactor model enhances the performance of feedback feedback control, producing higher concentrations, and yields, three of the most important parameters in industrial higher concentrations, rates, and yields, three of the most important parameters in industrial that scaling of a bioreactor model enhances the performance of feedback control, producing higher concentrations, rates, and yields, three of the most important parameters in industrial microbial biocatalysis. microbial biocatalysis. higher concentrations, microbial biocatalysis. rates, and yields, three of the most important parameters in industrial © 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. microbial biocatalysis. Keywords: Bioprocess Bioprocess control, control, Industrial Industrial biotechnology, biotechnology, fed-batch, fed-batch, feedback control, control, parameter Keywords: Keywords: Bioprocess control, Industrial biotechnology, fed-batch, feedback feedback control, parameter parameter estimation, fermentation process, dynamic model. estimation, fermentation process, dynamic model. Keywords: Bioprocess control, Industrial fed-batch, feedback control, parameter estimation, fermentation process, dynamicbiotechnology, model. estimation, fermentation process, dynamic model. 1. INTRODUCTION INTRODUCTION and and examine examine the the resulting resulting control control performance. performance. Herein, Herein, 1. 1. INTRODUCTION and examine the resulting control non-structured performance. Herein, we turned a previously published, model we turned a previously published, non-structured model 1. INTRODUCTION and examine the resulting control performance. Herein, we turned a previously published, non-structured model of a microbial bioreactor to a dimensionless model, and of aaturned microbial bioreactorpublished, to aa dimensionless dimensionless model, and we a previously non-structured model Microbial biocatalysis biocatalysis gained gained large large momentum momentum in in recent recent of microbial bioreactor to model, and compared the control performance of a real CorynebacMicrobial compared the control control performance of aa real real model, CorynebacMicrobial biocatalysis large metabolic momentum in recent compared of a microbial bioreactor to aproducing dimensionless decades due due to genetic genetic gained engineering, engineering the performance of Corynebacterium glutamicum bioreactor the amino amino acidand Ldecades to engineering, metabolic engineering terium glutamicum bioreactor producing the acid LMicrobial biocatalysis large metabolic momentum in recent decades due to genetic gained engineering, engineering compared the control performance of a real Corynebacand novel technologies (Bornscheuer et al. (2012)). Howterium glutamicum bioreactor producing the amino acid Lglutamate with these two models. In this paper, we looked and novel technologies (Bornscheuer et al. (2012)). Howwe acid looked glutamate with these these two models. models. In this this the paper, decades due to genetic engineering, engineering and technologies (Bornscheuer et al. (2012)). How- terium glutamicum bioreactor producing amino Lever,novel it is is still a challenge challenge to find find aametabolic reasonable trade-off glutamate with two In paper, we looked into model model definition, batch simulations, simulations, local elasticity ever, it still a to reasonable trade-off into definition, batch local elasticity and novel technologies (Bornscheuer et al. (2012)). However, it isbiomass, still a challenge to find a reasonable trade-off glutamate with these two models. In this paper, we looked between product formation and substrate coninto model definition, batch simulations, local elasticity analysis, parameter parameter estimation, estimation, continuous continuous optimal optimal operoperbetween biomass, product formation formation and substrate substrate concon- analysis, ever, it isbiomass, still a challenge to findneed a reasonable between product and definition, batch and simulations, local elasticity sumption. Fermentation processes fine-tuningtrade-off of con- into analysis, estimation, continuous optimal ation,model stepparameter response analysis, the performance performance ofopera PI PI sumption. Fermentation processes need fine-tuning of ation, step response analysis, and the of between biomass, product formation substrate sumption. Fermentation processes needand fine-tuning ofetconanalysis, estimation, optimal ditions and an adequate control strategy (Stanbury al. ation, stepparameter response analysis, andcontinuous the performance ofoperaathat PI controller with these two models. The results show ditions and an adequate control strategy (Stanbury et al. controller with theseanalysis, two models. models. The results show show that sumption. Fermentation processes need fine-tuning ofetconditions an adequate control strategy (Stanbury al. ation, response and the performance of athat PI (2013)).and For that, traditional bioreactor models consider controller with these two The results scalingstep of our our bioreactor model (dimensionless model) led (2013)). For that, traditional bioreactor models consider scaling of bioreactor model (dimensionless model) led ditions and an adequate control strategy (Stanbury et al. (2013)). For that, traditional bioreactor models consider controller with these two models. The results show that both: scaling of our bioreactor model (dimensionless model) led to aa reduction reduction in in the the parameters, parameters, yielded yielded aa more more accurate accurate both: to (2013)). For that, traditional bioreactor models consider scaling both: of our bioreactor model) to a reduction in the parameters, yielded a more accurate parameter estimation andmodel most (dimensionless importantly, resulted inledaa • A reactor balance that analyses mass and transport parameter estimation and most importantly, resulted in both: A reactor reactor balance balance that that analyses analyses mass mass and and transport transport to a reduction incontrol the parameters, yielded a faster more accurate parameter estimation and most importantly, resulted in a better feedback performance with biomass •• A phenomena. better feedback feedback control performance with faster faster biomass phenomena. parameter estimation and most importantly, resulted in a better control performance with biomass growth and substrate consumption than feedback control •• A reactor balance that analyses mass and transport phenomena. A set set of of kinetic kinetic equations equations that that take take into into account account mimi- growth feedback and substrate substrate consumption than feedback control A control performance withfeedback faster biomass growth consumption than control with the theand non-scaled model (NS model). model). phenomena. •• A set of kinetic equations that take intoinaccount mi- better croorganism growth and concentrations concentrations the specific specific with non-scaled model (NS croorganism growth and in the substratemodel consumption than feedback control with theand non-scaled (NS model). • A set of kinetic equations that take intoinaccount mi- growth croorganism growth and concentrations the specific media (Moser (2012)). media (Moser (2012)). croorganism growth and concentrations in the specific with the non-scaled model (NS model). media (Moser (2012)). The main main control objectives in fermentation fermentation processes processes are are mediacontrol (Moserobjectives (2012)). in The The main stable controloperation, objectives desired in fermentation processes are 2. MODEL MODEL to ensure growth rate and high 2. to ensure ensure stable operation, desired growth rate rate and high high The main stable control objectives in fermentation processes are 2. MODEL to operation, desired growth and productivity (Patel and Padhiyar (2017)). Previous studproductivity (Patel and Padhiyar Padhiyar (2017)). Previous stud2. MODEL to stable operation, desired growth rate and studhigh productivity (Patel and (2017)). Previous ies ensure have scrutinized scrutinized different reactor configurations and 2.1 Reactor Model ies have different reactor configurations and productivity (Patelamong and Padhiyar (2017)). Previous stud2.1 Reactor Reactor Model Model ies havestrategies, scrutinized different reactor configurations and 2.1 control other examples are the scientific control strategies, among otherreactor examples are the the scientific scientific ies have scrutinized different configurations and 2.1 Reactor Model control strategies, among other examples are works with real-time monitoring, simulation and control control works with real-time monitoring, simulation and control strategies, among other examples are the scientific The general general nonlinear nonlinear bioreactor bioreactor model model is: is: works with real-time monitoring, simulation