Linearizing control of the anaerobic digestion with...

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Control Engineering Practice 14 (2006) 799–810

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Linearizing control of the anaerobic digestion with addition of acetate(control of the anaerobic digestion)

I. Simeonova, I. Queinnecb,�

aInstitute of Microbiology, Bulgarian Academy of Sciences, Acad. G. Bonchev St., Block 26, Sofia 1113, BulgariabLaboratoire d’Analyse et d’Architecture des Systemes (LAAS/CNRS) 7, Avenue du Colonel Roche, 31077 Toulouse cedex 4, France

Received 15 October 2002; accepted 4 April 2005

Available online 15 June 2005

Abstract

In this paper the principle of linearizing control was applied to anaerobic digestion of organic wastes with addition of a

stimulating substance (acetate). The objective consisted of regulating the biogas flow rate in the case of variations of the inlet organic

pollutant. For this purpose, a new control input was introduced in the fourth order model of the process, which reflects the acetate

addition. Laboratory experiments were done with step changes of this new input. New values of the model coefficients were

obtained. Input–output characteristics and optimal steady states were derived analytically using different optimality criteria. The

results obtained may be useful for industrial biogas plants operating with mixtures of organic wastes, where organic waste rich in

acetate (e.g., vinasse) will be added as a stimulating substance.

r 2005 Elsevier Ltd. All rights reserved.

Keywords: Anaerobic digestion; Acetate addition; Non-linear mathematical model; Parameter estimation; Steady-state analysis; Linearizing control

1. Introduction

Biological anaerobic wastewater treatment processes(anaerobic digestion) have been widely used in lifeprocess and has been confirmed as a promising methodfor solving some energy and ecological problems inagriculture and agro-industry. In such processes, gen-erally carried out in continuously stirred tank bioreac-tors (CSTR), the organic matter is depolluted bymicroorganisms into biogas (methane and carbondioxide) and fertilizer in the absence of oxygen. Thebiogas is an additional energy source and can alsoreplace fossil fuel sources and therefore has a directpositive effect on the greenhouse gas reduction. Un-fortunately, this process is very complex and may

e front matter r 2005 Elsevier Ltd. All rights reserved.

nengprac.2005.04.011

ing author. Tel.: +335 61 33 64 77;

69 69.

esses: [email protected] (I. Simeonov),

fr (I. Queinnec).

sometimes become very unstable. It then needs moreinvestigations.

The first step concerns mathematical modelling of theprocess. It represents a very attractive tool for studyingthis process. Angelidaki, Ellegaard, and Ahring (1999)developed a model involving 16 variables with six mainstages. The IWA Anaerobic Digestion Modelling Taskgroup has federated the energy to produce the IWA-ADM1 (Batstone et al., 2002), involving 24 variablesand many parameters. Such models are, however, notappropriate for control purposes due to their complex-ity. Numerous studies on special cases are also presentedin the literature, but only few of them focus on modelsappropriate for state observation and control. A simplemass-balance model involving five variables has beenproposed by Bernard, Hadj-Sadok, and Dochain (1999)to design a software sensor, with particular emphasis toalkalinity balance. Simeonov (1999) developed a second-order non-linear model based on a mere stage to beuseful for control purposes. Haag, Vande Wouwer,and Queinnec (2003) recently proposed a three-stage

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Nomenclature (list of symbols)

S00 concentration of the inlet diluted organics, g/L

X 1 concentration of acidogenic bacteria, g/LS1 concentration of substrate for acidogenic

bacteria (mainly glucose), g/LX 2 concentration of methane-producing (metha-

nogenic) bacteria, g/LS2 concentration of substrate for methane-pro-

ducing (methanogenic) bacteria (acetate), g/LS000 concentration of the acetate added in the

influent liquid, g/L (a new control input)Q biogas flow rate, L/dayS � COD Chemical Oxygen Demand

m1 specific growth rate of the acidogenic bacter-ia, day�1

m2 specific growth rate of the methanogenicbacteria, day�1

k1; k2; k3; k4;mmax 1;mmax 2; kS1and kS2

coefficientsD1 dilution rate for the inlet diluted organics,

day�1

D2 dilution rate for the acetate added in theinfluent liquid, day�1

D ¼ D1 þD2 the total dilution rate, day�1

xT ¼ ½X 1 S1 X 2 S2� the state vectoruT ¼ ½D S000 � the input vectory ¼ Q the measured output vectorpH acidity/alkalinity index

I. Simeonov, I. Queinnec / Control Engineering Practice 14 (2006) 799–810800

dynamic model (hydrolysis, acidogenesis and methano-genesis) involving seven variables but only two biomasscompounds to cope with identifiability problems relatedto the hydrolysis part.

