356 K. Stamatelatou et al. / Control strategies for anaerobic digesters Application of optimal control theory to fermentation • Acetate is the main constituent of the fatty acids.processes has been an intriguing issue for numerous inves- • Acetate is the key organic compound fed to the digester,tigators [5–7,13,16,17,19,21–24,29,32]. Many algorithms so that hydrogen utilizing methanogenic bacteria can beand other methods have also been developed, in order to neglected. This assumption will be exact if acidogene-overcome the difﬁculties entailed in applying optimal con- sis and methanogenesis occur separately in a two-stagetrol theory. Most researchers, however, have focused on anaerobic process [1,33]. In this case, the inﬂuent ofbatch and semi-batch bioreactors. In an attempt to opti- the methanogenic digester can be regulated to consistmize continuous throughput, as opposed to batch fermenta- mainly of acetate.tion processes, a different and interesting approach has been • The biomass loss due to bacterial decay is insigniﬁcantfollowed by D’Ans et al.  who applied Green’s theorem compared with the biomass loss due to the operatingto maximize bacterial growth during a transient state. conditions of the reactor (biomass in the efﬂuent), so A particular issue of importance when it comes to opti- that the endogenous decay term may be neglected .mization of continuous processes, is the fact that the aris-ing optimal control problem is singular because the control The above assumptions are necessary if we are to be ablevariable, i.e., the dilution rate, is linearly included in the to formulate a singular optimal control problem with an el-state equations [9,10]. Whenever a singular optimal control egant solution being possible. Deviations between real ap-policy can be explicitly determined, it is rather impractical, plication and the hereby developed theoretical results maysince it usually involves a complicated function of the state be attributed to one or more of the above assumptions notvariables, often not readily measurable on-line. From this being valid.aspect it is more useful to derive a suboptimal control law We distinguish two cases of disturbances affecting theexpressed in a simpler explicit form, in terms of quantities digester.which may be measured on-line, and resulting in an imper- (i) Disturbance caused by inhibitor intrusion to the systemceptible loss of performance. Pullammanappallil et al. considered the case of inhibitors entering with the feed in Consider the case where a waste, to be treated anaerobi-this perspective and presented a suboptimal solution of the cally, contains an inhibitory substance such as chloroformproblem, much simpler in form than the optimal one and or ammonia. Mass balances of biomass, substrate and in-easy to implement. In this work, we thoroughly examine hibitor constitute the following model equations:the conventional optimal control problem and its simpli- dXﬁed, suboptimal version when the normal operation of an = −DX + µX, (1)anaerobic digester is upset by entry of an inhibitor, a feed dtoverload or underload. dS 1 = D(S0 − S) − µX, (2) dt YX/S2. Modeling for optimization dI = D(I0 − I), (3) dt Application of optimal control theory requires a mathe- where X, S, I represent the methanogen, volatile fatty acidmatical model for the described process. Detailed modeling and inhibitor concentrations, respectively, D is the dilutionof anaerobic digestion has been the objective of many re- rate, and µ is the speciﬁc growth rate. The latter is a com-searchers for whom their main concern has been a deeper plicated function of the state variables and is assumed tounderstanding of the biochemical steps involved in the follow Andrews’ kinetics which predicts inhibition for highanaerobic processes [3,4,11,20]. Despite the usefulness of substrate concentrations. This is the case for methanogenicsuch models, their large dimensionality constitutes a seri- microorganisms since they are strongly inhibited by highous disadvantage in handling these models for application concentrations of what they metabolize, i.e., fatty acids:purposes, especially when development of control schemes µmaxis involved. In such cases, the insight of the control law µ = µ(S) = . (4) 1 + Ks /S + S/Kipthe researcher gets while using a simpliﬁed model is moreimportant than the model itself. In this expression µmax is the maximum speciﬁc growth This is also the case of the present work which necessi- rate, Ks is the saturation constant, and Kip is the substratetates the use of a simpliﬁed model based on the following inhibition constant.assumptions: Given that the presence of an inhibitor affects only the speciﬁc growth rate, µ, and the feed substrate concentra-• All the feed substrate converts into organic acid rapidly, tion (S0 ) does not vary, stoichiometry may be used to which permits us to neglect the dynamics of all the express volatile fatty acid concentration (S) in terms of reaction steps except for the rate limiting growth of methanogen concentration, X: methanogens and the associated production of methane [27,30]. X = YX/S (S0 − S), (5)• pH is assumed to be controlled to remain neutral. 