and control in penicillin penicillin production production (Golabgir (Golabgir and and Herwig Herwig (2016); (2016); The in The general nonlinear bioreactor model is: works with real-time monitoring, simulation and control in penicillin production (Golabgir and Herwig (2016); Goldrick et et al. al. (2015)), (2015)), model model predictive predictive control control (Macharia (Macharia Goldrick x˙˙ = =bioreactor f (x, (x, u, u, d, d, model p, t) t) is: (1) in (Golabgir andcontrol Herwig (2016); x p, (1) Goldrick al.production (2015)), model predictive (Macharia andpenicillin Tay et (2013)), integration of process process engineering, fer- The general nonlinear x ˙ = ff (x, u, d, p, t) (1) and Tay (2013)), integration of engineering, fern k Goldrick et al. (2015)), model predictive control (Macharia and Tay (2013)), of process engineering, fer- Where x ∈ Rn is xthe mentation, enzymeintegration and metabolic metabolic engineering in ethanol ethanol of variables, u ˙ = fvector (x, u, d, t) (1) the vector of p, variables, u ∈ ∈ R Rk are are mentation, enzyme and engineering in Where x x ∈ ∈ R Rnn is and Tay (2013)), integration of process engineering, fer- Where is jj the vector of variables,p u∈ ∈RiiRiskk the are mentation, enzyme and metabolic engineering in ethanol optimization (Menon and Rao (2012)), and in baker’s yeast the inputs, d ∈ R are the disturbances, the inputs, ∈ optimizationenzyme (Menonand andmetabolic Rao (2012)), (2012)), and in in baker’s baker’s yeast Where n Rjj are the disturbances, p ∈ Rii is is the variables, u∈ ∈R (Figure Risk the are mentation, engineering inemployed ethanol x parameters ∈d the inputs, dR∈ R are the the optimization (Menon and Rao and yeast production (G´ eelinas (2014)). While engineers vector of andvector t is disturbances, theoftime. Our p model production (G´ linas (2014)). While engineers employed vector of parameters j and t is the time. Our model i(Figure inputs, done ∈ R are the p ∈ R (Figure is the optimization (Menon and Rao (2012)), and in such baker’s yeast the vector of parameters and(F tinis),disturbances, the time. Our content model production (G´ efields linas in (2014)). While engineers employed scaling in other chemical engineering as (Bird 1) contains input with sugar (S in ), contains one input (Ftinis),the with sugar content (Sin ), scaling in in other other fields in chemical engineering suchemployed as (Bird (Bird 1) vector ofoutput parameters and time. Our model (Figure production (G´eto linas (2014)). While engineers 1) contains one(Finput (Fin scaling fields in chemical engineering such as ), with sugar content (S(V in et al. (2007)) improve analysis however, none of the and one ). The model states are volume in in ), out et al. (2007)) to improve analysis however, none of the and one output (F ). The model states are volume (V ), out contains one (Finand scaling instudies otherto fields in chemical engineering such asof(Bird ), with sugar(P content (S(V ), et al. (2007)) analysis none the 1) and one (X), output (Finput model states are volume in ), out previous onimprove fermentation we however, mentioned considered biomass substrate (S) product ). The full list out ). The biomass (X), substrate (S) and product (P ). The full list previous studies on fermentation we mentioned considered et al. (2007)) toonimprove analysis however, none ofform the and one (X), output (F model states(P are).Table volume (Vlist ), previous studies fermentation we to mentioned considered out ). The biomass substrate (S) and product The full scaling their models, turning these dimensionless of variables and parameters are provided in 1. scaling their models, turning these to dimensionless form of variables and parameters are provided in Table 1. previous studies on fermentation we to mentioned considered biomass (X),and substrate (S) and product (P The full scaling their models, turning these form of variables parameters are provided in).Table 1. list dimensionless scaling their models, turning these to dimensionless variables and Ltd. parameters provided in Table 1. 2405-8963 © 2019, IFAC (International Federation of Automatic form Control) of Hosting by Elsevier All rights are reserved.
Copyright © 2019 237 Copyright © under 2019 IFAC IFAC 237 Control. Peer review responsibility of International Federation of Automatic Copyright © 2019 IFAC 237 10.1016/j.ifacol.2019.06.068 Copyright © 2019 IFAC 237
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1 1 RXS − RP S YXS YP S YP X 1 µX − µX =− YXS YP S YP S + YP X YXS µX =− YXS YP S µX (5) =− κ where κ is a parameter that combines in one expression the three yields YP S , YP X , YXS and reduces the number of parameters of the dynamic model. The definition of κ is similar to a previous report (Sun et al. (2011)). Table 2 summarizes the reactor and kinetic equations. RS = −
Fig. 1. Diagram of a continuous bioreactor. Nomenclature: composition set-point (Csp ), composition controller (CC), volume set-point (Vsp ), volume controller (VC). Table 1. Variable, units and description. Variable V Fin Fout S Sin S0 X X0 P P0 RX RP RS µm µ KS Kd κ η YP X YXS YP S ωc KcV KcX τIV τIX
Units m3 m3 · h−1 m3 · h−1 kg · m−3 kg · m−3 kg · m−3 kg · m−3 kg · m−3 kg · m−3 kg · m−3 kg · m−3 · h−1 kg · m−3 · h−1 kg · m−3 · h−1 h−1 h−1 kg · m−3 h−1 [-] [-] kg · kg −1 kg · kg −1 kg · kg −1 h−1 [-] [-] [-] [-]
Table 2. Summary of NS reactor model and its kinetic equations
Description Reactor volume Flow in the bioreactor Flow out the bioreactor Substrate concentration Substrate concentration inlet flow Substrate initial concentration Biomass concentration Biomass initial concentration Product concentration Product initial concentration Rate of biomass formation Rate of product formation Rate of substrate consumption Maximum specific growth rate Specific growth rate Monod growth constant Cell death rate constant Ratio of yields Toxic power Product from Biomass yield Biomass from Substrate yield Product from Substrate yield Cross over frequency Volume controller gain Biomass controller gain Volume controller integral time Biomass controller integral term
State V X S P
Reactor model V˙ = Fin − Fout
in X˙ = − XF + RX V (S −S)F S˙ = in V in + RS P˙ = − P Fin + RP
V
Kinetic equations S µ = Kµm+S (1 − P P )η S
max
RX = µX
RS =
YP S +YP X YXS YXS YP S
µX
RP = YP X µX
2.3 Dimensionless variables A dimensionless model permits effect comparison (Bird et al. (2007)), model reduction, and is a tool for adequate controller design (Skogestad and Postlethwaite (2007)). We present a set of normalization variables similar to the previous works of Ingham et al. (2008) and Sun et al. (2011). In our set of differential equations we have more parameters than the systems studied by Ingham et al. (2008) (i.e. product inhibition term, see equation (4)) and therefore our proposal of dimensionless variables (7) is particular to our system. The product inhibition term shows a better correlation to fermentation data from experiments of Khan et al. (2005); Sun et al. (2011). The general description of the dimensionless model is:
2.2 Model kinetics
x˙ = f (x , u , d , pq , t)
We used an unstructured model of C. glutamicum growth, which implies that we ignored variations of bacteria composition in response to environmental changes (Ingham et al. (2008); Esener et al. (1983)), described by the following equation (Dey and Pal (2013); Pal et al. (2016)): RX = µX (2) The specific growth rate has a Monod growth shape with a product inhibition term described by the following equation (Khan et al. (2005); Sun et al. (2011)): µm S P η (1 − ( ) ) (3) µ= KS + S Pmax We assumed that product formation in a cell culture is proportional to the growth-associated biomass term (Doran (2013)): P η µm S (1 − ( ) )X (4) RP = YP X µX = YP X KS + S Pmax Thus, product and biomass formation diminishes substrate concentration (Ingham et al. (2008); Villadsen et al. (2011)) in the following manner: 238
(6)
where the prime denotes the dimensionless variable in the system, x = [V , S , X , P ]T is the dimensionless vector of states, and pq denotes the vector of parameters of the scaled model. Different methods to normalize are available in literature (Bird et al. (2007)), but we proposed a scaling that takes into account the reactor configuration, the growth kinetics, the maximum attainable concentration, while it removes the yields from the dynamics: V S X V = , S = , X = V0 S0 S0 κ Ks Fin Fout Fin = , Fout = Ks = S0 V0 µ m V0 µ m P µ Rs P = , µ = Rs = YP X κS0 µm µ m S0 RX RP t , RP , t = RX = = (7) µ m S0 κ µ m Y P X S0 κ µm Substituting the reactor model and the kinetic equations (Table 2) with the expressions in (7), we obtained the
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dimensionless model in equations (8-12). Our scaling procedure reduces the number of parameters in the set of differential equations. S µ = (1 − P )η (8) KS + S − Fout (9) V˙ = Fin