Moreover, because of very restrictive on-line informa-tion, the control of such a process is often reduced to theregulation of the biogas production rate (energy supply)or of the concentration of polluting organic matter(depollution control) at a desired value in presence ofperturbations (Bastin & Dochain, 1991; Steyer, Buffiere,Rolland, & Molleta, 1999). According to the stronglynon-linear input–output characteristics of the process,classical linear controllers have good performances onlyin a locally linear zone related to small variations of thedilution. More sophisticated robust and variable struc-ture controllers (VSC) may be applied (Simeonov &Stoyanov, 1995; Zlateva & Simeonov, 1995) but even inthat case, the performances of the closed loop systemmay be degraded due to the strongly non-lineardynamics of the process. On the contrary, linearizingalgorithms for control of the anaerobic digestion provedto have very good performances (Bastin & Dochain,1991; Dochain, 1995).

Moreover, recent investigations have shown thataddition of stimulating substances (acetate or glucose)in appropriate concentrations allow to stabilize theprocess and to increase the biogas flow rate (Simeonov& Galabova, 2000; Simeonov, Galabova, & Queinnec,2001). The aim of this paper is then to design andinvestigate different algorithms for linearizing control ofthe anaerobic digestion using the addition of acetate as acontrol action. The control algorithms are based on arelatively simple model developed for this purpose. Theoutline of the paper is as follows. Section 2 concerns theprocess modelling. Experimental studies used formathematical modelling and parameter estimation arepresented, so as identifiability properties and identifica-tion procedure. In Section 3 steady-state analysis and

optimal steady states following different criteria on thebasis of the developed model are performed. The controlproblem is formulated in Section 4, and two linearizedalgorithms for regulation of the biogas flow rate Q aredesigned on the basis of the reduced model (obtained forthe particular case when only acetate is added) for theprocess. Both approaches are evaluated by simulation inSection 5. Finally, Section 6 contains some concludingremarks.

2. Process modelling and parameter estimation

2.1. Experimental studies

Laboratory experiments have been carried out inCSTR with highly concentrated organic pollutants(cattle wastes) at mesophillic temperature and withaddition of acetate in low concentrations (Simeonov &Galabova, 2000). The laboratory experimental set-upincludes an automated bioreactor of a 3-l glass vesseldeveloped and adapted to fulfil the requirements foranaerobic digestion. It is mechanically stirred byelectrical drive and maintained at a constant tempera-ture (34� 0:5 �C) by computer controller. The monitor-ing of the methane reactor is carried out by dataacquisition computer system of on-line sensors, whichprovide the following measurements: pH, temperature,redox, speed of agitation and biogas flow rate (Q). Aschematic diagram of the experimental laboratory-scaleset-up is shown in Fig. 1, where 1 is the bioreactor; 2 theDC drive; 3 the biogas flow-meter; 4 the heating system;5 the peristaltic pump; 6 the gas holder; 7 the convertersunit; 8 the gas chromatograph; 9 the biogas flame; 10 thewatt-hour meter; 11 the personal computer and 12 theprinter.

It is well known that anaerobic digestion is a self-stabilization process as long as disturbance magnitude

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Inlet Substrate,Acetate,Glucose5

nt t pH6 3

9

8 10 220 V

VFA

Outlet

1 2

4

7

11 12

CH4

Fig. 1. Experimental set-up.

Q (

L/d

ay),

S0"

(g/

L)

00.20.40.60.8

11.21.41.6

35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69

Fig. 2. Evolution of Q in the case of step addition of acetate S000.

Table 1

The effect of acetate on the methane fermentation (in steady-state)

Date (day) From 1st to

34th

From 35th

to 40th

From 41st

to 50th

From 51st

to 90th

Feeding

D1 (day�1) 0.0375 0.0375 0.0375 0.0375

D2 (day�1) 0.0125 0.0125 0.0125 0.0125

S00 (g/L) 68 68 68 68

S000 (g/L) 0 25 50 75

Average

value of Q

(L/day) at

steady state

0.35 0.5 0.9 1.2

I. Simeonov, I. Queinnec / Control Engineering Practice 14 (2006) 799–810 801

does not exceed the buffer capacity of the medium,which would results in pH breakdown related toaccumulation of volatile fatty acids (VFA) produced(Angelidaki et al., 1999; Batstone et al., 2002). In theapproach proposed in the paper, addition of acetate actsas a control input. Then biogas production will increase,but pH breakdown may occur. To prevent from such afailure, Simeonov and Galabova (2000) have shown thatpH regulation (correction till pH 7.5) has to be done inthe inlet mixture ðsubstrateþ acetateÞ rather than in thebioreactor.

