optimal and suboptimal control of anaerobic digesters

Environmental Modeling and Assessment 2 (1997) 355–363 355

Optimal and suboptimal control of anaerobic digesters

K. Stamatelatou a, G. Lyberatos a, C. Tsiligiannis a, S. Pavlou a, P. Pullammanappallil b and S.A. Svoronos c

a Department of Chemical Engineering, University of Patras, and Institute of Chemical Engineering and High Temperature Chemical Processes,P.O. Box 1414, GR-26500 Patras, Greece

b Department of Chemical Engineering, University of Queensland, Australiac Department of Chemical Engineering, University of Florida, Gainesville, FL 32601, USA

Received June 1996; revised January 1997

Anaerobic digester failure due to entry of inhibitors or sudden changes in the feed substrate concentration may be encompassedbeneficially by applying optimal control theory. An almost proportional relationship between the dilution rate and the methane productionrate leads to a simple suboptimal control law with only minor loss in performance, after the occurrence of the above mentioned events.

Keywords: anaerobic digester, optimal control, suboptimal control law

0. Nomenclature

a, b constants (l/mg)D dilution rate (day−1)F feed rate (l/day)H HamiltonianHD partial derivative of Hamiltonian

with respect to dilution rate (∂H/∂D)I inhibitors concentration (mg/l)J performance measureKip inhibition constant (mg/l)Ks saturation constant (mg/l)QCH4 methane production rate (l/day)S substrate concentration (mg/l)t time (day)V reactor volume (l)X methanogen concentration (mg/l)YX/S yield coefficient (mg biomass/mg substrate)YCH4/X yield coefficient (l methane/mg biomass)

Greek letters

µ specific growth rate (day−1)λi co-state variables

Subscripts and superscripts

∗ optimalmax maximum0 feedf final

1. Introduction

Anaerobic digestion is a process that converts organicmatter into a gaseous mixture mainly composed of methaneand carbon dioxide through the concerted action of a close

knit community of bacteria [25]. It has been tradition-ally used for waste treatment but there is also considerableinterest in plant-biomass-fed digesters, since the producedmethane is a useful source of energy.

Methanogenesis, the ultimate step in anaerobic digestionis rate limiting and is usually the most sensitive step, sinceit is easily inhibited, by the substrate (i.e., volatile fattyacids) or by various other inhibitors (chloroform, oxygen,heavy metals, etc.). As a result, anaerobic digesters areeasily imbalanced causing abnormal increases in concentra-tions of organic acids. Such imbalances arise from suddenchanges in the feed substrate concentration (feed overloador underload) or the presence of inhibitory substances, suchas chloroform, ammonia, etc., which enter the reactor withthe feed. Every disturbance of this kind, with concomi-tant organic acid accumulation, ultimately brings about adrop of the specific growth rate of the methanogenic pop-ulation below the operating dilution rate, which leads themethanogenic bacteria to wash out. This is a serious prob-lem because lengthy startup periods are required to repop-ulate the digester with a healthy culture.

Process control strategies have been developed to solveor reduce the extent of this problem [12,15,17,26,28,30,31,34,35]. Pullammanappallil et al. [28] developed an in-tegrated expert system restoring balance in anaerobic di-gesters, having taken into consideration all possible typesof disturbances mentioned above. Whatever the cause ofthe imbalance, the digester in principle can be saved if thedilution rate is lowered below the ultimate value of themethanogen specific growth rate. The usual practice is tooperate in a batch mode (zero dilution rate), until the di-gester has sufficiently recovered. However, if our mainobjective, besides restoring digester operation, is to mini-mize the cost of imbalance caused in the system during thetransient, the question of how the dilution rate should belowered forms an interesting optimal control problem.