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- The total methane production over the interval [0, tf ] canquently, the model equations may be reduced to only two, be expressed asone for the methanogen concentration, X, or the substrate tfconcentration, S, and one for the inhibitor concentration, I. J D(t) = QCH4 (t) dt, (8)As a consequence, equations (1) or (2) along with (3) are 0adequate to describe the process. where QCH4 (t) is the methane production rate, D(t) is The impact of an inhibitory substance upon anaerobic the dilution rate, and tf is the ﬁnal time (chosen sufﬁ-digestion kinetics is expressed by a multiplying factor, i.e., ciently large, so that a new steady state is reached). Thef (I), which generally affects µmax or Ks of the speciﬁc methane production rate is assumed to be proportional togrowth rate . The latter is modiﬁed as follows: the methanogen growth rate [27,30] µmax f (I) QCH4 = V YCH4 /X µX, (9) µ = µ(X, I) = S0 −X/YX/S , (6) Ks 1+ S0 −X/YX/S + Kip where V is the volume of the reactor, YCH4 /X is a yield coefﬁcient, µ is the speciﬁc growth rate, and X is thewhere it is assumed that inhibition results simply in the methanogen concentration.reduction of the maximum speciﬁc growth rate, i.e., µmax .A variety of expressions have been suggested for f (I) ,among which two are considered in this work: 4. Optimization f (I) = e−aI (7a) The Hamiltonian function for case (i), i.e., the presence of an inhibitor in the feed, isor H = V YCH4 /X µ(X, I)X + λ1 −DX + µ(X, I)X 1 f (I) = . (7b) + λ2 D(I0 − I), (10) 1 + bIRelationships (7a) and (7b) are known expressions which while for case (ii), i.e., the feed overload or underload, ishave been reported to account for substrate and/or product H = V YCH4 /X µ(X, S)X + λ1 −DX + µ(X, S)Xinhibition . Additionally, the latter is already known todescribe the noncompetitive inhibition of enzyme-catalyzed 1 + λ2 D(S0 − S) − µ(X, S)X , (11)reaction kinetics . YX/S where λ1 , λ2 are the co-state variables. The time deriva-(ii) Disturbance caused by feed overload or feed tive of λi is deﬁned for each case as the negative partialunderload derivative of Hamiltonian with respect to the state variables (X or S). In this case, the feed substrate concentration is subjected It is evident that we are dealing with a singular controlto a step change (increase or decrease, respectively), so problem since the Hamiltonian, in both cases, is linear withthat efﬂuent biomass and substrate concentrations are not respect to the manipulated variable, D(t), and therefore forrelated stoichiometrically at all times. The model equations all t the optimal D is either on a singular arc or on a boundare restricted to (1) and (2) with the speciﬁc growth rate (either 0 or Dmax , depending on the initial conditions) untilsimply given by (4). it reaches a singular arc and remains on that singular arc until tf . On the singular arc the following equations are valid:3. Performance measure HD = 0, (12) dHD The selection of an appropriate performance measure = 0, (13)for an optimal control problem is the most important fac- dttor in an optimization problem deﬁnition. It enables us to d2 HD = 0. (14)get the most out of a transition state as it arises from the dt2dynamic conditions prevailing in the digester. In anaero- Equations (12) and (13) are independent of D and may bebic digestion, if one is primarily interested in the amount of used for expressing the co-state variables in terms of thegenerated methane, the total methane production during the state variables, while from equation (14) an expression forperiod of transition between two steady states constitutes an D can be derived, which is the optimal (feedback) controlappropriate performance measure to be maximized. In ad- policy on the singular arc (appendix B).dition, it is an indication of the system robustness, since it In order to solve the optimal control problem, the pointmonitors the methanogenic activity throughout the change. where the switching between the bound and the singularWhat is more, it is based on the methane production rate arc occurs is of crucial importance. For this reason, wewhich can be easily measured on-line. need to have an expression for the state variables which is
358 K. Stamatelatou et al. / Control strategies for anaerobic digestersvalid 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 /X µ∗ X ∗ , (17) Figure 1. Inhibitors entering with the feed: Optimal trajectory.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 Fmax Dmax = , (18) Vwhere 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 suboptimal control laws An arithmetic example following the procedure de- Figure 2. Step change in the feed substrate concentration: Optimal trajec- tory.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- leads the system optimally to the new optimal steady state,dix A. The analytical forms of the basic relationships of by changing the dilution rate according to equations (B4)the problem are presented in appendix B. or (B8). Figures 1 and 2 present the transition phase plane for the A typical time evolution of the optimal dilution rate andcases of the intrusion of an inhibitor, the effect of which on the corresponding methane production rate for the casesthe speciﬁc growth rate is given by (7a), and a disturbance examined is depicted in ﬁgures 3 and 4.at the feed substrate concentration, respectively. Immedi- It should be mentioned that if instead of (7a) we mod-ately following an imbalance, the digester operates either eled the effect of the inhibitors on the speciﬁc growth rateas a batch reactor (D = 0) or as a CSTR at its maximum through (7b), we would obtain qualitatively similar resultscapacity (D = Dmax ), until it reaches the singular arc that with those presented in ﬁgures 1 and 3.