Experimental design has then been developed con-sisting in appropriate (by amplitude and time) step andpulse changes of the acetate addition (S000) and measure-ments of the responses of the biogas flow rate ðQÞ and ofthe acetate concentration in the anaerobic bioreactor (bygas chromatography). Some rather repeatable results areshown in Fig. 2 (for step changes of (S000) from 0 to 25 g/L at 35th day, from 25 to 50 g/L at 41st and to 75 g/L at51st day) and the steady state of biogas flow rate afterstep changes are given in Table 1. It is seen that thesettling time for each step response (new steady-state) isabout 5–6 days.

The reported data offer the suggestion that acetateaddition positively affects the methane production andincreased levels of acetate as electron donor result infaster rates of methanogenesis (the second importantphase of the methane fermentation) (Simeonov &Galabova, 2000).

2.2. Mathematical modelling of the process

On the basis of the above-presented experimentalinvestigations and following the so-called two-stagebiochemical scheme of the methane fermentation(Bastin & Dochain, 1991), the following simplestrealistic non-linear model with two control inputs isproposed:

dX 1

dt¼ ðm1 �DÞX 1, (1)

dS1

dt¼ �k1m1X 1 þD1S00 �DS1, (2)

dX 2

dt¼ ðm2 �DÞX 2, (3)

dS2

dt¼ �k2m2X 2 þ k3m1X 1 þD2S

000 �DS2, (4)

Q ¼ k4m2X 2. (5)

In this mass balance model, Eq. (1) describes thegrowth and changes of the acidogenic bacteria (X 1),consuming the appropriate substrate (S1), where the firstterm in the right side reflects the growth of theacidogenic bacteria and the second one reflects theeffluent flow rate of liquid. The mass balance for thissubstrate is described by (2), where the first term reflectsthe consumption by the acidogenic bacteria, the secondterm reflects the influent flow rate of liquid withconcentration of diluted organics S00, and the third onethe effluent flow rate of liquid. Eq. (3) describes thegrowth and changes of the methane-producing (metha-nogenic) bacteria, with concentration X 2, consumingacetate, with concentration S2, where the first term inthe right side reflects the growth of the methanogenicbacteria and the second one reflects the effluent flow rateof liquid. The mass balance equation for acetate (4) hasfour terms in his right side. The first one reflects theconsumption of acetate by the methanogenic bacteria,the second one the acetate formed as a result of the

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Table 2

Values obtained for the coefficients of the fourth order model with

acetate addition

mmax 1

ðday�1Þ

mmax 2

ðday�1Þ

kS1

ðg=LÞ

kS2

ðg=LÞ

k1 k2 k3 k4

ðL.L=gÞ

0.2 0.25 0.3 0.87 6.7 4.2 5 4.35

0 5 10 15 20 25 30

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Time (days)

Q e x p

X1

Q 0.5 S2

5 S1

0.1 X2

Fig. 3. Evolution of the main variables in the case of step addition of

acetate S000.


activity of acidogenic bacteria, the third one the directaddition of acetate, with concentration S000 g/L, (a new

control input) and the last one the acetate in the effluentliquid. The algebraic equation (5) describes the forma-tion of biogas with flow rate Q.

The specific growth rate of the acidogenic bacteria(m1), and the specific growth rate of the methanogenicbacteria (m2), are described by Monod type structures asfollows:

m1 ¼mmax 1S1

kS1þ S1

m2 ¼mmax 2S2

kS2þ S2

. (6)

Generally S00 is an unmeasurable (in real time)perturbation, while S000 is a known constant or controlinput. In all cases, the washout of microorganisms isundesirable, that is why changes of the total dilutionrate D ¼ D1 þD2 and the perturbation S00 are possibleonly in some admissible ranges (for fixed value of S000):

0pDpDsup; S00 infpS00pS00sup. (7)

To summarize, the process is characterized by the statexT ¼ ½X 1 S1 X 2 S2�, the input vector uT ¼ ½D S000� oruT ¼ ½D2 S000� and the measured output vector y ¼ Q.

2.3. Parameter estimation

For parameter estimation, the value of eight para-meters has to be determined, so as the initial statevariable X 1ð0Þ;S1ð0Þ and X 2ð0Þ. S2ð0Þ is directly relatedto the measured biogas flow rate Qð0Þ. An identifiabilitytest quickly establishes that this whole set of parameterscannot be identified in one step. A sensitivity analysiswith respect to the eight kinetic parameters allows,however, to separate these parameters into two groups.The first one, composed of the yields k1; k2; k3 and k4, isthe most sensitive group, i.e., small variations of theseparameters involve strong variation of the simulatedbehaviour of the process model (1)–(5). The second one,composed of the parameters of Monod expressions, isless sensitive, as much as some standard values may befixed. Since prior knowledge about initial parametervalues is essential in solving non-linear estimationproblems (to avoid biased estimates to a large extent),parameter identification has started with initial valuesknown from our previous work (Simeonov, 2000).Applying the methodology from Simeonov (2000)estimation then starts with the first (more sensitive)group of coefficients with arbitrary known othercoefficients using optimization method; estimation ofthe second group of coefficients with the above-determined values of the first group in the followingstep, etc. The identification procedure has been initiatedin the present case with mmax 1 ¼ 0:2 day�1,mmax 2 ¼ 0:25 day�1, kS1

