optimal control strategies for simultaneous saccharification and fermentation of starch

Process Biochemistry 36 (2001) 713–722

Optimal control strategies for simultaneous saccharification andfermentation of starch

Sudip Roy, Ravindra D. Gudi *, K.V. Venkatesh, Sunil S. ShahDepartment of Chemical Engineering, Indian Institute of Technology, Mumbai 400 076, India

Received 28 August 2000; received in revised form 7 September 2000; accepted 7 November 2000

Abstract

Design and analysis of optimal control strategies for three types of inhibitory fed-batch bioprocesses have been discussed. Theseare simple saccharification (SS) of starch to glucose, simple fermentation (SF) of derived glucose to lactic acid (LA) andsimultaneous saccharification and fermentation (SSF) of starch to LA. Various optimal feeding strategies have been investigatedfor the SSF process by manipulating starch addition rates. To avoid the complexity of solving a singular problem, the starchaddition rates are expressed in terms of the broth volume, which is used as a control variable. The optimization strategy is thussolved in a nonsingular framework. Experimental studies carried out using the results of the optimization demonstrated theaccuracy and utility of the approach. An increase of 20% in lactate productivity was obtained by operating the SSF process ina fed-batch mode. The focus of all the optimization studies has been to improve the performance of the SSF process. Optimalcontrol of starch additions in the fed-batch process gave improved performance of the SSF process. © 2001 Elsevier Science Ltd.All rights reserved.

Keywords: Optimal control; Simultaneous saccharification and fermentation; Fed-batch fermentation; Multiple feed optimization; End pointconstraint

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1. Introduction

Bioprocesses are popularly operated in a fed-batchmode to regulate substrate or product concentrations inthe reactor. The design of optimal control strategies forfed-batch fermentation process is a growing field inbiotechnology. Optimal fed-batch operation also provesits superiority when there are demands for higherproduct purities, productivity rates and yield. Thus, inthe majority of cases, designing an optimal controlstrategy is a problem of determining a substrate feedingstrategy which maximizes or minimizes a given perfor-mance index subject to several physical constraintsimposed due to the nature of the problem. The opti-mization thus requires a reliable dynamic model of theprocess, which typically includes rate equations forbiomass formation, substrate consumption and productformation.

Several approaches have been taken up by re-searchers to tackle this kind of optimization problem.Some of the earlier methods were based on Green’stheorem [1], Pontryagin’s maximum principle [2,3], andorthogonal collocation techniques. More recently theuse of a differential evolution technique [4], which usesgenetic algorithm and evolution strategies, have alsobeen proposed. Although there are various methodsavailable for solving optimization problems, each hascertain prerequisites and shortcomings.

Fed-batch optimization problems for fermentationprocesses are often formulated with substrate feed rateas the control variable, which appears linearly in thestate equations as well as in the Hamiltonian. This maylead to a singular control problem, which is extremelydifficult to solve when four or more mass balanceequations describe the system. The algorithm comesacross a more complex situation when one or morestate constraints bound the problem. Therefore, choiceof control variable(s) is a critical step towards makingthe problem computationally feasible. Yamane et al. [5]

* Corresponding author. Tel.: +91-22-5782545; fax: +91-22-5783480.

E-mail address: [email protected] (R.D. Gudi).

0032-9592/01/$ - see front matter © 2001 Elsevier Science Ltd. All rights reserved.

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S. Roy et al. / Process Biochemistry 36 (2001) 713–722714

considered specific growth rate as the control variableand converted the problem into a nonsingular one. Thesuccess of this approach largely depended on the func-tional behaviour of the specific growth rate. Ohno et al.[6] used proportionality between specific substrate con-sumption rate and specific growth rate to reduce theproblem dimension to consist of only two variables, viz.the substrate concentration and culture volume. Theyused Green’s theorem to find the optimal feeding pro-gram. Cazzador [7] also employed Green’s theorem todetermine the optimal feeding policies that maximizebiomass production and also included time so that atmaximum time, the solution lies on or near the singulararc along which the substrate concentration is main-tained constant. Hong [8] presented the optimal feedingstrategies for substrate inhibited growth kinetics andshowed the strong dependence of the initial conditionsof the reactor to the final product concentration. Animportant contribution has been made by Weigand etal. [9], wherein they used Pontryagin’s maximum princi-ple in a singular control framework in which the timeto switch from the maximum rate to the singular feedrate was iterated.