Baltzer Science Publishers BV

356 K. Stamatelatou et al. / Control strategies for anaerobic digesters

Application of optimal control theory to fermentationprocesses has been an intriguing issue for numerous inves-tigators [5–7,13,16,17,19,21–24,29,32]. Many algorithmsand other methods have also been developed, in order toovercome the difficulties entailed in applying optimal con-trol theory. Most researchers, however, have focused onbatch and semi-batch bioreactors. In an attempt to opti-mize continuous throughput, as opposed to batch fermenta-tion processes, a different and interesting approach has beenfollowed by D’Ans et al. [5] who applied Green’s theoremto maximize bacterial growth during a transient state.

A particular issue of importance when it comes to opti-mization of continuous processes, is the fact that the aris-ing optimal control problem is singular because the controlvariable, i.e., the dilution rate, is linearly included in thestate equations [9,10]. Whenever a singular optimal controlpolicy can be explicitly determined, it is rather impractical,since it usually involves a complicated function of the statevariables, often not readily measurable on-line. From thisaspect it is more useful to derive a suboptimal control lawexpressed in a simpler explicit form, in terms of quantitieswhich may be measured on-line, and resulting in an imper-ceptible loss of performance. Pullammanappallil et al. [29]considered the case of inhibitors entering with the feed inthis perspective and presented a suboptimal solution of theproblem, much simpler in form than the optimal one andeasy to implement. In this work, we thoroughly examinethe conventional optimal control problem and its simpli-fied, suboptimal version when the normal operation of ananaerobic digester is upset by entry of an inhibitor, a feedoverload or underload.

2. Modeling for optimization

Application of optimal control theory requires a mathe-matical model for the described process. Detailed modelingof anaerobic digestion has been the objective of many re-searchers for whom their main concern has been a deeperunderstanding of the biochemical steps involved in theanaerobic processes [3,4,11,20]. Despite the usefulness ofsuch models, their large dimensionality constitutes a seri-ous disadvantage in handling these models for applicationpurposes, especially when development of control schemesis involved. In such cases, the insight of the control lawthe researcher gets while using a simplified model is moreimportant than the model itself.

This is also the case of the present work which necessi-tates the use of a simplified model based on the followingassumptions:

• All the feed substrate converts into organic acid rapidly,which permits us to neglect the dynamics of all thereaction steps except for the rate limiting growth ofmethanogens and the associated production of methane[27,30].

• pH is assumed to be controlled to remain neutral.

• Acetate is the main constituent of the fatty acids.

• Acetate is the key organic compound fed to the digester,so that hydrogen utilizing methanogenic bacteria can beneglected. This assumption will be exact if acidogene-sis and methanogenesis occur separately in a two-stageanaerobic process [1,33]. In this case, the influent ofthe methanogenic digester can be regulated to consistmainly of acetate.

• The biomass loss due to bacterial decay is insignificantcompared with the biomass loss due to the operatingconditions of the reactor (biomass in the effluent), sothat the endogenous decay term may be neglected [2].

The above assumptions are necessary if we are to be ableto formulate a singular optimal control problem with an el-egant solution being possible. Deviations between real ap-plication and the hereby developed theoretical results maybe attributed to one or more of the above assumptions notbeing valid.

We distinguish two cases of disturbances affecting thedigester.

(i) Disturbance caused by inhibitor intrusion to the system

Consider the case where a waste, to be treated anaerobi-cally, contains an inhibitory substance such as chloroformor ammonia. Mass balances of biomass, substrate and in-hibitor constitute the following model equations:

dXdt

= −DX + µX , (1)

dSdt

= D(S0 − S)− 1YX/S

µX , (2)

dIdt

= D(I0 − I), (3)

whereX , S, I represent the methanogen, volatile fatty acidand inhibitor concentrations, respectively, D is the dilutionrate, and µ is the specific growth rate. The latter is a com-plicated function of the state variables and is assumed tofollow Andrews’ kinetics which predicts inhibition for highsubstrate concentrations. This is the case for methanogenicmicroorganisms since they are strongly inhibited by highconcentrations of what they metabolize, i.e., fatty acids:

µ = µ(S) =µmax

1 +Ks/S + S/Kip. (4)

In this expression µmax is the maximum specific growthrate, Ks is the saturation constant, and Kip is the substrateinhibition constant.