¼ 0:3 g=L and kS2¼ 0:37 g=L

and initial value of the state vector corresponding to theinitial steady state Simeonov (2000). These values had

been determined from previous experiments withoutacetate addition. A simplex method has been used foreach step of the estimation procedure. Parameteridentification has been done with experimental datafor Q provided from experiments with known values ofthe influent (S00 ¼ 75 g=L; D1 ¼ 0:0375 day�1;D2 ¼ 0:0125 day�1). The experimental data presentedin Fig. 2 (with step addition of acetate) served forparameter identification. They involve 44 measurementsof the biogas flow rate Q. The first period, from t ¼ 35to 41 days, is with S000 ¼ 25 g=L, the second one, fromday 42 to day 90, is with S000 ¼ 50 g=L, and the third oneis with S000 ¼ 75 g=L. The parameter identification stepresulted in the estimates given in Table 2. Experimentaldata and model simulation results for the same case arepresented in Fig. 3. Experimental data and modelsimulation results with pulse addition of acetate (4pulses with amplitudes of 0.5, 0.75, 1.0 and 1.5 g/L) arepresented in Fig. 4 and served for model validation.They involve 64 measurements of the biogas flow rate Q.Good fit between biogas flow rate measurement (o) andsimulated Q (solid line) confirms the quality of themodelling step. Comparing the results from Simeonov(2000) and Angelidaki et al. (1999) (without acetateaddition) with the new obtained parameter values theconclusion is, as expected that differences exist only forvalues of kS2

and k4, related with the methanogenic stepof the process.

ARTICLE IN PRESSQ

m [L

/day

], Q

exp[

L/d

ay]

0 5 10 15 20 25 30

0.5

1.0

1.5

2.0

Time (days)

Qm

Qexp

Fig. 4. Evolution of Q (from experiments and simulations) in the case

of step addition of acetate S000.

S = C1.S1 + C2.S2

0 0.05 0.1 0.15 0.2 0.250

0.5

1

1.5

2

2.5

3

D [day-1]

Q[L

/day

], S

[g/

L]

50

70

90

110

_

Fig. 5. Input–output characteristics Q ¼ QðDÞ and S ¼ SðDÞ.


3. Input–output characteristics

In this section, the steady-state analysis and optim-ality conditions determination are achieved under thehypothesis of identical maximum specific rates for bothacidogenic and methanogenic bacteria ðmmax 1 ¼

mmax 2 ¼ mmÞ. This hypothesis could be relaxed and isonly considered to simplify the presentation.

3.1. Optimal steady states

From industrial point of view, operating conditionshave to be searched such that the process runs nearbysome static optimal points. However, there exist severaloptimal operating conditions related to several optim-ality criteria. The main optimality criteria may beformulated as:

1.
Maximal amount of biogas production Q. This is anenergetical criterion, in the sense that the controlobjective rather concerns the production of addi-tional energy source than the reduction of wastes.
2.
Maximal depollution effect. This criterion aims atreducing the concentration of organic matter at theoutlet of the process. It then corresponds to anecological criterion and may be expressed as min S,S ¼ C1S1 þ C2S2;C1 and C2 being given constants.S is generally associated with the COD (ChemicalOxygen Demands) of the outlet substance.
3.
Compromise between energetical and depollution
criteria. In this case, the criterion is expressed as acombination of Q and S to maximize, for example(Q� kS) or Q=kS, k40. This criterion may be veryuseful for a good efficiency of big anaerobic plants.

To illustrate these criteria, the input–output char-acteristics QðDÞ and SðDÞ are shown in Fig. 3, forvarious values of the influent organics S00. It brings tothe fore the opposite effect of the dilution rate D on thebiogas flow rate (which has to be maximized) and on S

(which has to be minimized). Indeed, S is minimized asmuch as D decreases. Then, for a small value of thedilution rate, the retention time related to the inverse ofD is very large, which induces poor efficiency of theprocess and small production of biogas. On thecontrary, when high quantity of biogas is wanted itresults in poor depollution effect. This justifies thenecessity of a mixed criterion both on Q and S.