X Fin + µ X X˙ = − V (S − S )Fin S˙ = in − µ X V P Fin P˙ = − + µ X V
(10) (11) (12)
2.4 Batch simulations To cover the whole space of initial conditions in the fermentation process, the original NS model from the previous reports (Sun et al. (2011); Khan et al. (2005), presented in Table 2) requires countless experiments. To account for that, we scaled the experimental data in order to compare different sources of information and different initial conditions. The procedure to couple the model with experimental data is: (1) Normalize the experimental data with the following equations: Sexp Pexp S = P = (13) S0 Pmax (2) Assume no inputs (Fin = 0, Fout = 0), integrate the differential expressions (10-12), and use the dimensionless definitions (7), to obtain the batch mass balance equations (Sun et al. (2011)): P˙ = X˙ = −S˙ X − X0 S − S0 P = =− (14) S0 Y P X κ S0 κ S0 (3) From equation (14), when the substrate concentration is zero (S = 0), we obtain the maximum product and biomass formation because there is no more sugar source to form product and/or biomass, and mathematically it allows to compute the yield YP X and κ: Pmax = S0 YP X κ
(15)
X0 ) Xmax = S0 κ(1 + S0 κ
(16)
Figure 2 presents data points taken from two different experimental conditions (Khan et al. (2005); Zhang et al. (1998)) and scaled to a dimensionless model using the equations in (7). The figure also shows the simulated dimensionless model of equations 9-12. Firstly, we verified that our dimensionless model manages to reproduce two independent experimental data. Secondly, it enables easy comparison of biomass growth from experiments with different initial conditions, that otherwise will need an elaborated analysis to compare. 3. LOCAL ELASTICITY ANALYSIS The local elasticity analysis gives information about function response with perturbations in the parameters (Tor239
Fig. 2. Growth of biomass from simulations of our dimensionless model (equations 8-12), using the parameters provided by Khan et al. (2005) and Zhang et al. (1998). The dimensionless model is able to reproduce the results of both experiments. torelli and Michaleris (1994)). Unlike the sensitivity analysis, the elasticity function is normalized, a feature that allows comparison between different models (Benton and Grant (1999)). In our model, concentrations can have values close to zero, producing numerical instability of the function. Therefore, dividing the sensitivity function of each state by its respective constant term (i.e. xiref = S0 , Pmax , Xmax ) in eq (17) removes this numerical issue: pj ∂xi pj ∆xi ≈ , xi = X, S, P (17) Eij = xiref ∂pj xiref ∆pj We used the infinity norm of the elasticity function to obtain information of the maximum response in a state with a parameter perturbation: Eij (t) ∞ = max(|Eij (t)|) i = X, S, P (18) Figure 3 presents the elasticity function norms. The sensitivity of the parameters Ks , η, µmax is similar for both models (gray bar), and Ks has the lowest elasticity function in both models. The dimensionless model reduces the three yields (YXS , YP S , YP X ) to one parameter κ (blue bar) defined in (5). As we expected, the biomass concentration (X) is highly sensitive to perturbations in YP X , but interestingly, the biomass X, the substrate S and the product concentrations P are all sensitive to Pmax . More importantly, the same variables in the dimensionless model are only moderately sensitive to κ ( Eiκ (t) ∞ ≈ 0.3), making the model and the corresponding control application more robust to errors in parameter estimation. 4. PARAMETER ESTIMATION To estimate the model parameters we used a nonlinear least square optimization technique to minimize the error between the experimental data and the solution of the differential equations (8-12), in the following manner: ˆ (19) e(p) = X(p) − Xexp ˆ are the simulated values of the biomass with Where X a parameter vector, Xexp is the measured biomass from
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Table 3. Parameter estimation results Parameter KS [kg · m−3 ] η[−] µm [h−1 ] κ[kg · kg −1 ] YXS [kg · kg −1 ] YP S [kg · kg −1 ] YP X [kg · kg −1 ]
Fig. 3. The NS model (top) is highly sensitive (expressed by elasticity norm, Equation 17) to four parameters while the dimensionless model (bottom) only to one. The NS model is particularly sensitive to the maximum product concentration Pmax and to the yields while the dimensionless is moderately sensitive only to κ. Parameters (Khan et al. (2005)) are in Table 3. . experiments, and e is the estimated error. Table 3 presents the parameters previously reported (Khan et al. (2005)), and our estimated results for the dimensionless and NS models. For the NS model the parameter that is associated with Monod growth constant (KS ) is over 25 times (24.3/0.8) the reported value (Khan et al. (2005)), but the same parameter in the dimensionless model has a similar order of magnitude (0.8 vs. 0.98). Despite the 25 fold deviation in KS the model simulation are in agreement with experimental data, since the set of differential equations are insensitive to estimation error of Ks . Figure 4 compares the response of both models with the estimated parameters, and both models can describe the biomass concentration. We computed the relative error norm obtaining values in the same order of magnitude, for the dimensionless model (L2 = 0.3) and for the NS model (L2 = 0.2). Moreover, the dimensionless model reproduces, with less parameters, the substrate concentration more accurately (L2 = 1) than the NS model (L2 = 8).
Khan et al. (2005) 0.8 1 0.21 0.0746 0.149 0.48 3.216
NS 24.3 1.09 0.32 0.0859 0.09 0.74 0.71
Dimensionless 0.98 1.2 0.23 0.0754 -
Fig. 4. The NS model and dimensionless model reproduce previous biomass concentration as a function of time. However, the parameter estimation of the dimensionless model gives a better representation of the substrate profile. Literature data from (Khan et al. (2005)) . J(x , u , d ) min u
s.t.x˙ = f (x , u , d , pq ) = 0
(20)
where J is steady state objective function and u = [Fin Sin ]. Substrate consumption is a key performance indicator for biochemical industry because it translates directly to cost (e.g. sugar feedstock) (Villadsen et al. (2011)). Previous work (Zhao and Skogestad (1997)) focused on biomass productivity, which we can define in the following manner: J1 = −X Fin (21) However, this objective function did not consider the sugar consumption. We accounted for this important key performance indicator by defining the next objective function: J2 = −(X − S )Fin . (22) The objective function J2 takes into consideration the difference between biomass productivity and substrate consumption. Figure 5 shows J1 and J2 as a function of: a) the inlet flow Fin , and b) the inlet substrate concentration. We solved the objective function minimization problems (J1 , J2 ) with a nonlinear least square minimization technique. Figure 5c, depicts the contour plot of J2, and together with d) show that the surface of J2 is convex, similar plots for J1 can be obtained. The results of the minimum of the objective functions, the manipulated variables (Fin , Sin ), and the states (X , S , P ) are in table 4.
5. STEADY STATE BEHAVIOR
6. STEP RESPONSE ANALYSIS
We defined the steady state of the biochemical process by the following set of equations:
We conducted a step response analysis in order to understand the system behavior with changes of manipulated
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states. The flow Fin contains substrate in a concentration Sin that activates biomass growth and the reaction kinetics. Based on the previous observations, we decided to con trol biomass concentration with Fin and the volume with Fout . Moreover, in-situ and accurate biomass estimations are desirable, because the state of the art measurement with chromatographic methods for substrates (sugars) and products (amino-acids or organic acids) require long and costly sampling time (ca 20-30 min).
7. PI TUNNING We chose to control the system with a proportionalintegral (PI) controller. To tune the PI controller, we used the internal mode controller (IMC), a common tunning technique. IMC can handle common challenges such as decouplers, dead-time, and anti-reset windup (Rivera et al. (1986)). We used Skogestad IMC rules (SIMC) for PI controllers tuning (Grimholt and Skogestad (2018)) around the set-point of J2 (Table 4). We used a smooth control approach to define the level controller tuning parameters, and a first order plus time delay transfer function to model the response of biomass concentration with Fin step (Skogestad (2006)): Fig. 5. Model based optimization functions with: a) Con stant Sin = 1 b) Constant Fin = 0.1 c) Countour and d) Surface plot of J2 as a function of Fin and Sin . Parameters from (Khan et al. (2005)) and in Table 3.
