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Linearizing control of the anaerobic digestion with addition of acetate (control of the anaerobic digestion)
ARTICLE IN PRESS
Control Engineering Practice 14 (2006) 799–810 www.elsevier.com/locate/conengprac
Linearizing control of the anaerobic digestion with addition of acetate (control of the anaerobic digestion) I. Simeonova, I. Queinnecb, a
Institute of Microbiology, Bulgarian Academy of Sciences, Acad. G. Bonchev St., Block 26, Sofia 1113, Bulgaria Laboratoire d’Analyse et d’Architecture des Syste`mes (LAAS/CNRS) 7, Avenue du Colonel Roche, 31077 Toulouse cedex 4, France
b
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 life process and has been confirmed as a promising method for solving some energy and ecological problems in agriculture and agro-industry. In such processes, generally carried out in continuously stirred tank bioreactors (CSTR), the organic matter is depolluted by microorganisms into biogas (methane and carbon dioxide) and fertilizer in the absence of oxygen. The biogas is an additional energy source and can also replace fossil fuel sources and therefore has a direct positive effect on the greenhouse gas reduction. Unfortunately, this process is very complex and may
Corresponding author. Tel.: +33 5 61 33 64 77; fax: +35 5 61 33 69 69. E-mail addresses: [email protected] (I. Simeonov), [email protected] (I. Queinnec).
0967-0661/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.conengprac.2005.04.011
sometimes become very unstable. It then needs more investigations. The first step concerns mathematical modelling of the process. It represents a very attractive tool for studying this process. Angelidaki, Ellegaard, and Ahring (1999) developed a model involving 16 variables with six main stages. The IWA Anaerobic Digestion Modelling Task group has federated the energy to produce the IWAADM1 (Batstone et al., 2002), involving 24 variables and many parameters. Such models are, however, not appropriate for control purposes due to their complexity. Numerous studies on special cases are also presented in the literature, but only few of them focus on models appropriate for state observation and control. A simple mass-balance model involving five variables has been proposed by Bernard, Hadj-Sadok, and Dochain (1999) to design a software sensor, with particular emphasis to alkalinity balance. Simeonov (1999) developed a secondorder non-linear model based on a mere stage to be useful for control purposes. Haag, Vande Wouwer, and Queinnec (2003) recently proposed a three-stage
ARTICLE IN PRESS 800
I. Simeonov, I. Queinnec / Control Engineering Practice 14 (2006) 799–810
Nomenclature (list of symbols) S00
concentration of the inlet diluted organics, g/ L X1 concentration of acidogenic bacteria, g/L S1 concentration of substrate for acidogenic bacteria (mainly glucose), g/L X2 concentration of methane-producing (methanogenic) bacteria, g/L S2 concentration of substrate for methane-producing (methanogenic) bacteria (acetate), g/L S000 concentration of the acetate added in the influent liquid, g/L (a new control input) Q biogas flow rate, L/day S COD Chemical Oxygen Demand
dynamic model (hydrolysis, acidogenesis and methanogenesis) involving seven variables but only two biomass compounds to cope with identifiability problems related to the hydrolysis part. Moreover, because of very restrictive on-line information, the control of such a process is often reduced to the regulation of the biogas production rate (energy supply) or of the concentration of polluting organic matter (depollution control) at a desired value in presence of perturbations (Bastin & Dochain, 1991; Steyer, Buffiere, Rolland, & Molleta, 1999). According to the strongly non-linear input–output characteristics of the process, classical linear controllers have good performances only in a locally linear zone related to small variations of the dilution. More sophisticated robust and variable structure controllers (VSC) may be applied (Simeonov & Stoyanov, 1995; Zlateva & Simeonov, 1995) but even in that case, the performances of the closed loop system may be degraded due to the strongly non-linear dynamics of the process. On the contrary, linearizing algorithms for control of the anaerobic digestion proved to have very good performances (Bastin & Dochain, 1991; Dochain, 1995). Moreover, recent investigations have shown that addition of stimulating substances (acetate or glucose) in appropriate concentrations allow to stabilize the process and to increase the biogas flow rate (Simeonov & Galabova, 2000; Simeonov, Galabova, & Queinnec, 2001). The aim of this paper is then to design and investigate different algorithms for linearizing control of the anaerobic digestion using the addition of acetate as a control action. The control algorithms are based on a relatively simple model developed for this purpose. The outline of the paper is as follows. Section 2 concerns the process modelling. Experimental studies used for mathematical modelling and parameter estimation are presented, so as identifiability properties and identification procedure. In Section 3 steady-state analysis and
specific growth rate of the acidogenic bacteria, day1 m2 specific growth rate of the methanogenic bacteria, day1 k1 ; k2 ; k3 ; k4 ; mmax 1 ; mmax 2 ; kS1 and kS2 coefficients D1 dilution rate for the inlet diluted organics, day1 D2 dilution rate for the acetate added in the influent liquid, day1 D ¼ D1 þ D2 the total dilution rate, day1 xT ¼ ½X 1 S 1 X 2 S 2 the state vector uT ¼ ½D S000 the input vector y ¼ Q the measured output vector pH acidity/alkalinity index
m1
optimal steady states following different criteria on the basis of the developed model are performed. The control problem is formulated in Section 4, and two linearized algorithms for regulation of the biogas flow rate Q are designed on the basis of the reduced model (obtained for the particular case when only acetate is added) for the process. Both approaches are evaluated by simulation in Section 5. Finally, Section 6 contains some concluding remarks.