Guthke and Knorre [10] used substrate concentrationas the control variable but ignored the physical con-straints on feeding rate and volume. However,Stephanopoulos and San [11] followed the same ap-proach with constraints imposed on the control vari-able and state variables and solved for the feedconcentration instead of substrate concentration in thereactor.

In this context, the contribution of Modak et al. [12]is very important. They have presented the generalcomputational algorithms and hence deduced feedingstrategies for different characteristic behaviour of thespecific growth rate and product formation rate.Modak and Lim [13] proposed solutions for feed rate asa nonlinear function of the state variables and coupledan open loop solution procedure in a feed back loop.Subsequent work by the same authors [14] also pro-posed transformation of the state vectors with culturevolume as the control variable, reformulation of theproblem in the framework of nonsingular control, andderivation of a reliable solution for the substrate feedrate from volume profiles. The technique that theypropose has been quite useful for problems describedby a large number of dynamic mass balance equations.

The use of an orthogonal collocation technique insolving optimal control problems for fed-batch fermen-tation has been successfully demonstrated by Kurtanjek[15] thus avoiding the difficulty of using split boundaryconditions in the optimization problem. The controlvariables were determined by optimization of theHamiltonian at the collocation points with the DFPmethod.

On the process side, there are various industriallyimportant fermentation products for which fed-batchtechniques are adopted. The products include antibi-otics, enzymes, organic acids, microbial cells, etc.Parulekar and Lim [16] have documented some ofthese. Among the various products, lactic acid (LA) isa very important commodity chemical, which findsmajor uses in food, drug, pharmaceutical, agro andcosmetic industries. Typical raw materials for LA pro-duction through fermentation are molasses (glucose)and whey (lactose). These sources are not abundantlyfound and are expensive. Recently, starch as a biore-source has been used to produce fermentative products.Glucose can also be enzymically derived from sacchar-ification of starch but glucose (the product) inhibits theenzyme. Anuradha et al. [17] has shown experimentallythat starch itself can be used as a raw material for LAproduction by coupling saccharification (of starch) andfermentation (of derived glucose) in a process termed assimultaneous saccharification and fermentation (SSF).This process is much more economical not only interms of saving overall fermentation time but also inreducing reactor volume. Thus the SSF process usesstarch as the primary raw material, which is cheap,abundant and present in variety of agricultural re-sources, e.g., potato tubers and pearl tapioca. A directbenefit of the SSF process is a decrease in inhibitioncaused by glucose accumulation leading to an increasein the LA productivity. An unstructured kinetic modelof the SSF process has also been developed [17]. It hasalso been experimentally demonstrated that intermittentaddition of starch during the fermentation (instead ofadding all the starch at the start of a batch) improvesthe productivity further [18].

The primary objective of this work is to analyse andestablish optimal conditions of operation for SSF pro-cess using optimal control theory. While preliminaryresults [18] do establish experimentally the possibility ofimproved productivity, in this work we attempt toachieve higher productivities and explore the potentialfor optimization more fully through the application ofoptimal control.

2. Process model description

To obtain a clear insight of the three processes, i.e.,enzymic saccharification of starch, microbial fermenta-tion of glucose and SSF of starch to LA, the kineticmodels are described independently for each of theprocesses as follows:

2.1. Saccharification

The kinetic model, which represents enzymic conver-sion of starch by glucoamylase allowing competitiveinhibitions from glucose, is given by

S. Roy et al. / Process Biochemistry 36 (2001) 713–722 715

dGdt

=6mS

Km(1+G/KG)+S, (1)

where KG is the term, which contributes to the glucoseinhibition on saccharification rate. The starch concen-tration S can be expressed in terms of the glucoseconcentration G using stoichiometry, since on completehydrolysis 1 g of starch gives 1.11 g of glucose. Thus,the rate expression becomes

dGdt

=6mS0−G/1.11

Km(1+G/KG)+ (S0−G/1.11). (2)