Given that the presence of an inhibitor affects only thespecific growth rate, µ, and the feed substrate concentra-tion (S0) does not vary, stoichiometry may be used toexpress volatile fatty acid concentration (S) in terms ofmethanogen concentration, X :

X = YX/S(S0 − S), (5)

where YX/S is the yield constant.

K. Stamatelatou et al. / Control strategies for anaerobic digesters 357

In view of equation (5), the state variables and, conse-quently, the model equations may be reduced to only two,one for the methanogen concentration, X , or the substrateconcentration, S, and one for the inhibitor concentration, I .As a consequence, equations (1) or (2) along with (3) areadequate to describe the process.

The impact of an inhibitory substance upon anaerobicdigestion kinetics is expressed by a multiplying factor, i.e.,f (I), which generally affects µmax or Ks of the specificgrowth rate [2]. The latter is modified as follows:

µ = µ(X , I) =µmaxf (I)

1 + KsS0−X/YX/S +

S0−X/YX/SKip

, (6)

where it is assumed that inhibition results simply in thereduction of the maximum specific growth rate, i.e., µmax.A variety of expressions have been suggested for f (I) [14],among which two are considered in this work:

f (I) = e−aI (7a)

or

f (I) =1

1 + bI. (7b)

Relationships (7a) and (7b) are known expressions whichhave been reported to account for substrate and/or productinhibition [14]. Additionally, the latter is already known todescribe the noncompetitive inhibition of enzyme-catalyzedreaction kinetics [8].

(ii) Disturbance caused by feed overload or feedunderload

In this case, the feed substrate concentration is subjectedto a step change (increase or decrease, respectively), sothat effluent biomass and substrate concentrations are notrelated stoichiometrically at all times. The model equationsare restricted to (1) and (2) with the specific growth ratesimply given by (4).

3. Performance measure

The selection of an appropriate performance measurefor an optimal control problem is the most important fac-tor in an optimization problem definition. It enables us toget the most out of a transition state as it arises from thedynamic conditions prevailing in the digester. In anaero-bic digestion, if one is primarily interested in the amount ofgenerated methane, the total methane production during theperiod of transition between two steady states constitutes anappropriate performance measure to be maximized. In ad-dition, it is an indication of the system robustness, since itmonitors the methanogenic activity throughout the change.What is more, it is based on the methane production ratewhich can be easily measured on-line.

The total methane production over the interval [0, tf] canbe expressed as

J(D(t)

)=

∫ tf

0QCH4 (t) dt, (8)

where QCH4 (t) is the methane production rate, D(t) isthe dilution rate, and tf is the final time (chosen suffi-ciently large, so that a new steady state is reached). Themethane production rate is assumed to be proportional tothe methanogen growth rate [27,30]

QCH4 = V YCH4/XµX , (9)

where V is the volume of the reactor, YCH4/X is a yieldcoefficient, µ is the specific growth rate, and X is themethanogen concentration.

4. Optimization

The Hamiltonian function for case (i), i.e., the presenceof an inhibitor in the feed, is

H = V YCH4/Xµ(X , I)X + λ1[−DX + µ(X , I)X

]+λ2D(I0 − I), (10)

while for case (ii), i.e., the feed overload or underload, is

H = V YCH4/Xµ(X ,S)X + λ1[−DX + µ(X ,S)X

]+λ2

[D(S0 − S)− 1

YX/Sµ(X ,S)X

], (11)

where λ1, λ2 are the co-state variables. The time deriva-tive of λi is defined for each case as the negative partialderivative of Hamiltonian with respect to the state variables(X or S).

It is evident that we are dealing with a singular controlproblem since the Hamiltonian, in both cases, is linear withrespect to the manipulated variable, D(t), and therefore forall t the optimal D is either on a singular arc or on a bound(either 0 or Dmax, depending on the initial conditions) untilit reaches a singular arc and remains on that singular arcuntil tf .