2. Process modelling and parameter estimation 2.1. Experimental studies Laboratory experiments have been carried out in CSTR with highly concentrated organic pollutants (cattle wastes) at mesophillic temperature and with addition of acetate in low concentrations (Simeonov & Galabova, 2000). The laboratory experimental set-up includes an automated bioreactor of a 3-l glass vessel developed and adapted to fulfil the requirements for anaerobic digestion. It is mechanically stirred by electrical drive and maintained at a constant temperature (34 0:5 C) by computer controller. The monitoring of the methane reactor is carried out by data acquisition computer system of on-line sensors, which provide the following measurements: pH, temperature, redox, speed of agitation and biogas flow rate (Q). A schematic diagram of the experimental laboratory-scale set-up is shown in Fig. 1, where 1 is the bioreactor; 2 the DC drive; 3 the biogas flow-meter; 4 the heating system; 5 the peristaltic pump; 6 the gas holder; 7 the converters unit; 8 the gas chromatograph; 9 the biogas flame; 10 the watt-hour meter; 11 the personal computer and 12 the printer. It is well known that anaerobic digestion is a selfstabilization process as long as disturbance magnitude
ARTICLE IN PRESS I. Simeonov, I. Queinnec / Control Engineering Practice 14 (2006) 799–810
Inlet
5
Table 1 The effect of acetate on the methane fermentation (in steady-state)
Substrate,Acetate,Glucose 11
1
4
nt t pH 2 6
12
CH4
9
8
10
220 V
VFA Outlet
Fig. 1. Experimental set-up.
Q (L/day), S0" (g/L)
Date (day)
7 3
801
1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0
Feeding D1 (day1 ) D2 (day1 ) S00 (g/L) S000 (g/L) Average value of Q (L/day) at steady state
From 1st to 34th
From 35th to 40th
From 41st to 50th
From 51st to 90th
0.0375 0.0125 68 0 0.35
0.0375 0.0125 68 25 0.5
0.0375 0.0125 68 50 0.9
0.0375 0.0125 68 75 1.2
2.2. Mathematical modelling of the process
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 .
does not exceed the buffer capacity of the medium, which would results in pH breakdown related to accumulation of volatile fatty acids (VFA) produced (Angelidaki et al., 1999; Batstone et al., 2002). In the approach proposed in the paper, addition of acetate acts as a control input. Then biogas production will increase, but pH breakdown may occur. To prevent from such a failure, Simeonov and Galabova (2000) have shown that pH regulation (correction till pH 7.5) has to be done in the inlet mixture ðsubstrate þ acetateÞ rather than in the bioreactor. Experimental design has then been developed consisting in appropriate (by amplitude and time) step and pulse changes of the acetate addition (S000 ) and measurements of the responses of the biogas flow rate ðQÞ and of the acetate concentration in the anaerobic bioreactor (by gas chromatography). Some rather repeatable results are shown in Fig. 2 (for step changes of (S 000 ) from 0 to 25 g/ L at 35th day, from 25 to 50 g/L at 41st and to 75 g/L at 51st day) and the steady state of biogas flow rate after step changes are given in Table 1. It is seen that the settling time for each step response (new steady-state) is about 5–6 days. The reported data offer the suggestion that acetate addition positively affects the methane production and increased levels of acetate as electron donor result in faster rates of methanogenesis (the second important phase of the methane fermentation) (Simeonov & Galabova, 2000).
On the basis of the above-presented experimental investigations and following the so-called two-stage biochemical scheme of the methane fermentation (Bastin & Dochain, 1991), the following simplest realistic non-linear model with two control inputs is proposed: dX 1 ¼ ðm1 DÞX 1 , dt
(1)
dS 1 ¼ k1 m1 X 1 þ D1 S00 DS 1 , dt
(2)
dX 2 ¼ ðm2 DÞX 2 , dt
(3)
dS 2 ¼ k2 m2 X 2 þ k3 m1 X 1 þ D2 S 000 DS 2 , dt
(4)
Q ¼ k4 m2 X 2 .