3.2. Steady-states analysis for optimal biogas flow rate

criterion

Algebraic equations corresponding to set all thederivatives in the model (1)–(4) to zero are solved todetermine the analytical steady-state values:

S1 ¼kS1

D

mmax 1 �D, (8)

X 1 ¼1

k1

D1S00

D�

kS1D

mmax 1 �D

� �, (9)

S2 ¼kS2

D

mmax 2 �D, (10)

X 2 ¼1

k2

k3

k1

D1S00D�

kS1D

mmax 1 �D

� ��

�kS2

D

mmax 2 �Dþ

D2S000

D

�, ð11Þ

which results for the steady-state values of the biogasflow rate in:

Q ¼k4

k2D

k3

k1

D1S00

D�

kS1D

mmax 1 �D

� ��

�kS2

D

mmax 2 �Dþ

D2S000

D

�. ð12Þ

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Table 3

Upper bounds and optimal values of the dilution rate for various

acetate influent conditions

S000 (g/L) 25 50 75

Dsup 1 (day�1) 0.166 0.166 0.166

Dsup 2 (day�1) 0.358 0.5 0. 647

DQopt (day

�1) 0.159 0.17 0.176

DJopt (day

�1) 0.0695 0.075 0.077


Under hypothesis ðmmax 1 ¼ mmax 2 ¼ mmÞ and taking intoaccount that D2 ¼ D�D1 (D1 is assumed to beconstant), it is shown in Appendix A that

DQopt ¼ mm 1�

ffiffiffiffiffiffiWp� �

(13)

with

W ¼k3kS1

þ k1kS2

k3kS1þ k1kS2

þ k1S000

; Wo1 (14)

It results in an optimal biogas flow rate Qopt is given by:

Qopt ¼k3k4

k1k2S00D1 þ

k4

k2mm 1�


�D1

� �S000

�mmk4ðk3kS1

þ k1kS2Þ 1�

ffiffiffiffiffiffiWp� �2

k1k2

ffiffiffiffiffiffiWp . ð15Þ

Moreover, the optimum Qopt is larger that the onewhich would be obtained in the case without acetateaddition.

3.3. Steady-state analysis for combination of energetical

and ecological criteria

The same procedure as in the previous section maynow be applied to determine an optimal Dopt maximiz-ing some mixed criterion on Q and S. Let us consider thecriterion J ¼ Q=kS; k40. As previously, it is shown inAppendix B that

DJopt ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðk1S000 � k3S00ÞD1mm

k3kS1þ k1kS2

þ k1S000

s(16)

is an optimum, with existence condition given by

k1S000 � k3S0040. (17)

Remark. It may be checked that, according to thenumerical value for k1 and k3, all the simulatedand experimental evaluations respect the existencecondition (17).

3.4. Physical admissibility of optimal dilution rates

According to model (1)–(5) and definition of thedilution rate, the first existence condition is classically

0oDommax. (18)

Moreover, according to (8), it may be verified that thebiomass concentration X 1 is positive implies that

DoDsup 1 ¼D1S

00

2kS1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ

mmkS1

D1S00

s� 1

!. (19)

In the same way, the condition on the dilution ratesuch that X 2 is positive expresses as:

DoDsup 2 ¼k3D1S

00 þ k1D2S000

2k1ðkS1þ kS2

Þ

�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ

4mmk1ðkS1þ kS2

Þ

k3D1S00 þ k1D2S

000

s� 1

!. ð20Þ

Finally, Dsup ¼MinðDsup 1;Dsup 2Þ.Let us consider the kinetic parameters given in Table

2 and the following condition for organic waste additionS00 ¼ 75 g=L and D1 ¼ 0:0375 day�1. The conditions(19), (20) and optimal values for D

Qopt (day

�1) and DJopt

(day�1) are given in Table 3 for various acetate influentconditions.

From the table, it may be concluded that the optimaldilution rate which would maximize the production ofbiogas, D

Qopt, is only admissible for S000 ¼ 25 g=L, but

cannot be reached for larger values of the influentacetate concentration. The optimal value of DJ

opt

maximizing a mixed criterion on Q and S is alwaysattainable, and, as it was expected from Fig. 5, is muchsmaller than for the case of D

Qopt.

4. Linearizing control

4.1. Formulation of the control problem

The problem of optimal control of anaerobic diges-tion may be decomposed in three subproblems:

(a)
static optimization; (b) optimal start-up; (c) dynamic optimization.
The static optimization of the process was presentedin the previous section. The problem for optimal start-up of the process with the new defined control input (D2)is a very promising one. The problem of the dynamicoptimization is reduced to regulation of:

(1)
the biogas production rate Q (energy supply), or (2) the organics concentration S (depollution control),

ARTICLE IN PRESSI. Simeonov, I. Queinnec / Control Engineering Practice 14 (2006) 799–810 805

at a prescribed value (Q and S , respectively) by actingupon the dilution rate D ¼ D1 þD2. The value of Q�

� �

may be calculated from (12).In this paper our attention is focused on the

linearizing control (Bastin & Dochain, 1991; Van Impe,Vanrolleghem, & Iserentant, 1998) of Q in the case ofunmeasured variations of the inlet soluble organics S00using the addition of stimulating substance (dilution rateD2 or influent acetate concentration S000) as the controlinput.