2.2. Fermentation

The growth kinetics of Lactobacillus delbrueckii(which is the most preferred species for fermentation ofglucose to LA) is given by

RX=dXdt

=mX, (3)

where the specific growth rate m is given by

m=mm exp(−KLP)G

Ks+G+G2/KI

. (4)

The above expression includes both the substrate inhi-bition term KI and product inhibition term KL. LAformation is modelled taking into account both thegrowth-associated and non-growth associated productformation and is given by the Leudeking–Piret rela-tionship as:

RP=dPdt

=adXdt

+bX. (5)

Similarly, the rate expression for glucose consumptionincorporates both the growth associated and mainte-nance terms as

RG=dGdt

= −�

adXdt

+bX�

. (6)

2.3. Simultaneous saccharification and fermentation

Simultaneous saccharification and fermentation ismodelled by integrating the independent kinetic modelsof saccharification and fermentation (described earlier)together. The only distinction made is in specifying theglucose derived from starch by saccharification (de-noted by G*) and the actual glucose concentration thataccumulates in the system and inhibits saccharification(denoted by G), separately. Thus, the rate of saccharifi-cation in SSF is given by

RG �=dG�

dt=6m

S0−G�/1.11Km(1+G/KG)+ (S0−G�/1.11)

, (7)

where the effect of accumulation of fermentationproduct on saccharification rate can be expressed as6m=6m

0 exp(−kP)n. The values of the two inhibition

parameters for LA are k=0.02 and n=0.04 [17]. Thus,the rate of glucose accumulation is given by

dGdt

=RG �−RG

=6mS0−G�/1.11

Km(1+G/KG)+ (S0−G�/1.11)

−�

adXdt

+bX�

. (8)

The growth rate of L. delbrueckii (RX) and the rate ofLA production (RP) remain the same as that given byEqs. (3) and (5).

3. Optimal control problem formulation

As shown in Refs. [14,15], design of an optimalcontrol policy using nutrient feed as the control vari-able leads to a singular control problem which is rela-tively difficult to solve especially for a complex processmodel. To overcome that problem variable transforma-tions are carried out [14] which result in a nonsingularproblem involving culture volume as the control vari-able. The same approach has been used in this work toreformulate the control problems for each of the threeprocesses, viz. saccharification, fermentation and SSF.

3.1. Saccharification

With starch feed rate as the control variable, the stateequations for the optimal control problem are obtainedby dynamic mass balance on both starch and glucose as

dSdt

= −1

1.11dGdt

+Fin

V(Sin−S)

= −6m1

1.11S

Km(1+G/KG)+S+

Fin

V(Sin−S), (9)

dGdt

=6mS0−G/1.11

Km(1+G/KG)+ (S0−G/1.11)−

Fin

VG. (10)

Now the problem is reformulated by making variabletransformation in the following way:

dSdt

= −6m1

1.11S

Km(1+G/KG)+S+

1V

dVdt

(Sin−S)

(refer Eq. (9))

or

VdSdt

+SdVdt

=Sin

dVdt

−6m1

1.11SV

Km(1+G/KG)+S

or

d(SV)dt

=d(SinV)

dt−6m

11.11

SVKm(1+G/KG)+S

or


d[(S−Sin)V ]dt

= −6m1

1.11SV

Km(1+G/KG)+S.

Now, if the terms [(S−Sin)V ] and (GV) are taken as newstate variables x1, and x2, respectively, then the abovestate equations can be written as

dx1

dt= −6m

(x1+SinV)VKm(V+x2/KG)+ (x1−SinV)

. (11)

Similarly, glucose balance equation (Eq. (10)) can betransformed into another state equation in the followingway:

dGdt

=6mS0−G/1.11

Km(1+G/KG)+ (S0−G/1.11)−

1V

dVdt

G

or

VdGdt

+GdVdt

=6mVS0−G/1.11

Km(1+G/KG)+ (S0−G/1.11)

or

d(GV)dt

=6m VS0V−GV/1.11

Km(V+GV/KG)+ (S0V−GV/1.11)

or

dx2

dt=6m

S0V−x2/1.11Km(V+x2/KG)+ (S0V−x2/1.11)

. (12)

Here the two state variables are related stoichiometricallyas x1= (S0−Sin)V−x2/1.11.