K. Stamatelatou et al. / Control strategies for anaerobic digesters 359Figure 3. Inhibitors entering with the feed: Optimal dilution rate and methane 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 dilution Figure 4. Step change in the feed substrate concentration: (a) Optimal dilution rate versus time; (b) optimal methane production rate versus time.rate to methane production rate under optimal operatingconditions at the new steady state. Equation (19) is not only easy to enforce but is also Table 1a very good suboptimal control law as indicated by the Performance measure values indicating the total methane produced whilecomparison of the performance estimated for both control implementing the optimal or the suboptimal control law.policies which have been implemented for each type of dis- Kind of disturbance Performance measure, Jturbance. This comparison is presented in table 1, where (total methane, l)the insigniﬁcant difference between the optimal and the sub-optimal solution of the problem establishes the credibility Optimal Suboptimaland value of the suboptimal control law. Inhibitor intrusion 302.19 302.17 In the examples with a step change in the feed substrate Overload 852.09 852.08concentration presented here, the optimal dilution rate val- Underload 561.08 561.07ues of the initial and the ﬁnal steady state are almost iden-
360 K. Stamatelatou et al. / Control strategies for anaerobic digesters Figure 7. Feed substrate underload (from 1000 to 100 mg/l): MethaneFigure 5. Inhibitors entering with the feed: Methane production rate production rate versus dilution rate. versus dilution rate. here but also in a wide variety of situations. A feature of the cases considered in this work, involves a permanent ef- fect (step change) of a selected disturbance on the system so that its operation has to be established at a new steady state. Even if this disturbance lasted only for a ﬁnite pe- riod of time (pulse change), implementation of (19) would lead the digester to its original steady state in an almost optimal way. The suboptimal control law given by (19) has been incorporated in an expert system developed by Pullammanappallil et al.  for stabilizing anaerobic di- gesters. This work, therefore, really comes to justify that particular choice made. 6. Conclusions Operation of anaerobic digesters is sensitive to a variety of disturbances, which may lead the digester to wash out. An optimal control policy should be addressed to avoid the impeding digester failure and restore its normal operation or lead it to a new optimal steady state. This has been accom-Figure 6. Step change in the feed substrate concentration: Methane pro- duction rate versus dilution rate. plished by using a simpliﬁed model of anaerobic digestion to determine the optimal dilution rate as a function of time,tical due to the fact that the initial and ﬁnal feed substrate in response to the entry of an inhibitor with the feed or aconcentration values were speciﬁcally high. Nevertheless, sudden change in the feed substrate concentration. By ex-even in the case this almost identity does not occur, the amining the essential features of the digester key operatingrelationship between methane production rate and dilution variables, a simpler and easily implementable suboptimalrate on the singular arc is still almost linear as can be seen control law was derived, according to which the dilutionin ﬁgure 7, and consequently the suboptimal control law rate should be changed proportionally to the methane pro-can be used instead. duction rate, with the gain determined from the new optimal The concept of this suboptimal control policy is not only steady state values. This suboptimal controller, with a widevalid in the speciﬁc formulation of the problem presented ﬁeld of enforcement, leads to almost optimal performance.