4.2. Control algorithm for regulation of Q

The model (1)–(5) may be decomposed into two partsfollowing the two stages of the process:

(a)
the ‘‘acidogenic stage’’, described by Eqs. (1) and (2)is not influenced by the control input;
(b)
the ‘‘methanogenic stage’’, described by Eqs. (3)–(5),is influenced by the control input.
Then, for design purposes, only the second part isneeded.

Proposition. The regulation of the biogas flow rate Q may

be achieved through linearizing control, where the control

input is given as:

(1)
D2ðtÞ ¼1
S000

lyðQ� �QÞ

Q�

1

yðm2 �DÞ þDS2

�

�k3m1X 1 þk2

k4Q

�; ðS000 ¼ const:Þ ð21Þ

with 0oD2ðtÞoDsup

or

(2)
S000ðtÞ ¼1
D2

lyðQ� �QÞ

Q�

1

yðm2 �DÞ þDS2

�

�k3m1X 1 þk2

k4Q

�; ðD2 ¼ const:Þ ð22Þ

with 0oS000ðtÞoS00sup0

depending on the experimental strategy for actuators. l is

a tuning parameter which represents the desired behaviour

of the closed-loop dynamics.

Proof. We consider the following linear stable first-order closed-loop dynamics:

dQ

dt� lðQ� �QÞ ¼ 0, (23)

where the first time-derivative of Q is given bydifferentiation of (5) as

dQ

dt¼ k4

dm2dt

X 2 þQðm2 �DÞ. (24)

Manipulation of time-derivative of m2 may cause manycomputation problems, and it is much more careful toconsider an algebraic expression of dm2=dt derived fromthe non-linear expression for m2 (6) and the time-derivative of S2, i.e.,

dm2dt¼

kS2

mmax 2

1

S22

m22dS2

dt¼

kS2

mmax 2

1

S22

m22 �k2

k4Q

�

þ k3m1X 1 þD2S000 �DS2

�.

ð25Þ

Then after substitutions of (24) and (25) in (23), oneobtains:

D2S000 ¼

ly

Q� �Q

Q�

1

yðm2 �DÞ þDS2 � k3m1X 1 þ

k2

k4Q,

(26)

where

y ¼kS2

m2mmax 2S

22

.

From expression (26), two input variables may beconsidered. Either the dilution rate D2 (S000 beingconstant) related to the addition of influent acetate orthe concentration of the influent acetate S000 (D2 beingconstant) may be used as control input, which results inthe two proposed control laws (21) and (22).

Remark. The laws (21) and (22) do not directly dependon the influent diluted organics S00. They depend,however, on the acidogenic reaction rate m1X 1 whichhas to be estimated on-line. This can be done by usingan observer-based estimator (Lubenova, Simeonov, &Queinnec, 2002).

5. Simulation studies and discussion

The designed algorithms are evaluated by simulation.Some results of the simulations with the controlalgorithm (21) are shown in Fig. 6 (for l ¼ 0:4) andFig. 7 (for l ¼ 0:1). In both figures the simulationconditions are as follows: step changes of the set pointQ� (L/day) (0.6 from 0 to 30th day, 1.2 from 30th to60th day, then 0.6 after 60th day); step and sinusoidalchanges of the disturbance S00 (7.5 g/L between day 0and day 20, then again between day 40 and day 60, 15 g/L between day 20 and day 40, then again between day 60and day 80 and a sinusoidal signal of 20% of amplitudewith period of 8 h is added on the step disturbance);D1 ¼ 0:0325 day�1 ¼ const:, S000 ¼ 25 g=L ¼ const:; 10%of noise under Q (in L/day). Biomasses X 1 and X 2 areplotted in subplot (a), S1 (in g/L), S2 (in g/L), Q, S00 andD2 are plotted in subplots (b), (c), (d), (e) and (f),respectively.

ARTICLE IN PRESS

0 20 40 60 800

1

2

3

4

biom

asse

s X

1, X

2

X1 : solid line

X2 : dashdot line

(a) 0 20 40 60 800

0.1

0.2

0.3

0.4

subs

trat

e S

1

(b)

0 20 40 60 800.05

0.1

0.15

0.2

0.25

subs

trat

e S

2

(c) 0 20 40 60 800

0.5

1

1.5

Q

(d)

0 20 40 60 805

10

15

20

S0’

time (d)(e)0 20 40 60 80

0

0.02

0.04

0.06

D2

time (d)(f)

Fig. 7. Simulations with control action D2 for l ¼ 0:1.