(5)
In this mass balance model, Eq. (1) describes the growth and changes of the acidogenic bacteria (X 1 ), consuming the appropriate substrate (S1 ), where the first term in the right side reflects the growth of the acidogenic bacteria and the second one reflects the effluent flow rate of liquid. The mass balance for this substrate is described by (2), where the first term reflects the consumption by the acidogenic bacteria, the second term reflects the influent flow rate of liquid with concentration of diluted organics S 00 , and the third one the effluent flow rate of liquid. Eq. (3) describes the growth and changes of the methane-producing (methanogenic) bacteria, with concentration X 2 , consuming acetate, with concentration S2 , where the first term in the right side reflects the growth of the methanogenic bacteria and the second one reflects the effluent flow rate of liquid. The mass balance equation for acetate (4) has four terms in his right side. The first one reflects the consumption of acetate by the methanogenic bacteria, the second one the acetate formed as a result of the
ARTICLE IN PRESS I. Simeonov, I. Queinnec / Control Engineering Practice 14 (2006) 799–810
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activity of acidogenic bacteria, the third one the direct addition of acetate, with concentration S000 g/L, (a new control input) and the last one the acetate in the effluent liquid. The algebraic equation (5) describes the formation of biogas with flow rate Q. The specific growth rate of the acidogenic bacteria (m1 ), and the specific growth rate of the methanogenic bacteria (m2 ), are described by Monod type structures as follows: m1 ¼
mmax 1 S 1 kS1 þ S 1
m2 ¼
mmax 2 S2 . kS2 þ S 2
(6)
Generally S00 is an unmeasurable (in real time) perturbation, while S 000 is a known constant or control input. In all cases, the washout of microorganisms is undesirable, that is why changes of the total dilution rate D ¼ D1 þ D2 and the perturbation S00 are possible only in some admissible ranges (for fixed value of S 000 ): 0pDpDsup ;
S00 inf pS 00 pS00sup .
(7)
To summarize, the process is characterized by the state xT ¼ ½X 1 S1 X 2 S2 , the input vector uT ¼ ½D S 000 or uT ¼ ½D2 S 000 and the measured output vector y ¼ Q. 2.3. Parameter estimation For parameter estimation, the value of eight parameters has to be determined, so as the initial state variable X 1 ð0Þ; S1 ð0Þ and X 2 ð0Þ. S 2 ð0Þ is directly related to the measured biogas flow rate Qð0Þ. An identifiability test quickly establishes that this whole set of parameters cannot be identified in one step. A sensitivity analysis with 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 , is the most sensitive group, i.e., small variations of these parameters involve strong variation of the simulated behaviour of the process model (1)–(5). The second one, composed of the parameters of Monod expressions, is less sensitive, as much as some standard values may be fixed. Since prior knowledge about initial parameter values is essential in solving non-linear estimation problems (to avoid biased estimates to a large extent), parameter identification has started with initial values known 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 other coefficients using optimization method; estimation of the second group of coefficients with the abovedetermined values of the first group in the following step, etc. The identification procedure has been initiated in the present case with mmax 1 ¼ 0:2 day1 , mmax 2 ¼ 0:25 day1 , kS1 ¼ 0:3 g=L and kS2 ¼ 0:37 g=L and initial value of the state vector corresponding to the initial steady state Simeonov (2000). These values had
been determined from previous experiments without acetate addition. A simplex method has been used for each step of the estimation procedure. Parameter identification has been done with experimental data for Q provided from experiments with known values of D1 ¼ 0:0375 day1 ; the influent (S 00 ¼ 75 g=L; 1 D2 ¼ 0:0125 day ). The experimental data presented in Fig. 2 (with step addition of acetate) served for parameter identification. They involve 44 measurements of the biogas flow rate Q. The first period, from t ¼ 35 to 41 days, is with S 000 ¼ 25 g=L, the second one, from day 42 to day 90, is with S 000 ¼ 50 g=L, and the third one is with S 000 ¼ 75 g=L. The parameter identification step resulted in the estimates given in Table 2. Experimental data and model simulation results for the same case are presented in Fig. 3. Experimental data and model simulation results with pulse addition of acetate (4 pulses with amplitudes of 0.5, 0.75, 1.0 and 1.5 g/L) are presented 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) and simulated Q (solid line) confirms the quality of the modelling step. Comparing the results from Simeonov (2000) and Angelidaki et al. (1999) (without acetate addition) with the new obtained parameter values the conclusion is, as expected that differences exist only for values of kS2 and k4 , related with the methanogenic step of the process.
Table 2 Values obtained for the coefficients of the fourth order model with acetate addition mmax 1 ðday1 Þ
mmax 2 ðday1 Þ
k S1 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
1.6 Qexp
1.4 1.2
X1
1 0.8 Q
0.6
5 S1
0.5 S2
0.4 0.2
0.1 X2 0
5
10
15 20 T i m e (days)
25
30
Fig. 3. Evolution of the main variables in the case of step addition of acetate S000 .
ARTICLE IN PRESS I. Simeonov, I. Queinnec / Control Engineering Practice 14 (2006) 799–810
Qm [L/day], Qexp[L/day]
2.0 Qexp 1.5
1.0 Qm 0.5
0
5
10
15 20 Time (days)
25
30
Fig. 4. Evolution of Q (from experiments and simulations) in the case of step addition of acetate S000 .