K. Stamatelatou et al. / Control strategies for anaerobic digesters 361Appendix A Feed overloadConstant and parameter values Step change in the feed substrate concentration from 20 to 30 g/l; a = 1 l/mg, b = 1.77 l/mg, S0 = 30 000 mg/l. Kip = 4 000 mg/l, Initial conditions: Ks = 20 mg/l, V = 5 l, S = 254.887 mg/l, YX/S = 0.035 mg biomass/mg substrate, X = 691.079 mg/l. YCH4 /X = 0.009 l methane/mg biomass, µmax = 0.36 day−1 . Feed underloadInhibitors entering with the feed Step change in the feed substrate concentration from 30 to 20 g/l; I0 = 1 mg/l, S0 = 20 000 mg/l. S0 = 25 000 mg/l.Initial conditions: Initial conditions: I = 0 mg/l, S = 263.342 mg/l, X = 856.905 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: µ(X, I) λ1 = V YCH4 /X −1 , (B1) ∂µ(X,I) ∂I (I0 − I) − ∂µ(X,I) ∂X X µ(X, I) X λ2 = V YCH4 /X −1 . (B2) ∂µ(X,I) ∂I (I0 − I) − ∂µ(X,I) ∂X X I0 − I(ii) Equation of singular arc: µ(X, I)2X V YCH4 /X = constant. (B3) ∂µ(X,I) ∂I (I0 − I) − ∂µ(X,I) X ∂X(iii) Dilution rate expression: 2 ∂ 2 µ(X, I) ∂µ(X, I) D= µ(X, I)2 − 2 µ(X, I) X 2 ∂X 2 ∂X ∂µ(X, I) ∂µ(X, I) ∂ 2 µ(X, I) ∂µ(X, I) + 2 µ(X, I) − µ(X, I)2 (I0 − I)X + µ(X, I)2 (I0 − I) ∂X ∂I ∂X∂I ∂I 2 ∂ 2 µ(X, I) ∂µ(X, I) ∂µ(X, I) ∂µ(X, I) ∂ 2 µ(X, I) × µ(X, I) − 2 X2 + 4 −2 µ(X, I) (I0 − I)X ∂X 2 ∂X ∂X ∂I ∂X∂I 2 −1 ∂ 2 µ(X, I) ∂µ(X, I) + µ(X, I) − 2 (I0 − I) 2 . (B4) ∂I 2 ∂I
362 K. Stamatelatou et al. / Control strategies for anaerobic digesters(II) Step change in the feed substrate concentration (i) Costate variables: µ(S) − ∂µ(S) ∂S (S0 − S) λ1 = V YCH4 /X , (B5) ∂µ(S) ∂S (S0 − S) − X YX/S µ(S) − ∂µ(S) ∂S (S0 − S) λ2 = V YCH4 /X X. (B6) ∂µ(S) ∂S (S0 − S) (S0 − S) − X YX/S(ii) Equation of singular arc: µ(S)2 V YCH4 /X X = constant. (B7) ∂µ(S) ∂S (S0 − S)(iii) Dilution rate expression: 2 ∂µ(S) ∂ 2 µ(S) ∂µ(S) X ∂µ(S) D= 2 µ(S)(S0 − S) − µ(S)2 (S0 − S) + µ(S)2 − µ(S)2 (S0 − S) ∂S ∂S 2 ∂S YX/S ∂S 2 −1 ∂µ(S) ∂ 2 µ(S) × 2 − µ(S) (S0 − S)2 . (B8) ∂S ∂S 2Appendix C. Proof that the new optimal steady state Consequently, the optimal steady state is one for whichlies on the singular arc (C1)–(C3) are satisﬁed. On substitution of these expres- sions into equations (12)–(14) it can be observed that they This is shown for the case where an inhibitor intrudes are all satisﬁed. Moreover, the optimal steady state also sat-into the digester. The same applies for the underload or ˙ ˙ isﬁes X = 0 and I = 0 and thus all optimality conditionsoverload case. are met. The state equations are: X = −DX + µ(X, I)X, ˙ References I = D(I0 − I). ˙  A. Aivasidis, Proceedings of Conference on Industrial Wastewater Treatment and Disposal, University of Patras, Greece (21–22 No-At steady state, vember 1996) pp. 127–137.  I.M. Alatiqi, A.A. Dadkhah and N.M. Jabr, Chem. Engrg. J. 43 D = µ(X, I), (C1) (1990) B81.  J.F. Andrews and S.F. Graef, in: Anaerobic Biological Treatment I = I0 . (C2) Processes, Advances in Chemistry Series, Vol. 105 (American Chemical Society, Washington, 1971) p. 126.For optimality we have:  I. Angelidaki, L. Ellegaard and B.K. Ahring, Biotech. Bioengrg. 42 (1993) 159. dJSS dµ(X, I)X  G.D. Ans, P. Kokotovic and D. Cottlieb, IEEE Trans. Automat. Con- = V YCH4 /X =0 dD dD trol 16 (1971) 341.  G.D. Ans, P. Kokotovic and D. Cottlieb, Journal of Optimization ∂µ(X, I)X ∂µ(X, I)X dI ⇒ + Theory and Applications 7 (1971) 61. ∂D ∂I dD  G.D. Ans, P. Kokotovic and D. Cottlieb, Automatica 8 (1972) 729. ∂µ(X, I)X dX  J.E. Bailey and D.F. Ollis, Biochemical Engineering Fundamentals + = 0. (McGraw Hill, Singapore, 1986). ∂X dD  D.J. Bell and D.H. Jacobson, in: Singular Optimal Control Prob-In the above equation lems, Mathematics in Science and Engineering, Vol. 117 (Academic Press, 1975). ∂µ(X, I)X  A.E. Bryson and Y.J. Ho, Applied Optimal Control (Halsted Press, = 0, Willey, 1975). ∂D  D.J. Costello, P.F. Greenﬁeld and P.L. Lee, Water Res. 25 (1991)and at steady state I is independent of D, i.e., dI/dD = 0. 847.Since dX/dD = 0,  D. Dochain, G. Bastin, A. Rozzi and A. Pauss, in: Adaptive Estima- tion and Control of Biotechnological Processes, eds. S.L. Dhah and ∂µ(X, I)X G. Dumont, Adaptive Control Strategies for Industrial Use (Springer, = 0. (C3) Berlin, 1989). ∂X
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