∆F V τ ≥ 4θ = 4 IV ∆Vmax Fin 1 τ1 τIX = min[τ1 , 4(τc + θ)] KcX = k τc + θ With the controller equations as: KcV ≥
(23)
t KcX −X )+ − X )dτ ( (Xsp τIX 0 t KcV = KcV (Vsp − V ) + ( − V )dτ (24) (Vsp Fout τIV 0 Fin
Fig. 6. Step response of the outlet affected only the volume (left column), whereas step response of the inlet affected the dynamics of all the states (right column). a) The dynamics of the states, and b) the input step, Fout = 0.1 at t=10 and Fin = 0.1 with a sugar inlet concentration Sin = 1 at t=10. Parameters from (Khan et al. (2005)) and in Table 3. Table 4. J1 and J2 function values with V’= 1. Function −X F −(X − S )F
Value 0.5 0.176
F 0.5 0.38
Sin 1 0.627
X 0.494 0.537
S 0.5 0.089
= KcX (Xsp
The dimensionless model resulted (Figure 7) in a lower integral of absolute error (IAE) and a lower response time compared to the NS model (2.5 vs. 4, respectively). The feedback response time was over 15% faster (Figure 7a), both indicating a significantly better controller performance. It is important to note that achieving accurate biomass and product formations are important to the fermentation industry, and faster biomass production has great economical consequences (Villadsen et al. (2011)). Further work of the present paper include additional states that represent substrate of different composition (e.g. glucose and other sugar compounds), consumption of these substrates, a controllability analysis, and set-point disturbance rejection analysis. Table 5. PI tunning Model Dimensionless NS
P 0.494 0.537
variables. The simulations of the open loop system (Figure 2) indicate that the microbial growth is internally stable. Figure 6 shows that a change in the manipulated variable Fout only affects the volume V , while Fin affects all the 241
KcX -1.1 -1.42
τIX 5 5
KcV -5 -5
τIV 10.3 10.3
8. CONCLUSIONS We showed in this article that scaling of a nonlinear fermentation process led to parameter reduction, an im-
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Fig. 7. Comparison between the control performance of the dimensionless model (D, solid curve) vs. the nonscaled (NS, dashed curve) model. a) The control set point for the biomass was 0.537 (see Table 4), achieved a response time of 2.5 by the PI controller for the scaled model, compared to 4 for the NS model. b) The overall control performance, measured by integral of absolute error (IAE) was significantly better for the dimensionless model. Parameters from Table 3 and controller values from Table 5. proved parameter estimation routine and significantly better control performance of a PI controller. Our results suggest that scaling of such models for control purposes can greatly improve the productivity and yields, two of the most important key performance indicators in industrial microbial biocatalysis. ACKNOWLEDGEMENTS The authors would like to thank prof S. Skogestad for his discussions and comments. REFERENCES Tim G. Benton and Alastair Grant. Elasticity analysis as an important tool in evolutionary and population ecology. Trends in Ecology & Evolution, 14(12):467 – 471, 1999. R Byron Bird, Warren E Stewart, and Edwin N Lightfoot. Transport Phenomena. John Wiley & Sons, 2007. UT Bornscheuer, GW Huisman, RJ Kazlauskas, S Lutz, JC Moore, and K Robins. Engineering the third wave of biocatalysis. Nature, 485(7397):185, 2012. P Dey and P Pal. Modelling and simulation of continuous l (+) lactic acid production from sugarcane juice in membrane integrated hybrid-reactor system. Biochemical engineering journal, 79:15–24, 2013. Pauline M. Doran. Chapter 12 - homogeneous reactions. In Bioprocess Engineering Principles (Second Edition), pages 599 – 703. Academic Press, London, 2013. A. A. Esener, J. A. Roels, and N. W. F. Kossen. Theory and applications of unstructured growth models: Kinetic and energetic aspects. Biotechnology and Bioengineering, 25(12):2803–2841, 1983. Pierre G´elinas. Fermentation control in baker’s yeast production: mapping patents. Comprehensive Reviews in Food Science and Food Safety, 13(6):1141–1164, 2014. Aydin Golabgir and Christoph Herwig. Combining mechanistic modeling and raman spectroscopy for real-time monitoring of fed-batch penicillin production. Chemie Ingenieur Technik, 88(6):764–776, 2016. Stephen Goldrick, Andrei Tefan, David Lovett, Gary Montague, and Barry Lennox. The development of 242
an industrial-scale fed-batch fermentation simulation. Journal of biotechnology, 193:70–82, 2015. Chriss Grimholt and Sigurd Skogestad. Optimal pi and pid control of first-order plus delay processes and evaluation of the original and improved simc rules. Journal of Process Control, 70:36 – 46, 2018. John Ingham, Irving J Dunn, Elmar Heinzle, Jir´ı E Prenosil, and Jonathan B Snape. Chemical engineering dynamics: an introduction to modelling and computer simulation. John Wiley & Sons, 2008. Noor Salam Khan, Indra Mani Mishra, RP Singh, and Basheshwer Prasad. Modeling the growth of Corynebacterium glutamicum under product inhibition in lglutamic acid fermentation. Biochemical Engineering Journal, 25(2):173–178, 2005. Maina A Macharia and Michael E Tay. Model predictive control of fermentation in biofuel production, October 29 2013. US Patent 8,571,689. Vishnu Menon and Mala Rao. Trends in bioconversion of lignocellulose: Biofuels, platform chemicals biorefinery concept. Progress in Energy and Combustion Science, 38(4):522 – 550, 2012. Anton Moser. Bioprocess technology: kinetics and reactors. Springer Science & Business Media, 2012. Parimal Pal, Ramesh Kumar, D VikramaChakravarthi, and Sankha Chakrabortty. Modeling and simulation of continuous production of l (+) glutamic acid in a membrane-integrated bioreactor. Biochemical Engineering Journal, 106:68–86, 2016. Narendra Patel and Nitin Padhiyar. Multi-objective dynamic optimization study of fed-batch bio-reactor. Chemical Engineering Research and Design, 119:160– 170, 2017. Daniel E. Rivera, Manfred Morari, and Sigurd Skogestad. Internal model control: Pid controller design. Industrial & Engineering Chemistry Process Design and Development, 25(1):252–265, 1986. Sigurd Skogestad. Tuning for smooth pid control with acceptable disturbance rejection. Industrial & Engineering Chemistry Research, 45(23):7817–7822, 2006. Sigurd Skogestad and Ian Postlethwaite. Multivariable feedback control: analysis and design, volume 2. Wiley New York, 2007. Peter F Stanbury, Allan Whitaker, and Stephen J Hall. Principles of fermentation technology. Elsevier, 2013. Kaibiao Sun, Andrzej Kasperski, Yuan Tian, and Lansun Chen. Modelling of the Corynebacterium glutamicum biosynthesis under aerobic fermentation conditions. Chemical engineering science, 66(18):4101–4110, 2011. D. A. Tortorelli and P. Michaleris. Design sensitivity analysis: Overview and review. Inverse Problems in Engineering, 1(1):71–105, 1994. John Villadsen, Jens Nielsen, and Eli Keshavarz-Moore. Bioreaction Engineering Principles. Springer, 2011. Xue-Wu Zhang, Tao Sun, Zhong-Yi Sun, Xin Liu, and DeXiang Gu. Time-dependent kinetic models for glutamic acid fermentation. Enzyme and microbial technology, 22 (3):205–209, 1998. Ying Zhao and Sigurd Skogestad. Comparison of various control configurations for continuous bioreactors. Industrial & Engineering Chemistry Research, 36(3):697–705, 1997.