3
110
Q[L/day ], S [g/L]
2.5 2
90
1.5 70
1 0.5 0
_ S = C1.S1 + C2.S2 0
0.05
0.1
0.15 D [day-1]
50 0.2
0.25
¯ Fig. 5. Input–output characteristics Q ¼ QðDÞ and S¯ ¼ SðDÞ.
2. Maximal depollution effect. This criterion aims at reducing the concentration of organic matter at the outlet of the process. It then corresponds to an ¯ ecological criterion and may be expressed as min S, ¯S ¼ C 1 S1 þ C 2 S 2 ; C 1 and C 2 being given constants. S¯ is generally associated with the COD (Chemical Oxygen Demands) of the outlet substance. 3. Compromise between energetical and depollution criteria. In this case, the criterion is expressed as a combination of Q and S¯ to maximize, for example ¯ or Q=kS, ¯ k40. This criterion may be very (Q kS) useful 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, for various values of the influent organics S 00 . It brings to the fore the opposite effect of the dilution rate D on the biogas flow rate (which has to be maximized) and on S¯ (which has to be minimized). Indeed, S¯ is minimized as much as D decreases. Then, for a small value of the dilution rate, the retention time related to the inverse of D is very large, which induces poor efficiency of the process and small production of biogas. On the contrary, when high quantity of biogas is wanted it results in poor depollution effect. This justifies the ¯ necessity 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 the derivatives in the model (1)–(4) to zero are solved to determine the analytical steady-state values:
3. Input–output characteristics S1 ¼ In this section, the steady-state analysis and optimality conditions determination are achieved under the hypothesis of identical maximum specific rates for both acidogenic and methanogenic bacteria ðmmax 1 ¼ mmax 2 ¼ mm Þ. This hypothesis could be relaxed and is only considered to simplify the presentation. 3.1. Optimal steady states From industrial point of view, operating conditions have to be searched such that the process runs nearby some static optimal points. However, there exist several optimal operating conditions related to several optimality criteria. The main optimality criteria may be formulated as: 1. Maximal amount of biogas production Q. This is an energetical criterion, in the sense that the control objective rather concerns the production of additional energy source than the reduction of wastes.
803
kS1 D , mmax 1 D
kS1 D 1 D1 S 00 X1 ¼ , mmax 1 D k1 D S2 ¼
kS2 D , mmax 2 D
kS1 D 1 k3 D1 S00 X2 ¼ mmax 1 D k2 k1 D kS2 D D2 S 000 þ , mmax 2 D D
(8)
(9)
(10)
ð11Þ
which results for the steady-state values of the biogas flow rate in: kS1 D k4 k3 D1 S 00 Q¼ D mmax 1 D k2 k1 D kS2 D D2 S 000 þ . ð12Þ mmax 2 D D
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804
Under hypothesis ðmmax 1 ¼ mmax 2 ¼ mm Þ and taking into account that D2 ¼ D D1 (D1 is assumed to be constant), it is shown in Appendix A that DQ opt
¼ mm
pffiffiffiffiffiffi 1 W
(13)
with k3 kS1 þ k1 kS2 W¼ ; k3 kS1 þ k1 kS2 þ k1 S 000
W o1
Qopt
ð15Þ
3.3. Steady-state analysis for combination of energetical and ecological criteria The same procedure as in the previous section may now be applied to determine an optimal Dopt maximiz¯ Let us consider the ing some mixed criterion on Q and S. ¯ k40. As previously, it is shown in criterion J ¼ Q=kS; Appendix B that DJopt
25
50
75
Dsup 1 (day1 ) Dsup 2 (day1 ) 1 DQ opt (day )
0.166 0.358 0.159
0.166 0.5 0.17
0.166 0. 647 0.176
DJopt (day1 )
0.0695
0.075
0.077
In the same way, the condition on the dilution rate such that X 2 is positive expresses as: DoDsup 2 ¼
Moreover, the optimum Qopt is larger that the one which would be obtained in the case without acetate addition.
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðk1 S000 k3 S00 ÞD1 mm ¼ k3 kS1 þ k1 kS2 þ k1 S 000
S000 (g/L)
(14)
It results in an optimal biogas flow rate Qopt is given by: pffiffiffiffiffiffi k3 k4 0 k4 ¼ S 0 D1 þ mm 1 W D1 S 000 k1 k2 k2 pffiffiffiffiffiffi2 mm k4 ðk3 kS1 þ k1 kS2 Þ 1 W p ffiffiffiffiffi ffi . k1 k2 W
Table 3 Upper bounds and optimal values of the dilution rate for various acetate influent conditions
(16)
k3 D1 S 00 þ k1 D2 S000 2k1 ðkS1 þ kS2 Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! 4mm k1 ðkS1 þ kS2 Þ 1 . 1þ k3 D1 S 00 þ k1 D2 S 000
ð20Þ
Finally, Dsup ¼ MinðDsup 1 ; Dsup 2 Þ. Let us consider the kinetic parameters given in Table 2 and the following condition for organic waste addition S 00 ¼ 75 g=L and D1 ¼ 0:0375 day1 . The conditions 1 J (19), (20) and optimal values for DQ opt (day ) and Dopt 1 (day ) are given in Table 3 for various acetate influent conditions. From the table, it may be concluded that the optimal dilution rate which would maximize the production of 00 biogas, DQ opt , is only admissible for S 0 ¼ 25 g=L, but cannot be reached for larger values of the influent acetate concentration. The optimal value of DJopt maximizing a mixed criterion on Q and S¯ is always attainable, and, as it was expected from Fig. 5, is much smaller than for the case of DQ opt .
is an optimum, with existence condition given by k1 S 000 k3 S00 40.