0 20 40 60 800

1

2

3

4

biom

asse

s X

1, X

2

X1 : solid line

X2 : dashdot line

0 20 40 60 800

0.1

0.2

0.3

0.4

subs

trat

e S

1

(b)

0 20 40 60 800

0.1

0.2

0.3

0.4

subs

trat

e S

2

0 20 40 60 800

0.5

1

1.5

Q

(d)

0 20 40 60 805

10

15

20

S0’

time (d)(e)0 20 40 60 80

0

0.02

0.04

0.06

D2

time (d)(f)

(a)

(c)

Fig. 6. Simulations with control action D2 for l ¼ 0:4.


Some results of the simulations with the controlalgorithm (22) are shown in Fig. 8 for l ¼ 0:4. Thesimulation conditions are the same as in the previouscase (except D2 ¼ 0:0125 day�1 ¼ const:, S000 ¼ var.).Biomasses X 1 and X 2 are plotted in subplot (a),

S1;S2;Q;S00 and S000 are plotted in subplots (b), (c), (d),

(e) and (f), respectively.Comparing Figs. 6 and 8 the conclusion is that there is

nearly no difference between evolutions of S2 and Q

whatever the control action is (D2 or S000), however

ARTICLE IN PRESS

0 20 40 60 800

2

4

6

biom

asse

s X

1, X

2

X1 : solid line

X2 : dashdot line

(a) 0 20 40 60 800

0.1

0.2

0.3

0.4

subs

trat

e S

1

(b)

0 20 40 60 800

0.1

0.2

0.3

0.4

subs

trat

e S

2

(c) 0 20 40 60 800

0.5

1

1.5

Q

(d)

0 20 40 60 805

10

15

20

S0

time (d)(e)0 20 40 60 80

0

50

100

S0

time (d)(f)

, "

Fig. 8. Simulations with control action S000 for l ¼ 0:4.

1.2 µ max1

0 10 20 30 40 50 60 70 800.2

0.4

0.6

0.8

1

1.2

1.4

1.6

time (day)

Q (

L/da

y)

nominal case0.8 µ

max1

Fig. 9. Influence of model error—error of þ or �20% on mmax 1.


differences exist between evolutions of X 1;X 2 and S1

due to the fact that changes of D2 result in changes of D.Simulation studies with the linearizing control algo-

rithms show that they present very good performancesof regulation with different values of the tuningparameter l. Some problems, however, arise when l istoo much increased. It may result into some vanishingoscillations (for l ¼ 1) or even instability due tosaturations on the actuators. This problem may beovercome with more precise determination of Dsup andS0 inf0 (in each particular case it is possible to measure

S0 inf0 and than to calculate Dsup) and the choice of an

optimal value of l. The practical realization of (20) and(21) is very realistic when all unmeasured variables areestimated by suitable observers (Lubenova et al., 2002).

Moreover, Figs. 9–16 show the influence on thecontrol of the biogas production Q of model errors.Simulations compare the biogas production Q in thenominal case (control law D2ðtÞ is computed by usingthe process parameters) with respect to the biogasproduction Q obtained when the control law iscomputed with a model error of þ or �20% on eachparameter. It may be checked from these figures that k1

has no influence on the quality of the control and thatmmax 1;mmax 2; kS1

; kS2and k3 have a small influence on

the quality of the control. On the other hand, theefficiency of the control is closely related to the qualityof estimates k2 and k4. This is an expected result sincethe steady-state value of D2 is closely related to thefactor k2=k4.

The algorithm (22) is slightly more difficult toimplement than algorithm (21) since it is generallyeasier to act on a pump, i.e., on a dilution rate than on aconcentration. This can, however, be done by using asystem of two pumps relied to two tanks allowing tocontrol (with constant flow rate) variations of theconcentration. But in spite of the technical difficulty,the algorithm (22) is theoretically more correct since D iskept constant. Then the control only acts on themethanogenic phase of the process.

ARTICLE IN PRESS

0 10 20 30 40 50 60 70 800.2

0.4

0.6

0.8

1

1.2

1.4

1.6

time (day)

Q (

L/da

y)

nominal case0.8 µmax21.2 µmax2

Fig. 10. Influence of model error—error of þ or �20% on mmax 2.

0 10 20 30 40 50 60 70 800.2

0.4

0.6

0.8

1

1.2

1.4

1.6

time (day)

Q (

L/da

y)

nominal case0.8kS1

1.2kS1

Fig. 11. Influence of model error—error of þ or �20% on kS1.

0 10 20 30 40 50 60 70 800.2

0.4

0.6

0.8

1

1.2

1.4

1.6

time (day)

Q (

L/da

y)

nominal case0.8kS21.2kS2

Fig. 12. Influence of model error—error of þ or �20% on kS2.

0 10 20 30 40 50 60 70 800.2

0.4

0.6

0.8

1

1.2

1.4

1.6

time (day)

Q (

L/da

y)

nominal case0.8k1 1.2k1

Fig. 13. Influence of model error—error of þ or �20% on k1.