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Bioreactor Scaling Enhances Feedback Bioreactor Bioreactor Scaling Scaling Enhances Enhances Feedback Feedback Control Of Concentration, Rates, and Bioreactor Scaling Enhances Feedback Concentration, Rates, Control Of Control Of Concentration, Rates, and and Yields. Control Of Concentration, Rates, and Yields. Yields. Yields. Pedro A. Lira Parada, Even Pettersen, Nadav Bar ∗∗
Pedro A. Lira Parada, Even Pettersen, Nadav Bar ∗∗ Pedro A. Lira Parada, Even Pettersen, Nadav Bar ∗ ∗ Pedro A. Lira Parada, Even Pettersen, Nadav Bar of Chemical Engineering, Norwegian University of ∗ Department Department of Chemical Engineering, Norwegian University of ∗ ∗ Department of Chemical N-7491 Engineering, Norwegian University of Science and Technology, Trondheim, Norway (e-mail: ∗ Science and Technology, N-7491 Trondheim, Norway (e-mail: Department of Chemical Engineering, Norwegian University of Science and Technology, N-7491 Trondheim, Norway (e-mail: [email protected], [email protected], [email protected]) [email protected], Science and Technology,[email protected], N-7491 Trondheim, [email protected]) Norway (e-mail: [email protected], [email protected], [email protected]) [email protected], [email protected], [email protected]) Abstract: Fermentation Fermentation processes processes are are biologically biologically complex, complex, multiple-input, multiple-input, multiple-output multiple-output Abstract: Abstract: processes biologically multiple-output systems, and andFermentation as such, such, selection selection of an anare adequate controlcomplex, strategymultiple-input, is often often an an elaborate elaborate task. The The systems, as of adequate control strategy is task. Abstract: processes are biologically multiple-input, multiple-output systems, andFermentation as such, selection of anare adequate controlcomplex, strategy is the often an elaborate task. The process and disturbance dynamics easily oversimplified, and controller design in such process and dynamics easily oversimplified, and controller design in systems, anddisturbance as such, selection of anare control strategy is the often an elaborate task. The process and disturbance dynamics areadequate easily oversimplified, and the controller design in such such cases seldom seldom yield adequate performance. In the the present study, we developed a non-structured, non-structured, cases yield adequate performance. In present study, we developed a process and disturbance dynamics are easily oversimplified, and the controller design in such cases seldom yield adequate performance. In the present study, we developed a non-structured, dimensionless model model of of microbial microbial fermentation fermentation production. production. Analysis Analysis and and simulations simulations show show that that dimensionless cases yield adequate performance. In theproduction. present study, we developed a non-structured, dimensionless model ofmodel microbial fermentation Analysis simulations show that scalingseldom of aa bioreactor bioreactor facilitates parameter estimation, yield aand straightforward objective scaling of model facilitates parameter estimation, yield a straightforward objective dimensionless modeland ofmodel microbial fermentation production. Analysis simulations show that scaling a bioreactor facilitates parameter estimation, yield aand straightforward objective functionofdefinition, definition, promote control decisions with better performance. The results results suggest function and promote control decisions with better performance. The suggest scaling of a bioreactor model facilitates parameter estimation, yield a straightforward objective function definition, and promote control decisions with better performance. The results suggest that scaling scaling of of a bioreactor bioreactor model model enhances enhances the the performance performance of of feedback control, control, producing producing that function definition, andrates, promote control decisions with better performance. The results suggest that scaling of a a bioreactor model enhances the performance of feedback feedback control, producing higher concentrations, and yields, three of the most important parameters in industrial higher concentrations, rates, and yields, three of the most important parameters in industrial that scaling of a bioreactor model enhances the performance of feedback control, producing higher concentrations, rates, and yields, three of the most important parameters in industrial microbial biocatalysis. microbial biocatalysis. higher concentrations, microbial biocatalysis. rates, and yields, three of the most important parameters in industrial © 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. microbial biocatalysis. Keywords: Bioprocess Bioprocess control, control, Industrial Industrial biotechnology, biotechnology, fed-batch, fed-batch, feedback control, control, parameter Keywords: Keywords: Bioprocess control, Industrial biotechnology, fed-batch, feedback feedback control, parameter parameter estimation, fermentation process, dynamic model. estimation, fermentation process, dynamic model. Keywords: Bioprocess control, Industrial fed-batch, feedback control, parameter estimation, fermentation process, dynamicbiotechnology, model. estimation, fermentation process, dynamic model. 1. INTRODUCTION INTRODUCTION and and examine examine the the resulting resulting control control performance. performance. Herein, Herein, 1. 1. INTRODUCTION and examine the resulting control non-structured performance. Herein, we turned a previously published, model we turned a previously published, non-structured model 1. INTRODUCTION and examine the resulting control performance. Herein, we turned a previously published, non-structured model of a microbial bioreactor to a dimensionless model, and of aaturned microbial bioreactorpublished, to aa dimensionless dimensionless model, and we a previously non-structured model Microbial biocatalysis biocatalysis gained gained large large momentum momentum in in recent recent of microbial bioreactor to model, and compared the control performance of a real CorynebacMicrobial compared the control control performance of aa real real model, CorynebacMicrobial biocatalysis large metabolic momentum in recent compared of a microbial bioreactor to aproducing dimensionless decades due due to genetic genetic gained engineering, engineering the performance of Corynebacterium glutamicum bioreactor the amino amino acidand Ldecades to engineering, metabolic engineering terium glutamicum bioreactor producing the acid LMicrobial biocatalysis large metabolic momentum in recent decades due to genetic gained engineering, engineering compared the control performance of a real Corynebacand novel technologies (Bornscheuer et al. (2012)). Howterium glutamicum bioreactor producing the amino acid Lglutamate with these two models. In this paper, we looked and novel technologies (Bornscheuer et al. (2012)). Howwe acid looked glutamate with these these two models. models. In this this the paper, decades due to genetic engineering, engineering and technologies (Bornscheuer et al. (2012)). How- terium glutamicum bioreactor producing amino Lever,novel it is is still a challenge challenge to find find aametabolic reasonable trade-off glutamate with two In paper, we looked into model model definition, batch simulations, simulations, local elasticity ever, it still a to reasonable trade-off into definition, batch local elasticity and novel technologies (Bornscheuer et al. (2012)). However, it isbiomass, still a challenge to find a reasonable trade-off glutamate with these two models. In this paper, we looked between product formation and substrate coninto model definition, batch simulations, local elasticity analysis, parameter parameter estimation, estimation, continuous continuous optimal optimal operoperbetween biomass, product formation formation and substrate substrate concon- analysis, ever, it isbiomass, still a challenge to findneed a reasonable between product and definition, batch and simulations, local elasticity sumption. Fermentation processes fine-tuningtrade-off of con- into analysis, estimation, continuous optimal ation,model stepparameter response analysis, the performance performance ofopera PI PI sumption. Fermentation processes need fine-tuning of ation, step response analysis, and the of between biomass, product formation substrate sumption. Fermentation processes needand fine-tuning ofetconanalysis, estimation, optimal ditions and an adequate