On the singular arc the following equations are valid:

HD = 0, (12)dHD

dt= 0, (13)

d2HD

dt2= 0. (14)

Equations (12) and (13) are independent of D and may beused for expressing the co-state variables in terms of thestate variables, while from equation (14) an expression forD can be derived, which is the optimal (feedback) controlpolicy on the singular arc (appendix B).

In order to solve the optimal control problem, the pointwhere the switching between the bound and the singulararc occurs is of crucial importance. For this reason, weneed to have an expression for the state variables which is


valid only on the singular arc, so that while on the bound,we will be able to check if we have reached the singulararc. Such an expression can be derived by the followingargument:

Since the Hamiltonian is not an implicit function of time,its first derivative with respect to time is zero:

H = 0. (15)

As a consequence,

H = constant (16)

throughout the singular arc. It can be easily proved that thenew optimal steady state singular arc is unique, regardlessof tf , upon which the new optimal steady state lies (ap-pendix C). Clearly, at the final time, tf , when the systemreaches a new optimal steady state corresponding to thenew feed conditions, the Hamiltonian will be equal to H∗

(from (10) or (11) at the steady state), which is the valueof the constant in (16):

H∗ = V YCH4/Xµ∗X∗, (17)

where the ∗ denotes the value of these quantities at thenew optimal steady state. In this way, equation (16) givesan expression for the singular arc with respect to the statevariables.

It should be mentioned that the manipulated variable, D,is constrained between two bounds – zero (equivalent tobatch operation) and Dmax, given by

Dmax =Fmax

V, (18)

where Fmax is the maximum flowrate the feeding pump mayprovide, and V is the volume of the reactor vessel. Theabove constraint has an implication for the case in whichthe first calculated value of the dilution rate on the singulararc is higher than Dmax. In such a case, the optimal controlpolicy is a bang–bang control policy (sequential switchingbetween the upper and lower bounds) until the calculateddilution rate on the singular arc lies within the limits.

5. Optimal and easily implementable suboptimalcontrol laws

An arithmetic example following the procedure de-scribed above for each type of disturbance considered ispresented in this section. The values of the constants andthe other parameters of the problem are given in appen-dix A. The analytical forms of the basic relationships ofthe problem are presented in appendix B.

Figures 1 and 2 present the transition phase plane for thecases of the intrusion of an inhibitor, the effect of which onthe specific growth rate is given by (7a), and a disturbanceat the feed substrate concentration, respectively. Immedi-ately following an imbalance, the digester operates eitheras a batch reactor (D = 0) or as a CSTR at its maximumcapacity (D = Dmax), until it reaches the singular arc that

Figure 1. Inhibitors entering with the feed: Optimal trajectory.

Figure 2. Step change in the feed substrate concentration: Optimal trajec-tory.

leads the system optimally to the new optimal steady state,by changing the dilution rate according to equations (B4)or (B8).

A typical time evolution of the optimal dilution rate andthe corresponding methane production rate for the casesexamined is depicted in figures 3 and 4.

It should be mentioned that if instead of (7a) we mod-eled the effect of the inhibitors on the specific growth ratethrough (7b), we would obtain qualitatively similar resultswith those presented in figures 1 and 3.


Figure 3. Inhibitors entering with the feed: Optimal dilution rate andmethane production rate versus time.

Still practical problems arise when it comes to enforcingthe optimal control law, since it requires the knowledge ofthe state variables which cannot easily be measured on-line.

As an alternative, a good suboptimal and easy to imple-ment control law has been formulated.

Figures 3 and 4 indicate that almost for the entire intervalof optimization, the optimal trajectory lies on the singulararc excluding a negligibly small initial interval when thecontrol variable is on a bound. The plot of the generatedmethane production rate versus dilution rate while on thesingular arc is almost a straight line which passes throughthe origin of the axes as can be seen from figures 5 and 6.Thus, the suboptimal control policy that results is to sim-ply change the dilution rate proportionally to the methaneproduction rate which can be readily measured on-line, i.e.,

D(t) = kQCH4 (t). (19)

Here k is the proportionality constant which can be deter-mined by estimating the slope of the straight line of fig-ures 5 and 6. A very good approximation of this optimalvalue of k can be easily determined by the ratio of dilutionrate to methane production rate under optimal operatingconditions at the new steady state.