(17)
Remark. It may be checked that, according to the numerical value for k1 and k3 , all the simulated and experimental evaluations respect the existence condition (17).
4.1. Formulation of the control problem The problem of optimal control of anaerobic digestion may be decomposed in three subproblems: (a) static optimization; (b) optimal start-up; (c) dynamic optimization.
3.4. Physical admissibility of optimal dilution rates According to model (1)–(5) and definition of the dilution rate, the first existence condition is classically 0oDommax .
4. Linearizing control
(18)
Moreover, according to (8), it may be verified that the biomass concentration X 1 is positive implies that sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! mm kS1 D1 S 00 sup 1 1þ 1 . (19) DoD ¼ 2kS1 D1 S00
The static optimization of the process was presented in the previous section. The problem for optimal startup of the process with the new defined control input (D2 ) is a very promising one. The problem of the dynamic optimization is reduced to regulation of: (1) the biogas production rate Q (energy supply), or (2) the organics concentration S¯ (depollution control),
ARTICLE IN PRESS I. Simeonov, I. Queinnec / Control Engineering Practice 14 (2006) 799–810 at a prescribed value (Q and S¯ , respectively) by acting upon the dilution rate D ¼ D1 þ D2 . 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 of unmeasured variations of the inlet soluble organics S 00 using the addition of stimulating substance (dilution rate D2 or influent acetate concentration S000 ) as the control input.
805
Manipulation of time-derivative of m2 may cause many computation problems, and it is much more careful to consider an algebraic expression of dm2 =dt derived from the non-linear expression for m2 (6) and the timederivative of S2 , i.e., kS2 1 2 dS2 kS2 1 2 dm2 k2 m m Q ¼ ¼ 2 2 2 2 dt mmax 2 S 2 dt mmax 2 S2 k4 þ k3 m1 X 1 þ D2 S 000 DS 2 . ð25Þ
4.2. Control algorithm for regulation of Q The model (1)–(5) may be decomposed into two parts following 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 is needed. Proposition. The regulation of the biogas flow rate Q may be achieved through linearizing control, where the control input is given as: (1) D2 ðtÞ ¼ 1 l ðQ QÞ 1 ðm2 DÞ þ DS 2 00 S0 y Q y k2 ð21Þ k3 m1 X 1 þ Q ; ðS000 ¼ const:Þ k4 with 0oD2 ðtÞoDsup
Then after substitutions of (24) and (25) in (23), one obtains: D2 S 000 ¼
l Q Q 1 k2 ðm2 DÞ þ DS 2 k3 m1 X 1 þ Q, y Q y k4 (26)
where y¼
kS2 m2 mmax 2 S 22
.
From expression (26), two input variables may be considered. Either the dilution rate D2 (S 000 being constant) related to the addition of influent acetate or the concentration of the influent acetate S 000 (D2 being constant) may be used as control input, which results in the two proposed control laws (21) and (22). Remark. The laws (21) and (22) do not directly depend on the influent diluted organics S00 . They depend, however, on the acidogenic reaction rate m1 X 1 which has to be estimated on-line. This can be done by using an observer-based estimator (Lubenova, Simeonov, & Queinnec, 2002).
or (2) S00 ðtÞ ¼ 1 l ðQ QÞ 1 ðm2 DÞ þ DS 2 0 D2 y Q y k2 k3 m1 X 1 þ Q ; ðD2 ¼ const:Þ k4
5. Simulation studies and discussion ð22Þ
with 0oS000 ðtÞoS 00sup 0 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 firstorder closed-loop dynamics: dQ lðQ QÞ ¼ 0, (23) dt where the first time-derivative of Q is given by differentiation of (5) as dQ dm ¼ k4 2 X 2 þ Qðm2 DÞ. dt dt
(24)
The designed algorithms are evaluated by simulation. Some results of the simulations with the control algorithm (21) are shown in Fig. 6 (for l ¼ 0:4) and Fig. 7 (for l ¼ 0:1). In both figures the simulation conditions are as follows: step changes of the set point Q (L/day) (0.6 from 0 to 30th day, 1.2 from 30th to 60th day, then 0.6 after 60th day); step and sinusoidal changes of the disturbance S 00 (7.5 g/L between day 0 and day 20, then again between day 40 and day 60, 15 g/ L between day 20 and day 40, then again between day 60 and day 80 and a sinusoidal signal of 20% of amplitude with period of 8 h is added on the step disturbance); D1 ¼ 0:0325 day1 ¼ const:, S000 ¼ 25 g=L ¼ const:; 10% of noise under Q (in L/day). Biomasses X 1 and X 2 are plotted in subplot (a), S1 (in g/L), S 2 (in g/L), Q, S 00 and D2 are plotted in subplots (b), (c), (d), (e) and (f), respectively.