3.2. Fermentation

The dynamic mass balance equations for the fed-batchfermentation with glucose being the only nutrient forfeeding can be written as

dXdt

= −Fin

VX+mX (for biomass), (13)

dGdt

=Fin

V(Gin−G)+

�a

dXdt

+bX�

(for glucose),

(14)

dPdt

= −FinP

V+�

adXdt

+bX�

(for LA). (15)

Now, if the term (XV) is taken as the new state variablex1, [(G−Gin)V ] as x2, and (PV) as x3 then on makingtransformations of the above three equations with cul-ture volume as the control variable, the state equationsare given as

dx1

dt=mmV exp(−KLx3/V)

(x2+GinV)x1

KsV2+ (x2+GinV)V+ (x2+GinV)2/KI

, (16)

dx2

dt= −

�a

dx1

dt+bx1

�, (17)

dx3

dt= −

�a

dx1

dt+bx1

�. (18)

The performance objective, which is to be maximized forthe above case may be either the productivity or the finalconcentration of LA, or simply the yield of LA. Theoptimal control problem is solved for the optimal culturevolume profile V�(t), which maximizes the performanceobjective, subjected to the following constraint on thecontrol vector:

V: E0.


If a feed primarily containing starch (with someamount of glucose being present due to starch autoclav-ing) is considered being added to an otherwise batchprocess, the kinetic equations for SSF become

dXdt

= −Fin

VX+mX (for biomass), (19)

dSdt

=Fin

V(Sin−S)−6m

S/1.11Km(1+G/KG)+S

(for starch), (20)

dGdt

=Fin

V(Gin−G)−

�a

dXdt

+bX�

+6mS

Km(1+G/KG)+S(for glucose), (21)

dPdt

= −FinP

V+�

adXdt

+bX�

(for LA). (22)

After variable transformations the above set of equationsgives the following state equations with four state vari-ables as x1 (=XV), x2 (= [(S−Sin)V), x3 (= [(G−Gin)V) and x4 (=PV).

dx1

dt=mmV exp(−KLx4/V)

(x3+GinV)x1

KsV2+ (x3+GinV)V+ (x3+GinV)2/KI

, (23)

dx2

dt= −6m

11.11

(x2+SinV)VKm[V+ (x3+GinV)/KG ]+ (x2+SinV)

,

(24)

dx3

dt= −1.11

dx2

dt−�

adx1

dt+bx1

�, (25)

dx4

dt=�

adx1

dt+bx1

�. (26)

The performance objective is set to be the same as thatfor fermentation, i.e., either of the final concentration orthe productivity or the yield of LA.

Optimization procedure. In each of the above cases, aperformance objective J can be maximized in the spaceof the control variables. This is done by formulating theHamiltonian H for the problem as


H=J+ %N

i=1

li

#xi

#t, (27)

while the dynamics of the co-state variables(l1,l2,…,lN) are written as

dli

dt= −

#H#xi

, i=1,…,N (28)

The problem may be solved for maximizing either ofthe multiple objectives as stated above. Using the max-imum principle and the initial condition for the statevariables, the final condition for the co-state variablesare obtained by the following condition:

li(tf)=#J#xi(tf)

, i=1,…,N (29)

where J is the performance objective. The open loopcontrol policy (i.e. the optimal time profile for theculture volume) can then be obtained using a variety ofmethods. We propose the use of the control vectoriteration procedure to solve the problem and includethe steps in Appendix A [19] for a systematic outline ofthe strategy).

4. Materials

4.1.1. MicroorganismHomofermentative L. delbrueckii (NCIM 2365), a

strain producing mainly L-(+ )-lactic acid, was ob-tained from National Collection of Industrial Microor-ganisms, National Chemical Laboratory, Pune, India.Cultures were maintained at 4°C on slants containing3% glucose along with other essential nutrients. LAbacteria were revived by two successive propagations at45°C for 12–18 h in the modified MRS broth.