0 10 20 30 40 50 60 70 800.2

0.4

0.6

0.8

1

1.2

1.4

1.6

time (day)

Q (

L/da

y)nominal case0.8k2 1.2k2


0 10 20 30 40 50 60 70 800.2

0.4

0.6

0.8

1

1.2

1.4

1.6

time (day)

Q (

L/da

y)




ARTICLE IN PRESS

0 10 20 30 40 50 60 70 800.2

0.4

0.6

0.8

1

1.2

1.4

1.6

time (day)

Q (

L/da

y)




6. Conclusion

Experimental and analytical studies have shown thataddition of acetate (with pH correction of the addedsubstrate) allows to stabilize the process with respect toload and activity disturbances and to increase theamount of biogas obtained from the anaerobic digestionof organic wastes. This fact is very promising forstabilization of the biogas plants in the case of strongvariations of the influent organic matter.

Theoretical studies and simulation results haveproven that the linearizing control design on the basisof an appropriate model of the anaerobic digestion witha new control input, the addition of acetate, may be veryuseful for the regulation of the amount of biogas in therealistic case of strong variations of the influent organicmatter. From practical point of view both linearizingcontrol algorithms ((21) and (22)) are easy to implement.However, even if it is easier to control variations of aflow rate (law (21)), the control of the influent acetateconcentration presents the advantage to keep theprocess dilution rate constant.

Acknowledgements

This work was supported by Contract no TH-1004/00of The Bulgarian National Found ‘‘Scientific re-searches’’ and by a CNRS-BAS exchange program.

Appendix A. Optimal biogas flow rate criterion

For sake of simplicity, we only consider the hypoth-esis mmax 1 ¼ mmax 2 ¼ mm. Taking into account thatD2 ¼ D�D1 (D1 is assumed to be constant), Eq. (12)

may be expressed in the form:

Q ¼ AD1 � BD2

mm �Dþ CD (27)

with

A ¼k4

k2

k3

k1S00 � S000

� �; B ¼

ðk3kS1þ k1kS2

Þk4

k1k2;

C ¼k4

k2S000.

It is then possible to evaluate an optimal biogas flowrate Qopt from the determination of an extremum of(27), obtained when the derivative of Q with respect to D

is equal to zero:

dQ

dD¼ C � B

2Dðmm �DÞ þD2

ðmm �DÞ2

¼ 0! D2 � 2mmDþm2mC

Bþ C¼ 0.

Only one root of this second-order equation isadmissible, i.e.,

DQopt ¼ mm 1�


; with

W ¼k3kS1

þ k1kS2

k3kS1þ k1kS2

þ k1S000; Wo1.

The other root, D ¼ mmð1þffiffiffiffiffiffiWpÞ, is not admissible

since D4mm is not physically admissible (it wouldresults in the washout of the process). From evaluationof the second derivative d2Q=dD2, it may be checkedthat:

Qopt ¼k3k4

k1k2S00D1 þ

k4

k2mm 1�


�D1

� �S000

�mmk4ðk3kS1

þ k1kS2Þ 1�

ffiffiffiffiffiffiWp� �2

k1k2

ffiffiffiffiffiffiWp

is a maximum.

Appendix B. Mixed energetical and ecological criterion

Let us consider the mixed criterion J ¼ Q=kS; k40.Similarly to Appendix A, we only present the casemmax 1 ¼ mmax 2 ¼ mm and taking into account that D2 ¼

D�D1 (D1 is assumed to be constant), the criterion J

may be expressed in the form:

J ¼a1 þ a2D� a3D

2

a4D(28)

with

a1 ¼ AD1mm ¼ A ¼k4

k2

k3

k1S00 � S000

� �D1mm;

a2 ¼k4

k1k2ðk1S

000ðmm þD1Þ � k3S00D1Þ;

ARTICLE IN PRESSI. Simeonov, I. Queinnec / Control Engineering Practice 14 (2006) 799–810810

a3 ¼ Bþ C ¼k4ðk3kS1

þ k1kS2Þ þ k1k4S000

k1k240;

a4 ¼ kðc1kS1þ c2kS2

Þ40.

It is then possible to evaluate an optimal value for thecriterion J from the determination of an extremum of(28), obtained when the derivative of J with respect to D

is equal to zero:

dJ

dD¼

a1 þ a3D2

a4D2¼ 0! D2 ¼

ðk3S00 � k1S

000ÞD1mm

k1S000 þ k3kS1

þ k1kS2

.

Only one root of this second-order equation isadmissible, i.e.,

DJopt ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðk1S000 � k3S00ÞD1mm

k3kS1þ k1kS2

þ k1S000

s,

where the existence condition is k1S000 � k3S0040. From

evaluation of the second derivative d2J=dD2, it may bechecked that Dopt is a maximum.

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