control strategy (Stanbury al. ation, stepparameter response analysis, andcontinuous the performance ofoperaathat PI controller with these two models. The results show ditions and an adequate control strategy (Stanbury et al. controller with theseanalysis, two models. models. The results show show that sumption. Fermentation processes need fine-tuning ofetconditions an adequate control strategy (Stanbury al. ation, response and the performance of athat PI (2013)).and For that, traditional bioreactor models consider controller with these two The results scalingstep of our our bioreactor model (dimensionless model) led (2013)). For that, traditional bioreactor models consider scaling of bioreactor model (dimensionless model) led ditions and an adequate control strategy (Stanbury et al. (2013)). For that, traditional bioreactor models consider controller with these two models. The results show that both: scaling of our bioreactor model (dimensionless model) led to aa reduction reduction in in the the parameters, parameters, yielded yielded aa more more accurate accurate both: to (2013)). For that, traditional bioreactor models consider scaling both: of our bioreactor model) to a reduction in the parameters, yielded a more accurate parameter estimation andmodel most (dimensionless importantly, resulted inledaa • A reactor balance that analyses mass and transport parameter estimation and most importantly, resulted in both: A reactor reactor balance balance that that analyses analyses mass mass and and transport transport to a reduction incontrol the parameters, yielded a faster more accurate parameter estimation and most importantly, resulted in a better feedback performance with biomass •• A phenomena. better feedback feedback control performance with faster faster biomass phenomena. parameter estimation and most importantly, resulted in a better control performance with biomass growth and substrate consumption than feedback control •• A reactor balance that analyses mass and transport phenomena. A set set of of kinetic kinetic equations equations that that take take into into account account mimi- growth feedback and substrate substrate consumption than feedback control A control performance withfeedback faster biomass growth consumption than control with the theand non-scaled model (NS model). model). phenomena. •• A set of kinetic equations that take intoinaccount mi- better croorganism growth and concentrations concentrations the specific specific with non-scaled model (NS croorganism growth and in the substratemodel consumption than feedback control with theand non-scaled (NS model). • A set of kinetic equations that take intoinaccount mi- growth croorganism growth and concentrations the specific media (Moser (2012)). media (Moser (2012)). croorganism growth and concentrations in the specific with the non-scaled model (NS model). media (Moser (2012)). The main main control objectives in fermentation fermentation processes processes are are mediacontrol (Moserobjectives (2012)). in The The main stable controloperation, objectives desired in fermentation processes are 2. MODEL MODEL to ensure growth rate and high 2. to ensure ensure stable operation, desired growth rate rate and high high The main stable control objectives in fermentation processes are 2. MODEL to operation, desired growth and productivity (Patel and Padhiyar (2017)). Previous studproductivity (Patel and Padhiyar Padhiyar (2017)). Previous stud2. MODEL to stable operation, desired growth rate and studhigh productivity (Patel and (2017)). Previous ies ensure have scrutinized scrutinized different reactor configurations and 2.1 Reactor Model ies have different reactor configurations and productivity (Patelamong and Padhiyar (2017)). Previous stud2.1 Reactor Reactor Model Model ies havestrategies, scrutinized different reactor configurations and 2.1 control other examples are the scientific control strategies, among otherreactor examples are the the scientific scientific ies have scrutinized different configurations and 2.1 Reactor Model control strategies, among other examples are works with real-time monitoring, simulation and control control works with real-time monitoring, simulation and control strategies, among other examples are the scientific The general general nonlinear nonlinear bioreactor bioreactor model model is: is: works with real-time monitoring, simulation and control in penicillin penicillin production production (Golabgir (Golabgir and and Herwig Herwig (2016); (2016); The in The general nonlinear bioreactor model is: works with real-time monitoring, simulation and control in penicillin production (Golabgir and Herwig (2016); Goldrick et et al. al. (2015)), (2015)), model model predictive predictive control control (Macharia (Macharia Goldrick x˙˙ = =bioreactor f (x, (x, u, u, d, d, model p, t) t) is: (1) in (Golabgir andcontrol Herwig (2016); x p, (1) Goldrick al.production (2015)), model predictive (Macharia andpenicillin Tay et (2013)), integration of process process engineering, fer- The general nonlinear x ˙ = ff (x, u, d, p, t) (1) and Tay (2013)), integration of engineering, fern k Goldrick et al. (2015)), model predictive control (Macharia and Tay (2013)), of process engineering, fer- Where x ∈ Rn is xthe mentation, enzymeintegration and metabolic metabolic engineering in ethanol ethanol of variables, u ˙ = fvector (x, u, d, t) (1) the vector of p, variables, u ∈ ∈ R Rk are are mentation, enzyme and engineering in Where x x ∈ ∈ R Rnn is and Tay (2013)), integration of process engineering, fer- Where is jj the vector of variables,p u∈ ∈RiiRiskk the are mentation, enzyme and metabolic engineering in ethanol optimization (Menon and Rao (2012)), and in baker’s yeast the inputs, d ∈ R are the disturbances, the inputs, ∈ optimizationenzyme (Menonand andmetabolic Rao (2012)), (2012)), and in in baker’s baker’s yeast Where n Rjj are the disturbances, p ∈ Rii is is the variables, u∈ ∈R (Figure Risk the are mentation, engineering inemployed ethanol x parameters ∈d the inputs, dR∈ R are the the optimization (Menon and Rao and yeast production (G´ eelinas (2014)). While engineers vector of andvector t is disturbances, theoftime. Our p model production (G´ linas (2014)). While engineers employed vector of parameters j and t is the time. Our model i(Figure inputs, done ∈ R are the p ∈ R (Figure is the optimization (Menon and Rao (2012)), and in such baker’s yeast the vector of parameters and(F tinis),disturbances, the time. Our content model production (G´ efields linas in (2014)). While engineers employed scaling in other chemical engineering as (Bird 1) contains input with sugar (S in ), contains one input (Ftinis),the with sugar content (Sin ), scaling in in other other fields in chemical engineering suchemployed as (Bird (Bird 1) vector ofoutput parameters and time. Our model (Figure production (G´eto linas (2014)). While engineers 1) contains one(Finput (Fin scaling fields in chemical engineering such as ), with sugar content (S(V in et al. (2007)) improve analysis however, none of the and one ). The model states are volume in in ), out et al. (2007)) to improve analysis however, none of the and one output (F ). The model states are volume (V ), out contains one (Finand scaling instudies otherto fields in chemical engineering such asof(Bird ), with sugar(P content (S(V ), et al. (2007)) analysis none the 1) and one (X), output (Finput model states are volume in ), out previous onimprove fermentation we however, mentioned considered biomass substrate (S) product ). The full list out ). The biomass (X), substrate (S) and product (P ). The full list previous studies on fermentation we mentioned considered et al. (2007)) toonimprove analysis however, none ofform the and one (X), output (F model states(P are).Table volume (Vlist ), previous studies fermentation we to mentioned considered out ). The biomass substrate (S) and product The full scaling their models, turning these dimensionless of variables and parameters are provided in 1. scaling their models, turning these to dimensionless form of variables and parameters are provided in Table 1. previous studies on fermentation we to mentioned considered biomass (X),and substrate (S) and product (P The full scaling their models, turning these form of variables parameters are provided in).Table 1. list dimensionless scaling their models, turning these to dimensionless variables and Ltd. parameters provided in Table 1. 2405-8963 © 2019, IFAC (International Federation of Automatic form Control) of Hosting by Elsevier All rights are reserved.