ARTICLE IN PRESS I. Simeonov, I. Queinnec / Control Engineering Practice 14 (2006) 799–810
(a)
4
0.4
3
substrate S1
biomasses X1, X2
806
X2 : dashdot line
2 X1 : solid line
1 0
0
20
40
60
80
0.3 0.2 0.1 0
(b)
S’0
(c)
40
60
80
0
20
40
60
80
0
20
40 time (d)
60
80
1.5
0.3 Q
1
0.2
0.5
0.1 0
20
0
20
40
60
80
0
(d)
20
0.06
15
0.04 D2
substrate S2
0.4
0
0.02
10 5
0
20
(e)
40 time (d)
60
0
80
(f)
(a)
4
0.4 X2 : dashdot line
3
substrate S1
biomasses X1, X2
Fig. 6. Simulations with control action D2 for l ¼ 0:4.
2 X1 : solid line 1 0
0
20
40
60
80
(b)
0
0
20
40
60
80
0
20
40
60
80
0
20
40 time (d)
60
80
1
0.15
0.5
0.1 0
20
40
60
80
0
(d) 0.06
15
0.04 D2
S0’
20
10 5
(e)
0.1
Q
substrate S2
0.2
0.05
0.2
1.5
0.25
(c)
0.3
0.02
0
20
40 time (d)
60
0
80
(f)
Fig. 7. Simulations with control action D2 for l ¼ 0:1.
Some results of the simulations with the control algorithm (22) are shown in Fig. 8 for l ¼ 0:4. The simulation conditions are the same as in the previous case (except D2 ¼ 0:0125 day1 ¼ const:, S 000 ¼ var.). Biomasses X 1 and X 2 are plotted in subplot (a),
S 1 ; S2 ; Q; S 00 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 S 000 ), however
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(d)
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S0"
,
S0
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10 5
(e)
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(f)
Fig. 8. Simulations with control action S000 for l ¼ 0:4.
1.6 nominal case 0.8 µ max1 1.2 µ max1
1.4 1.2 Q (L/day)
differences exist between evolutions of X 1 ; X 2 and S 1 due to the fact that changes of D2 result in changes of D. Simulation studies with the linearizing control algorithms show that they present very good performances of regulation with different values of the tuning parameter l. Some problems, however, arise when l is too much increased. It may result into some vanishing oscillations (for l ¼ 1) or even instability due to saturations on the actuators. This problem may be overcome with more precise determination of Dsup and 0 S 0inf (in each particular case it is possible to measure 0 S 0inf 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 are estimated by suitable observers (Lubenova et al., 2002). Moreover, Figs. 9–16 show the influence on the control of the biogas production Q of model errors. Simulations compare the biogas production Q in the nominal case (control law D2 ðtÞ is computed by using the process parameters) with respect to the biogas production Q obtained when the control law is computed with a model error of þ or 20% on each parameter. It may be checked from these figures that k1 has no influence on the quality of the control and that mmax 1 ; mmax 2 ; kS1 ; kS2 and k3 have a small influence on the quality of the control. On the other hand, the efficiency of the control is closely related to the quality of estimates k2 and k4 . This is an expected result since the steady-state value of D2 is closely related to the factor k2 =k4 .
1 0.8 0.6 0.4 0.2 0
10
20
30
40 50 time (day)
60
70
80
Fig. 9. Influence of model error—error of þ or 20% on mmax 1 .
The algorithm (22) is slightly more difficult to implement than algorithm (21) since it is generally easier to act on a pump, i.e., on a dilution rate than on a concentration. This can, however, be done by using a system of two pumps relied to two tanks allowing to control (with constant flow rate) variations of the concentration. But in spite of the technical difficulty, the algorithm (22) is theoretically more correct since D is kept constant. Then the control only acts on the methanogenic phase of the process.
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1.6
1.6 nominal case 0.8 µ max2 1.2 µ
1.4
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1.4
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0.6
0.4
0.4
0
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30
40 50 time (day)
60
70
0.2
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Fig. 10. Influence of model error—error of þ or 20% on mmax 2 .
30
40 50 time (day)
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80
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Q (L/day)
20
nominal case 0.8k2 1.2k2
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30
40 50 time (day)
60
70
80
1.6
nominal case 0.8kS2 1.2kS2
1.4
0
Fig. 14. Influence of model error—error of þ or 20% on k2 .