4.1.2. EnzymesCommercial a-amylase and glucoamylase from Novo

Nordisk, Denmark, were employed in the presentstudy.

4.1.3. Fermentation mediumShake flask experiments were carried out using MRS

broth containing yeast extract (0.5%), urea (0.5%),dipotassium hydrogen phosphate (0.1%), sodium ace-tate (0.5%), magnesium sulphate (0.3%), and varyingglucose concentrations.

4.1.4. Simultaneous saccharification and fermentationmedium

The SSF medium consisted of yeast extract (0.5%),

urea (0.5%), dipotassium hydrogen phosphate (0.1%),potassium phosphate (0.1%), sodium acetate (0.5%),magnesium sulphate (0.3%), and varying quantities (30,100 and 250 g/l of liquefied analytical grade potatostarch (containing 20% moisture).

4.2. Methods

The experimental methods followed for saccharifica-tion of starch, fermentation of glucose and batch SSFof starch with L. delbrueckii are stated in Ref. [17].

4.2.1. Fed-batch SSF of starch with Lactobacillusdelbrueckii

The medium containing starch was autoclaved alongwith a-amylase (0.15 ml/100 g starch) at 121°C and 15psi for 15 min. The starch concentration was keptinitially at 30 g/l and then fresh starch feed was addedas per the rate shown in Fig. 1 using a graded peristalticpump (Neolab graded pump – 0.1 ml to 100 ml/h).Yeast extract and urea were proportionately increasedto avoid nitrogen limitation. The medium was inocu-lated with the second generation of L. delbrueckii culti-vated at 45°C for about 14–18 h. The inoculum sizewas fixed at 10% and SSF was carried out after adding0.15% (based on dry starch) of glucoamylase.

4.3. Sample analysis

Samples withdrawn from either fermentation or SSFmedium were centrifuged at 10 000 rpm for 15 min andthe supernatants frozen for further analysis. Lactateand glucose concentrations were determined by theLDH and o-Toluidine method, respectively [17].

Fig. 1. Starch addition rates in the experiments carried out onfed-batch SSF.


5. Results and discussion

This work analyses various approaches to increasingLA productivity/concentration. It explores using anoptimal control technique to determine whether a scopeof optimization exists for each of the steps viz., simplesaccharification (SS) of starch, fermentation of glucoseand SSF of starch to glucose. The discussion beginswith numerical results involving SS of starch to glucose.Simple fermentation (SF) of the derived glucose to LAis considered next. Results of optimization for the SSFof starch to LA is discussed along with experimentalresults obtained for the optimal feed strategy evolved.In an earlier work from this group, the parameters forthe SSF process were evaluated from experimental dataand a dynamic model was proposed [17]. The samedynamic model has been used in all the simulationspresented here.

5.1. Simple saccharification

The optimum conditions for saccharification atwhich the rate is maximal is 60°C at 4 pH [18]. Underthese conditions the values of the saccharificationparameters are given as 6m=68 g/l h, Km=96.5 g/l,and KG=33 g/l (Eqs. (9) and (10)). Since autoclavingstarch solutions, especially in the presence of a-amylase,produces some glucose, a representative value of 15%of the starch concentrations was assumed as the initialconcentration of glucose. Under these operating condi-tions, the potential of fed-batch operation was exploredfor two initial starch concentrations (S0), viz., 30 g/l(considered as low), and 100 g/l considered as high).

The optimization study revealed that neither the pro-ductivity nor the final concentration of LA improves infed-batch operation when compared with the batchoperation. The reasons for this can be explained asfollows. The potential for higher productivity of glu-cose in fed batch exists because of decreased inhibitionof glucose due to dilution. However, in this case, thestarch that is added in the fed-batch mode typicallycontains glucose that is formed in the autoclaving stepby enzymic hydrolysis by a-amylase. Therefore, thedilution effect on the glucose concentrations due toaddition in the fed-batch mode is compensated for bythe addition of glucose formed during the autoclavingof the starch solution. Thus, no potential for fed-batchoperation exists. Fig. 2 shows the batch concentrationprofile for saccharification beginning with a initialstarch concentration of 100 g/l. As in the case of abatch process, the initial saccharification rate was fastbut decreased rapidly thereafter because of glucoseinhibition. The batch profile showed a stoichiometricyield of glucose with a productivity of 3.68 g/l h and afinal glucose concentration of 113 g/l.