Equation (19) is not only easy to enforce but is alsoa very good suboptimal control law as indicated by thecomparison of the performance estimated for both controlpolicies which have been implemented for each type of dis-turbance. This comparison is presented in table 1, wherethe insignificant difference between the optimal and the sub-optimal solution of the problem establishes the credibilityand value of the suboptimal control law.

In the examples with a step change in the feed substrateconcentration presented here, the optimal dilution rate val-ues of the initial and the final steady state are almost iden-

Figure 4. Step change in the feed substrate concentration: (a) Optimaldilution rate versus time; (b) optimal methane production rate versus time.

Table 1Performance measure values indicating the total methane produced while

implementing the optimal or the suboptimal control law.

Kind of disturbance Performance measure, J(total methane, l)

Optimal Suboptimal

Inhibitor intrusion 302.19 302.17

Overload 852.09 852.08

Underload 561.08 561.07


Figure 5. Inhibitors entering with the feed: Methane production rateversus dilution rate.

Figure 6. Step change in the feed substrate concentration: Methane pro-duction rate versus dilution rate.

tical due to the fact that the initial and final feed substrateconcentration values were specifically high. Nevertheless,even in the case this almost identity does not occur, therelationship between methane production rate and dilutionrate on the singular arc is still almost linear as can be seenin figure 7, and consequently the suboptimal control lawcan be used instead.

The concept of this suboptimal control policy is not onlyvalid in the specific formulation of the problem presented

Figure 7. Feed substrate underload (from 1000 to 100 mg/l): Methaneproduction rate versus dilution rate.

here but also in a wide variety of situations. A feature ofthe cases considered in this work, involves a permanent ef-fect (step change) of a selected disturbance on the systemso that its operation has to be established at a new steadystate. Even if this disturbance lasted only for a finite pe-riod of time (pulse change), implementation of (19) wouldlead the digester to its original steady state in an almostoptimal way. The suboptimal control law given by (19)has been incorporated in an expert system developed byPullammanappallil et al. [28] for stabilizing anaerobic di-gesters. This work, therefore, really comes to justify thatparticular choice made.

6. Conclusions

Operation of anaerobic digesters is sensitive to a varietyof disturbances, which may lead the digester to wash out.An optimal control policy should be addressed to avoid theimpeding digester failure and restore its normal operation orlead it to a new optimal steady state. This has been accom-plished by using a simplified model of anaerobic digestionto determine the optimal dilution rate as a function of time,in response to the entry of an inhibitor with the feed or asudden change in the feed substrate concentration. By ex-amining the essential features of the digester key operatingvariables, a simpler and easily implementable suboptimalcontrol law was derived, according to which the dilutionrate should be changed proportionally to the methane pro-duction rate, with the gain determined from the new optimalsteady state values. This suboptimal controller, with a widefield of enforcement, leads to almost optimal performance.


Appendix A

Constant and parameter values

a = 1 l/mg,b = 1.77 l/mg,Kip = 4 000 mg/l,Ks = 20 mg/l,V = 5 l,YX/S = 0.035 mg biomass/mg substrate,YCH4/X = 0.009 l methane/mg biomass,µmax = 0.36 day−1.

Inhibitors entering with the feed

I0 = 1 mg/l,S0 = 25 000 mg/l.

Initial conditions:

I = 0 mg/l,X = 856.905 mg/l.

Feed overload

Step change in the feed substrate concentration from 20to 30 g/l;

S0 = 30 000 mg/l.

Initial conditions:

S = 254.887 mg/l,X = 691.079 mg/l.