Fig. 11. Influence of model error—error of þ or 20% on kS1 .
nominal case 0.8k3 1.2k3
1.4
1.2
Q (L/day)
1.2
1 0.8
1 0.8
0.6
0.6
0.4
0.4
0.2
10
1.6
nominal case 0.8kS 1.2kS11
1.4
0.2
0
Fig. 13. Influence of model error—error of þ or 20% on k1 .
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0.6
0.2
nominal case 0.8k1 1.2k1
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10
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30
40 50 time (day)
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70
80
Fig. 12. Influence of model error—error of þ or 20% on kS2 .
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0
10
20
30
40 50 time (day)
60
70
Fig. 15. Influence of model error—error of þ or 20% on k3 .
80
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may be expressed in the form:
1.6 nominal case 0.8k4 1.2k4
Q (L/day)
1.4
Q ¼ AD1 B
1.2
with
1
A¼
D2 þ CD mm D
k4 k3 0 S 0 S000 ; k2 k1 k4 C ¼ S000 . k2
0.8 0.6
B¼
(27)
ðk3 kS1 þ k1 kS2 Þk4 ; k1 k2
It is then possible to evaluate an optimal biogas flow rate Qopt from the determination of an extremum of (27), obtained when the derivative of Q with respect to D is equal to zero:
0.4 0.2
809
0
10
20
30
40 50 time (day)
60
70
80
Fig. 16. Influence of model error—error of þ or 20% on k4 .
dQ 2Dðmm DÞ þ D2 ¼CB dD ðmm DÞ2 ¼ 0 ! D2 2mm D þ
6. Conclusion Experimental and analytical studies have shown that addition of acetate (with pH correction of the added substrate) allows to stabilize the process with respect to load and activity disturbances and to increase the amount of biogas obtained from the anaerobic digestion of organic wastes. This fact is very promising for stabilization of the biogas plants in the case of strong variations of the influent organic matter. Theoretical studies and simulation results have proven that the linearizing control design on the basis of an appropriate model of the anaerobic digestion with a new control input, the addition of acetate, may be very useful for the regulation of the amount of biogas in the realistic case of strong variations of the influent organic matter. From practical point of view both linearizing control algorithms ((21) and (22)) are easy to implement. However, even if it is easier to control variations of a flow rate (law (21)), the control of the influent acetate concentration presents the advantage to keep the process dilution rate constant.
Acknowledgements This work was supported by Contract no TH-1004/00 of The Bulgarian National Found ‘‘Scientific researches’’ and by a CNRS-BAS exchange program.
Only one root of this second-order equation is admissible, i.e., pffiffiffiffiffiffi W ; with DQ opt ¼ mm 1 W¼
k3 kS1 þ k1 kS2 ; k3 kS1 þ k1 kS2 þ k1 S000
W o1.
pffiffiffiffiffiffi The other root, D ¼ mm ð1 þ W Þ, is not admissible since D4mm is not physically admissible (it would results in the washout of the process). From evaluation of the second derivative d2 Q=dD2 , it may be checked that: pffiffiffiffiffiffi k3 k4 0 k4 Qopt ¼ S 0 D1 þ mm 1 W D1 S000 k1 k2 k2 pffiffiffiffiffiffi2 mm k4 ðk3 kS1 þ k1 kS2 Þ 1 W pffiffiffiffiffiffi k1 k2 W is a maximum.
Appendix B. Mixed energetical and ecological criterion ¯ k40. Let us consider the mixed criterion J ¼ Q=kS; Similarly to Appendix A, we only present the case mmax 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: J¼
a1 þ a2 D a3 D 2 a4 D
with Appendix A. Optimal biogas flow rate criterion a1 ¼ AD1 mm ¼ A ¼ For sake of simplicity, we only consider the hypothesis mmax 1 ¼ mmax 2 ¼ mm . Taking into account that D2 ¼ D D1 (D1 is assumed to be constant), Eq. (12)
m2m C ¼ 0. BþC
a2 ¼
k4 k3 0 S 0 S 000 D1 mm ; k2 k1
k4 ðk1 S 000 ðmm þ D1 Þ k3 S00 D1 Þ; k1 k2
(28)
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k4 ðk3 kS1 þ k1 kS2 Þ þ k1 k4 S000 40; k1 k2 þ c2 kS2 Þ40.