Fig. 2. Starch and glucose profiles for the saccharification step (batchsimulation).

5.2. Simple fermentation

The parameters chosen for the optimization of fer-mentation of glucose to LA using L. delbrueckii are thefermentation time (t), initial cell mass concentration(X0), feed glucose concentration (Gin) and the initialvolume of batch (V0). The optimization problem wassolved independently for each of these variables whilekeeping the others constant. Table 1 shows the valuesof the different objective functions for fermentationstarting with an initial concentration of 30 g/l glucoseand using a feed containing 200 g/l of glucose in thefed-batch mode. The table summarizes the results of theoptimization at three different fermentation times. Forthis study the values of all other parameters are keptconstant at X0=0.5 g/l and V0=0.1 l. Batch simula-tion with 200 g/l initial glucose shows a batch time of70 h at the end of which the LA concentration is 163 g/land hence a productivity of 2.28 g/l h. It is clear fromTable 1 that fed-batch operation for 40 h gives im-proved productivity and yield when compared with the200 g/l of batch operation. Thus fed-batch operationdefinitely saves fermentation time but at the cost of LAconcentration. Figs. 3 and 4 show that the fed-batch

Table 1Results of fed-batch fermentation of 30 g/l of glucose at various timewith X0=0.5 g/l, V0=0.1 l and Gin=200 g/l

Time, tf (h) Productivity (g/lEnd lactate Yieldconcentration (g/l) h)

15080 1.8750.81760 0.818 2.0120

2.940 116 0.819


Fig. 3. Concentration profiles for fed-batch fermentation starting withan initial concentration of 30 g/l of glucose and fed batch with 200 g/lof glucose.

stay at small values for a longer time. This can beexplained by noting that at this stage, there is alreadysome LA production in the broth, which inhibitsbiomass growth rates and hence the product evolutionrates. Thus in keeping with the optimizer objective ofminimizing inhibition by glucose and LA and towardsmaximizing final LA concentrations, the optimizer pre-dicts that fresh glucose additions be done so as to haveoptimal evolution and concentration of the LA at theend of the batch. The time gap between depletion ofglucose and addition of fresh glucose however reducesthe overall productivity, and hence a fed-batch mode ofoperation of 60 h was considered optimal.

It should be noted here that since the objective is tomaintain higher LA concentration with improved pro-ductivity and yield, the glucose concentration in thefeed (Gin) is kept as high as possible. The dilutedglucose on the other hand produces no improvement interms of concentration/productivity. Batch fermenta-tion of 200 g/l of glucose produces LA having a finalconcentration of 163 g/l whereas fermentation of 30 g/lof glucose fed with 200 g/l of glucose gives LA concen-tration of 116 g/l. Increasing the initial volume (V0) ofthe fed-batch operation can make up for this fall inconcentration. Instead of starting with 0.1 l, a start-upvolume of 1 l increases the LA concentration to 137 g/lwith a productivity of 2.83. With higher initial glucoseconcentration (G0=100 g/l), neither the productivitynor the concentration was found to give any improve-ment in the fed-batch operation. This is due to highglucose inhibition initially and then higher LA inhibi-tion towards the end of the fermentation.

Thus, a low initial concentration of glucose solutionfollowed by a fed-batch mode with concentrated glu-cose (200 g/l) improves the productivity of the batch bysaving the fermentation time. However, high final con-centrations of LA, seen in the batch mode of operation,cannot be attained here.