Copyright © 2019 237 Copyright © under 2019 IFAC IFAC 237 Control. Peer review responsibility of International Federation of Automatic Copyright © 2019 IFAC 237 10.1016/j.ifacol.2019.06.068 Copyright © 2019 IFAC 237
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1 1 RXS − RP S YXS YP S YP X 1 µX − µX =− YXS YP S YP S + YP X YXS µX =− YXS YP S µX (5) =− κ where κ is a parameter that combines in one expression the three yields YP S , YP X , YXS and reduces the number of parameters of the dynamic model. The definition of κ is similar to a previous report (Sun et al. (2011)). Table 2 summarizes the reactor and kinetic equations. RS = −
Fig. 1. Diagram of a continuous bioreactor. Nomenclature: composition set-point (Csp ), composition controller (CC), volume set-point (Vsp ), volume controller (VC). Table 1. Variable, units and description. Variable V Fin Fout S Sin S0 X X0 P P0 RX RP RS µm µ KS Kd κ η YP X YXS YP S ωc KcV KcX τIV τIX
Units m3 m3 · h−1 m3 · h−1 kg · m−3 kg · m−3 kg · m−3 kg · m−3 kg · m−3 kg · m−3 kg · m−3 kg · m−3 · h−1 kg · m−3 · h−1 kg · m−3 · h−1 h−1 h−1 kg · m−3 h−1 [-] [-] kg · kg −1 kg · kg −1 kg · kg −1 h−1 [-] [-] [-] [-]
Table 2. Summary of NS reactor model and its kinetic equations
Description Reactor volume Flow in the bioreactor Flow out the bioreactor Substrate concentration Substrate concentration inlet flow Substrate initial concentration Biomass concentration Biomass initial concentration Product concentration Product initial concentration Rate of biomass formation Rate of product formation Rate of substrate consumption Maximum specific growth rate Specific growth rate Monod growth constant Cell death rate constant Ratio of yields Toxic power Product from Biomass yield Biomass from Substrate yield Product from Substrate yield Cross over frequency Volume controller gain Biomass controller gain Volume controller integral time Biomass controller integral term
State V X S P
Reactor model V˙ = Fin − Fout
in X˙ = − XF + RX V (S −S)F S˙ = in V in + RS P˙ = − P Fin + RP
V
Kinetic equations S µ = Kµm+S (1 − P P )η S
max
RX = µX
RS =
YP S +YP X YXS YXS YP S
µX
RP = YP X µX
2.3 Dimensionless variables A dimensionless model permits effect comparison (Bird et al. (2007)), model reduction, and is a tool for adequate controller design (Skogestad and Postlethwaite (2007)). We present a set of normalization variables similar to the previous works of Ingham et al. (2008) and Sun et al. (2011). In our set of differential equations we have more parameters than the systems studied by Ingham et al. (2008) (i.e. product inhibition term, see equation (4)) and therefore our proposal of dimensionless variables (7) is particular to our system. The product inhibition term shows a better correlation to fermentation data from experiments of Khan et al. (2005); Sun et al. (2011). The general description of the dimensionless model is:
2.2 Model kinetics
x˙ = f (x , u , d , pq , t)
We used an unstructured model of C. glutamicum growth, which implies that we ignored variations of bacteria composition in response to environmental changes (Ingham et al. (2008); Esener et al. (1983)), described by the following equation (Dey and Pal (2013); Pal et al. (2016)): RX = µX (2) The specific growth rate has a Monod growth shape with a product inhibition term described by the following equation (Khan et al. (2005); Sun et al. (2011)): µm S P η (1 − ( ) ) (3) µ= KS + S Pmax We assumed that product formation in a cell culture is proportional to the growth-associated biomass term (Doran (2013)): P η µm S (1 − ( ) )X (4) RP = YP X µX = YP X KS + S Pmax Thus, product and biomass formation diminishes substrate concentration (Ingham et al. (2008); Villadsen et al. (2011)) in the following manner: 238
(6)
where the prime denotes the dimensionless variable in the system, x = [V , S , X , P ]T is the dimensionless vector of states, and pq denotes the vector of parameters of the scaled model. Different methods to normalize are available in literature (Bird et al. (2007)), but we proposed a scaling that takes into account the reactor configuration, the growth kinetics, the maximum attainable concentration, while it removes the yields from the dynamics: V S X V = , S = , X = V0 S0 S0 κ Ks Fin Fout Fin = , Fout = Ks = S0 V0 µ m V0 µ m P µ Rs P = , µ = Rs = YP X κS0 µm µ m S0 RX RP t , RP , t = RX = = (7) µ m S0 κ µ m Y P X S0 κ µm Substituting the reactor model and the kinetic equations (Table 2) with the expressions in (7), we obtained the
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239
dimensionless model in equations (8-12). Our scaling procedure reduces the number of parameters in the set of differential equations. S µ = (1 − P )η (8) KS + S − Fout (9) V˙ = Fin
X Fin + µ X X˙ = − V (S − S )Fin S˙ = in − µ X V P Fin P˙ = − + µ X V
(10) (11) (12)
2.4 Batch simulations To cover the whole space of initial conditions in the fermentation process, the original NS model from the previous reports (Sun et al. (2011); Khan et al. (2005), presented in Table 2) requires countless experiments. To account for that, we scaled the experimental data in order to compare different sources of information and different initial conditions. The procedure to couple the model with experimental data is: (1) Normalize the experimental data with the following equations: Sexp Pexp S = P = (13) S0 Pmax (2) Assume no inputs (Fin = 0, Fout = 0), integrate the differential expressions (10-12), and use the dimensionless definitions (7), to obtain the batch mass balance equations (Sun et al. (2011)): P˙ = X˙ = −S˙ X − X0 S − S0 P = =− (14) S0 Y P X κ S0 κ S0 (3) From equation (14), when the substrate concentration is zero (S = 0), we obtain the maximum product and biomass formation because there is no more sugar source to form product and/or biomass, and mathematically it allows to compute the yield YP X and κ: Pmax = S0 YP X κ
(15)
X0 ) Xmax = S0 κ(1 + S0 κ
(16)
Figure 2 presents data points taken from two different experimental conditions (Khan et al. (2005); Zhang et al. (1998)) and scaled to a dimensionless model using the equations in (7). The figure also shows the simulated dimensionless model of equations 9-12. Firstly, we verified that our dimensionless model manages to reproduce two independent experimental data. Secondly, it enables easy comparison of biomass growth from experiments with different initial conditions, that otherwise will need an elaborated analysis to compare. 3. LOCAL ELASTICITY ANALYSIS The local elasticity analysis gives information about function response with perturbations in the parameters (Tor239
Fig. 2. Growth of biomass from simulations of our dimensionless model (equations 8-12), using the parameters provided by Khan et al. (2005) and Zhang et al. (1998). The dimensionless model is able to reproduce the results of both experiments. torelli and Michaleris (1994)). Unlike the sensitivity analysis, the elasticity function is normalized, a feature that allows comparison between different models (Benton and Grant (1999)). In our model, concentrations can have values close to zero, producing numerical instability of the function. Therefore, dividing the sensitivity function of each state by its respective constant term (i.e. xiref = S0 , Pmax , Xmax ) in eq (17) removes this numerical issue: pj ∂xi pj ∆xi ≈ , xi = X, S, P (17) Eij = xiref ∂pj xiref ∆pj We used the infinity norm of the elasticity function to obtain information of the maximum response in a state with a parameter perturbation: Eij (t) ∞ = max(|Eij (t)|) i = X, S, P (18) Figure 3 presents the elasticity function norms. The sensitivity of the parameters Ks , η, µmax is similar for both models (gray bar), and Ks has the lowest elasticity function in both models. The dimensionless model reduces the three yields (YXS , YP S , YP X ) to one parameter κ (blue bar) defined in (5). As we expected, the biomass concentration (X) is highly sensitive to perturbations in YP X , but interestingly, the biomass X, the substrate S and the product concentrations P are all sensitive to Pmax . More importantly, the same variables in the dimensionless model are only moderately sensitive to κ ( Eiκ (t) ∞ ≈ 0.3), making the model and the corresponding control application more robust to errors in parameter estimation. 4. PARAMETER ESTIMATION To estimate the model parameters we used a nonlinear least square optimization technique to minimize the error between the experimental data and the solution of the differential equations (8-12), in the following manner: ˆ (19) e(p) = X(p) − Xexp ˆ are the simulated values of the biomass with Where X a parameter vector, Xexp is the measured biomass from
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Table 3. Parameter estimation results Parameter KS [kg · m−3 ] η[−] µm [h−1 ] κ[kg · kg −1 ] YXS [kg · kg −1 ] YP S [kg · kg −1 ] YP X [kg · kg −1 ]
Fig. 3. The NS model (top) is highly sensitive (expressed by elasticity norm, Equation 17) to four parameters while the dimensionless model (bottom) only to one. The NS model is particularly sensitive to the maximum product concentration Pmax and to the yields while the dimensionless is moderately sensitive only to κ. Parameters (Khan et al. (2005)) are in Table 3. . experiments, and e is the estimated error. Table 3 presents the parameters previously reported (Khan et al. (2005)), and our estimated results for the dimensionless and NS models. For the NS model the parameter that is associated with Monod growth constant (KS ) is over 25 times (24.3/0.8) the reported value (Khan et al. (2005)), but the same parameter in the dimensionless model has a similar order of magnitude (0.8 vs. 0.98). Despite the 25 fold deviation in KS the model simulation are in agreement with experimental data, since the set of differential equations are insensitive to estimation error of Ks . Figure 4 compares the response of both models with the estimated parameters, and both models can describe the biomass concentration. We computed the relative error norm obtaining values in the same order of magnitude, for the dimensionless model (L2 = 0.3) and for the NS model (L2 = 0.2). Moreover, the dimensionless model reproduces, with less parameters, the substrate concentration more accurately (L2 = 1) than the NS model (L2 = 8).