Feed underload

Step change in the feed substrate concentration from 30to 20 g/l;

S0 = 20 000 mg/l.

Initial conditions:

S = 263.342 mg/l,X = 1040.783 mg/l.

Appendix B. Optimal control law on the singular arc

(I) Inhibitors entering with the feed

(i) Costate variables:

λ1 = V YCH4/X

(µ(X , I)

∂µ(X,I)∂I (I0 − I)− ∂µ(X,I)

∂X X− 1

), (B1)

λ2 = V YCH4/X

(µ(X , I)

∂µ(X,I)∂I (I0 − I)− ∂µ(X,I)

∂X X− 1

)X

I0 − I. (B2)

(ii) Equation of singular arc:

V YCH4/Xµ(X , I)2X

∂µ(X,I)∂I (I0 − I)− ∂µ(X,I)

∂X X= constant. (B3)

(iii) Dilution rate expression:

D=

{[∂2µ(X , I)∂X2

µ(X , I)2 − 2

(∂µ(X , I)∂X

)2

µ(X , I)

]X2

+

(2∂µ(X , I)∂X

∂µ(X , I)∂I

µ(X , I)− ∂2µ(X , I)∂X∂I

µ(X , I)2

)(I0 − I)X +

∂µ(X , I)∂I

µ(X , I)2(I0 − I)

}

×{[

∂2µ(X , I)∂X2

µ(X , I)− 2

(∂µ(X , I)∂X

)2]X2 +

(4∂µ(X , I)∂X

∂µ(X , I)∂I

− 2∂2µ(X , I)∂X∂I

µ(X , I)

)(I0 − I)X

+

[∂2µ(X , I)∂I2

µ(X , I)− 2

(∂µ(X , I)∂I

)2](I0 − I)2

}−1

. (B4)


(II) Step change in the feed substrate concentration

(i) Costate variables:

λ1 = V YCH4/X

µ(S)− ∂µ(S)∂S (S0 − S)

∂µ(S)∂S

[(S0 − S)− X

YX/S

] , (B5)

λ2 = V YCH4/X

µ(S)− ∂µ(S)∂S (S0 − S)

∂µ(S)∂S (S0 − S)

[(S0 − S)− X

YX/S

]X. (B6)

(ii) Equation of singular arc:

V YCH4/Xµ(S)2

∂µ(S)∂S (S0 − S)

X = constant. (B7)

(iii) Dilution rate expression:

D=

{[2

(∂µ(S)∂S

)2

µ(S)(S0 − S)− ∂2µ(S)∂S2

µ(S)2(S0 − S) +∂µ(S)∂S

µ(S)2

]X

YX/S− ∂µ(S)

∂Sµ(S)2(S0 − S)

}

×{[

2

(∂µ(S)∂S

)2

− ∂2µ(S)∂S2

µ(S)

](S0 − S)2

}−1

. (B8)

Appendix C. Proof that the new optimal steady statelies on the singular arc

This is shown for the case where an inhibitor intrudesinto the digester. The same applies for the underload oroverload case.

The state equations are:

X =−DX + µ(X , I)X ,

I =D(I0 − I).

At steady state,

D = µ(X , I), (C1)

I = I0. (C2)

For optimality we have:

dJSS

dD= V YCH4/X

dµ(X , I)XdD

= 0

⇒ ∂µ(X , I)X∂D

+∂µ(X , I)X

∂I

dIdD

+∂µ(X , I)X

∂X

dXdD

= 0.

In the above equation

∂µ(X , I)X∂D

= 0,

and at steady state I is independent of D, i.e., dI/dD = 0.Since dX/dD 6= 0,

∂µ(X , I)X∂X

= 0. (C3)

Consequently, the optimal steady state is one for which(C1)–(C3) are satisfied. On substitution of these expres-sions into equations (12)–(14) it can be observed that theyare all satisfied. Moreover, the optimal steady state also sat-isfies X = 0 and I = 0 and thus all optimality conditionsare met.

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optimal and suboptimal control of anaerobic digesters

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