It is clear from the results of saccharification andfermentation that in order to build up high productconcentration it is first required to start with low sub-strate (starch) concentration and then starch should befed at higher concentrations. But before going into thefed-batch optimization for SSF, the results of batchmode operation for the SSF at different initial concen-trations of starch are shown in Table 2. It can be seenthat the yield is very low at high initial starch concen-tration (S0); this is due to high-glucose inhibition onbiomass growth and enzymic saccharification rates dueto which the overall SSF rate becomes small. Since thevalue of KI (the glucose inhibition constant) for saccha-rification is 33 g/l, there is substantial glucose inhibitionon the saccharification in SSF at higher initial starch

concentration profile is essentially a combination of twoconsecutive batches, one of 30 g/l and another of 125g/l. From the feed profiles shown in Fig. 4, it can beseen that the optimization suggests that a fresh input ofglucose be given as a pulse at the instant the glucoseconcentration depletes to a small value. Thus, a fed-batch operation similar to two successive batch opera-tions is predicted. With increasing fed-batch time to 60h or more, the optimizer suggests a time gap betweenthese two successive batches, i.e., glucose is allowed to

Fig. 4. Glucose feeding rate for the concentration profiles shown inFig. 3.


Table 2Batch results of SSF for various initial starch concentrations (S0) starting with X0=0.5 g/l

End lactate concentration, P(tf) Lactate productivity, P(tf)/(tf−t0) (g/l h)Initial starch concentration, S0 Lactate yield based on(g/l) starch(g/l)

1.325 0.530 40.280.597460.83 1.01100

200 0.6911139.47 1.29

concentrations. Therefore, the advantage of SSF interms of rate is partly lost at higher initial starchconcentrations. This indicates the utility of operatingthe SSF process in fed-batch mode.

The fed-batch optimal control problem is thussolved with a low initial starch concentration (S0=30g/l) and a high-feed starch concentration (Sin=200g/l) with some glucose being available in the feed(Gin=15% of Sin) due to autoclaving. The fed-batchoptimization problem is solved for different time in-tervals and the results are shown in Table 3. Thetable shows that fed-batch operation improves uponthe batch performance (shown in Table 2) not only interms of concentration but also in terms of LA yieldand productivity. The reason for this is the reductionof glucose inhibition rates by an optimal addition ofstarch thus not allowing the glucose to build up toinhibitory levels in the process (i.e. always maintain-ing at less than 33 g/l). The optimal profile for starchaddition is shown in Fig. 5. From the addition profi-les, it can be seen that the optimizer predicts a sub-stantial starch addition at the beginning and thensuggests only smaller addition rates towards the endso as to prevent the glucose levels from falling tozero. This ensures that only those glucose levels thatare necessary to maintain LA production and sustainthe biomass growth are permitted. The fed-batch con-centration profiles are shown in Fig. 6. It can be seenthat the maximum glucose concentration for the opti-mal profile was only 25 g/l, which is far lower thanthat obtained in the batch. Thus, the optimal controlstrategy helps in determining the best nutrient addi-tion profiles that maximizes the SSF rates resulting inhigher productivity of LA.

The fed-batch feed (starch) profile predicted by theoptimal control strategy was implemented in the fer-mentation runs carried out as outlined in Section 4.While the exact optimized feed profile predicted asseen in Fig. 5 was not implementable, the approxi-mated feeding profile as shown in Fig. 1 was used.Fig. 7 shows the comparison of the simulated LAprofiles with that obtained experimentally. It can beseen that the profiles agree quite well for most of thefermentation time. However, there is a small mis-match towards the end of the fermentation. These

could be attributed to the limitations in the experi-mental procedures for starch additions (especially to-wards the end, Figs. 1 and 5) as well as shortcomingsin the model description. Here, an unstructured modelis used to describe the product (LA) formation. Sincethe Leudeking–Piret model was used to predict LAformation, the stoichiometry of glucose consumptionto LA was not incorporated. The term describing thestationary phase production of LA (bX) is opera-tional even after the complete consumption of glu-cose. This is a major drawback of using unstructuredgrowth models and explains the mismatch betweenpredicted and experimental profiles.

The experimental comparison of lactate yield andproductivity for the batch case over the fed-batch ispresented in Table 4. It can be seen that the yield ofLA per unit of starch as well as the productivity washigher in the case of fed batch than for the batch. Anincrease of 20% was achieved in lactate productivityby utilising the fed-batch mode of operation.