Khan et al. (2005) 0.8 1 0.21 0.0746 0.149 0.48 3.216
NS 24.3 1.09 0.32 0.0859 0.09 0.74 0.71
Dimensionless 0.98 1.2 0.23 0.0754 -
Fig. 4. The NS model and dimensionless model reproduce previous biomass concentration as a function of time. However, the parameter estimation of the dimensionless model gives a better representation of the substrate profile. Literature data from (Khan et al. (2005)) . J(x , u , d ) min u
s.t.x˙ = f (x , u , d , pq ) = 0
(20)
where J is steady state objective function and u = [Fin Sin ]. Substrate consumption is a key performance indicator for biochemical industry because it translates directly to cost (e.g. sugar feedstock) (Villadsen et al. (2011)). Previous work (Zhao and Skogestad (1997)) focused on biomass productivity, which we can define in the following manner: J1 = −X Fin (21) However, this objective function did not consider the sugar consumption. We accounted for this important key performance indicator by defining the next objective function: J2 = −(X − S )Fin . (22) The objective function J2 takes into consideration the difference between biomass productivity and substrate consumption. Figure 5 shows J1 and J2 as a function of: a) the inlet flow Fin , and b) the inlet substrate concentration. We solved the objective function minimization problems (J1 , J2 ) with a nonlinear least square minimization technique. Figure 5c, depicts the contour plot of J2, and together with d) show that the surface of J2 is convex, similar plots for J1 can be obtained. The results of the minimum of the objective functions, the manipulated variables (Fin , Sin ), and the states (X , S , P ) are in table 4.
5. STEADY STATE BEHAVIOR
6. STEP RESPONSE ANALYSIS
We defined the steady state of the biochemical process by the following set of equations:
We conducted a step response analysis in order to understand the system behavior with changes of manipulated
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states. The flow Fin contains substrate in a concentration Sin that activates biomass growth and the reaction kinetics. Based on the previous observations, we decided to con trol biomass concentration with Fin and the volume with Fout . Moreover, in-situ and accurate biomass estimations are desirable, because the state of the art measurement with chromatographic methods for substrates (sugars) and products (amino-acids or organic acids) require long and costly sampling time (ca 20-30 min).
7. PI TUNNING We chose to control the system with a proportionalintegral (PI) controller. To tune the PI controller, we used the internal mode controller (IMC), a common tunning technique. IMC can handle common challenges such as decouplers, dead-time, and anti-reset windup (Rivera et al. (1986)). We used Skogestad IMC rules (SIMC) for PI controllers tuning (Grimholt and Skogestad (2018)) around the set-point of J2 (Table 4). We used a smooth control approach to define the level controller tuning parameters, and a first order plus time delay transfer function to model the response of biomass concentration with Fin step (Skogestad (2006)): Fig. 5. Model based optimization functions with: a) Con stant Sin = 1 b) Constant Fin = 0.1 c) Countour and d) Surface plot of J2 as a function of Fin and Sin . Parameters from (Khan et al. (2005)) and in Table 3.
∆F V τ ≥ 4θ = 4 IV ∆Vmax Fin 1 τ1 τIX = min[τ1 , 4(τc + θ)] KcX = k τc + θ With the controller equations as: KcV ≥
(23)
t KcX −X )+ − X )dτ ( (Xsp τIX 0 t KcV = KcV (Vsp − V ) + ( − V )dτ (24) (Vsp Fout τIV 0 Fin
Fig. 6. Step response of the outlet affected only the volume (left column), whereas step response of the inlet affected the dynamics of all the states (right column). a) The dynamics of the states, and b) the input step, Fout = 0.1 at t=10 and Fin = 0.1 with a sugar inlet concentration Sin = 1 at t=10. Parameters from (Khan et al. (2005)) and in Table 3. Table 4. J1 and J2 function values with V’= 1. Function −X F −(X − S )F
Value 0.5 0.176
F 0.5 0.38
Sin 1 0.627
X 0.494 0.537
S 0.5 0.089
= KcX (Xsp
The dimensionless model resulted (Figure 7) in a lower integral of absolute error (IAE) and a lower response time compared to the NS model (2.5 vs. 4, respectively). The feedback response time was over 15% faster (Figure 7a), both indicating a significantly better controller performance. It is important to note that achieving accurate biomass and product formations are important to the fermentation industry, and faster biomass production has great economical consequences (Villadsen et al. (2011)). Further work of the present paper include additional states that represent substrate of different composition (e.g. glucose and other sugar compounds), consumption of these substrates, a controllability analysis, and set-point disturbance rejection analysis. Table 5. PI tunning Model Dimensionless NS
P 0.494 0.537
variables. The simulations of the open loop system (Figure 2) indicate that the microbial growth is internally stable. Figure 6 shows that a change in the manipulated variable Fout only affects the volume V , while Fin affects all the 241
KcX -1.1 -1.42
τIX 5 5
KcV -5 -5
τIV 10.3 10.3
8. CONCLUSIONS We showed in this article that scaling of a nonlinear fermentation process led to parameter reduction, an im-
2019 IFAC DYCOPS 242 Pedro A. Lira Parada et al. / IFAC PapersOnLine 52-1 (2019) 237–242 Florianópolis - SC, Brazil, April 23-26, 2019
Fig. 7. Comparison between the control performance of the dimensionless model (D, solid curve) vs. the nonscaled (NS, dashed curve) model. a) The control set point for the biomass was 0.537 (see Table 4), achieved a response time of 2.5 by the PI controller for the scaled model, compared to 4 for the NS model. b) The overall control performance, measured by integral of absolute error (IAE) was significantly better for the dimensionless model. Parameters from Table 3 and controller values from Table 5. proved parameter estimation routine and significantly better control performance of a PI controller. Our results suggest that scaling of such models for control purposes can greatly improve the productivity and yields, two of the most important key performance indicators in industrial microbial biocatalysis. ACKNOWLEDGEMENTS The authors would like to thank prof S. Skogestad for his discussions and comments. REFERENCES Tim G. Benton and Alastair Grant. Elasticity analysis as an important tool in evolutionary and population ecology. Trends in Ecology & Evolution, 14(12):467 – 471, 1999. R Byron Bird, Warren E Stewart, and Edwin N Lightfoot. Transport Phenomena. John Wiley & Sons, 2007. UT Bornscheuer, GW Huisman, RJ Kazlauskas, S Lutz, JC Moore, and K Robins. Engineering the third wave of biocatalysis. Nature, 485(7397):185, 2012. P Dey and P Pal. Modelling and simulation of continuous l (+) lactic acid production from sugarcane juice in membrane integrated hybrid-reactor system. Biochemical engineering journal, 79:15–24, 2013. Pauline M. Doran. Chapter 12 - homogeneous reactions. In Bioprocess Engineering Principles (Second Edition), pages 599 – 703. Academic Press, London, 2013. A. A. Esener, J. A. Roels, and N. W. F. Kossen. Theory and applications of unstructured growth models: Kinetic and energetic aspects. Biotechnology and Bioengineering, 25(12):2803–2841, 1983. Pierre G´elinas. Fermentation control in baker’s yeast production: mapping patents. Comprehensive Reviews in Food Science and Food Safety, 13(6):1141–1164, 2014. Aydin Golabgir and Christoph Herwig. Combining mechanistic modeling and raman spectroscopy for real-time monitoring of fed-batch penicillin production. Chemie Ingenieur Technik, 88(6):764–776, 2016. Stephen Goldrick, Andrei Tefan, David Lovett, Gary Montague, and Barry Lennox. The development of 242
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