6. Conclusions

The principal conclusions derived from this workare as follows: (1) The performance of the saccharifi-cation process cannot be improved in fed batch whilethe batch saccharification shows a stoichiometric yieldunder optimum operating conditions; (2) In the caseof fermentation, it is better to start with low initialconcentrations of glucose and later feed concentratedglucose in the fed-batch mode. Improved LA produc-tivity is seen in the fed-batch mode but at the expenseof LA concentrations; (3) The performance of theSSF process can be improved substantially by operat-ing in fed-batch mode rather than batch in terms offinal LA concentration/productivity. The problem ofresidual starch at the end of the SSF process can betackled by augmenting the optimization strategy withan end point constraint on the starch concentrations.Our further studies on the use of optimal controlstrategies will focus on solving the resulting optimiza-tion problem to minimize residual starch concentra-tions and yet maintain the best possible productivity.


Table 3Fed-batch optimization results for SSF for various time intervals

Time, tf (h) End lactate concentration, P(tf) (g/l) Yield based on starch, Productivity of lactate, Pf/tf (g/l h)PfVf/[S0V0+Sin(Vf−V0)]

60 0.5753104.72 1.7450.753 1.7380 138.500.942174.28 1.743100

Fig. 5. Starch feed profiles predicted by the optimal control strategyin the fed-batch SSF for a total time of 100 h.

Appendix A. Control vector iteration procedure for solv-ing the optimal control problem [19]

Given the set of state equations for the culturevariables (i.e. biomass, substrate, etc.) as in Eqs. (23)–(26) and a performance objective J, the control vectoriteration solves for the time profile of the controlvariable V by an iterative procedure. The steps aregiven as follows:1. Guess a value of V(t), 00 t0 tf for the time profile

of the control variable;2. With value of V(t), integrate the state Eqs. (23)–

(26) forward in time from the given initial values toproduce the state variable profile;

3. Using the state profiles generated in step (2), inte-grate the co-state equations (28) backward in timeto produce the co-state variable profiles;

4. Correct V(t) using the equation dV(t)=o dH/dV,where o is chosen arbitrarily. Evaluate J for the newcontrol Vn(t)=V(t)+dV(t);

5. If J [Vn(t)]\J [V(t)], double o and repeat step (4),else halve o and repeat step (4). This is done till aconcave function of J is obtained and an optimalvalue of oopt is found;

6. Correct V(t) using the oopt found and return to step(2);

7. Iterate till the convergence is achieved.

Appendix B. Nomenclature

growth-associated glucose depletion parametera(g G)/(g X)non-growth associated glucose depletionbparameter (g G)/(g X)feed rate, (l/h)Fglucose concentration, (g/l)Ginlet glucose concentration, (g/l)Gin

stoichiometrically formed glucose by saccharifi-G*cation, (g/l)

H Hamiltonian in the optimal control problemperformance index in the optimal controlJproblem

Fig. 6. Concentration profiles for fed-batch SSF for 100 h. Theglucose levels do not reach inhibitory levels, hence enhancing theproductivity of the fermentation.


Fig. 7. Experimental verification of the optimized profiles for fed-batch SSF. The circles (�) show the experimental values of LA whilethe solid line shows the values arrived at from the simulations.

Table 4Experimental comparison of lactate yield and productivity for SSFwith 200 g/l of starch (for fermentation time of 100 h)

Lactate yieldMode of Lactate productivity (g/l h)operation

Batch 1.290.6911Fed batch 0.8135 1.56 (experimental)

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KG glucose inhibition constant for saccharification,(g/l)

KI glucose inhibition constant for fermentation, (g/l)KL lactate inhibition constant for fermentation, (g/l)

Michaelis Menten constant, (g/l)Km

Ks glucose saturation constant, (g/l)lactic acid concentration, (g/l)P

S starch concentration, (g/l)S0 initial starch concentration, (g/l)

inlet starch concentration, (g/l)Sin

V bioreactor volume, (l)Vi initial culture volume in fed batch, (l)6m maximum rate of saccharification, (g/l h)

cell mass concentration, (g/l)X

Greek symbolsmm maximum specific cell growth rate, (h−1)a growth-associated LA production parameter,

(g P)/(g X)b non-growth associated LA formation parameter,

(g P)/(g X) hl co-state variable for optimal control problemd set point

References

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optimal control strategies for simultaneous saccharification and fermentation of starch

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