a3 ¼ B þ C ¼ a4 ¼ kðc1 kS1
It is then possible to evaluate an optimal value for the criterion J from the determination of an extremum of (28), obtained when the derivative of J with respect to D is equal to zero: dJ a1 þ a3 D2 ðk3 S 00 k1 S 000 ÞD1 mm 2 ¼ ¼ 0 ! D ¼ . dD k1 S 000 þ k3 kS1 þ k1 kS2 a4 D2 Only one root of this second-order equation is admissible, i.e., sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðk1 S000 k3 S00 ÞD1 mm J Dopt ¼ , k3 kS1 þ k1 kS2 þ k1 S 000 where the existence condition is k1 S000 k3 S 00 40. From evaluation of the second derivative d2 J=dD2 , it may be checked that Dopt is a maximum. References Angelidaki, I., Ellegaard, L., & Ahring, B. (1999). A comprehensive model of anaerobic bioconversion of complex substrates to biogas. Biotechnology and Bioengineering, 63, 363–372. Bastin, G., & Dochain, D. (1991). On-line estimation and adaptive control of bioreactors. Amsterdam: Elsevier Science Publication. Batstone, D. J., Keller, J., Angelidaki, R. I., Kalyuzhny, S. V., Pavlostathis, S. G., Rozzi, A., et al. (2002). The IWA anaerobic digestion model No 1 (ADM1). Water Science and Technology, 45(10), 65–73. Bernard, O., Hadj-Sadok, Z., & Dochain, D. (1999). Dynamical modelling and state estimation of anaerobic wastewater treatment plants. European control conference, Karslru¨he, Germany.
Dochain, D. (1995). Recent approaches for the modelling, monitoring and control of anaerobic digestion processes. In Proceedings of the international workshop on monitoring and control of anaerobic digestion processes (pp. 23–29). Narbonne, France. Haag, J., Vande Wouwer, A., & Queinnec, I. (2003). Macroscopic modelling and identification of a biogas production process. Chemical Engineering Sciences, 58(9), 4307–4316. Lubenova, V., Simeonov, I., Queinnec, I. (2002). Two-step parameter and state estimation of the anaerobic digestion. In Proceedings of the 15th IFAC world congress. Barcelona, Spain. Simeonov, I. (1999). Mathematical modelling and parameters estimation of anaerobic fermentation processes. Bioprocess Engineering, 21, 377–381. Simeonov, I. (2000). Methodology for parameter estimation of nonlinear models of anaerobic wastewaters treatment processes in stirred tank bioreactors. Proceedings of the fifth international symposium on systems analysis and computing in water quality management— WATERMATEX 2000 (pp. 8.40–8.47). Gent, Belgium. Simeonov, I., & Galabova, D. (2000). Investigations and mathematical modelling of the anaerobic digestion of organic wastes. Proceedings of the fifth international conference on environmental pollution (pp. 285–295). Thessaloniki, Greece. Simeonov, I., Galabova, D., & Queinnec, I. (2001). Investigations and mathematical modelling of the anaerobic digestion of organic wastes with addition of electron acceptors. Proceedings of the ninth world congress on anaerobic digestion 2001 (pp. 381–383). Antwerpen, Belgium. Simeonov, I., & Stoyanov, S. (1995). Dynamic output compensator control of methane fermentation. Proceedings of the workshop on monitoring and control of anaerobic digesters (pp. 47–51). Narbonne, France. Steyer, J.-Ph., Buffiere, P., Rolland, D., & Molleta, R. (1999). Advanced control of anaerobic digestion processes through disturbances monitoring. Water Research, 33, 2059–2068. Van Impe, J. F. M., Vanrolleghem, P. A. V., & Iserentant, D. M. (1998). Advanced instrumentation, date interpretation and control of biotechnological processes. Dordrecht: Kluwer Academic Publication. Zlateva, P., & Simeonov, I. (1995). Variable structure control of methane fermentation. System, Modelling, Control, 2, 433–436.
Control Engineering Practice 14 (2006) 799–810 www.elsevier.com/locate/conengprac
Linearizing control of the anaerobic digestion with addition of acetate (control of the anaerobic digestion) I. Simeonova, I. Queinnecb, a
Institute of Microbiology, Bulgarian Academy of Sciences, Acad. G. Bonchev St., Block 26, Sofia 1113, Bulgaria Laboratoire d’Analyse et d’Architecture des Syste`mes (LAAS/CNRS) 7, Avenue du Colonel Roche, 31077 Toulouse cedex 4, France
b
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 life process and has been confirmed as a promising method for solving some energy and ecological problems in agriculture and agro-industry. In such processes, generally carried out in continuously stirred tank bioreactors (CSTR), the organic matter is depolluted by microorganisms into biogas (methane and carbon dioxide) and fertilizer in the absence of oxygen. The biogas is an additional energy source and can also replace fossil fuel sources and therefore has a direct positive effect on the greenhouse gas reduction. Unfortunately, this process is very complex and may
Corresponding author. Tel.: +33 5 61 33 64 77; fax: +35 5 61 33 69 69. E-mail addresses: [email protected] (I. Simeonov), [email protected] (I. Queinnec).
0967-0661/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.conengprac.2005.04.011
sometimes become very unstable. It then needs more investigations. The first step concerns mathematical modelling of the process. It represents a very attractive tool for studying this process. Angelidaki, Ellegaard, and Ahring (1999) developed a model involving 16 variables with six main stages. The IWA Anaerobic Digestion Modelling Task group has federated the energy to produce the IWAADM1 (Batstone et al., 2002), involving 24 variables and many parameters. Such models are, however, not appropriate for control purposes due to their complexity. Numerous studies on special cases are also presented in the literature, but only few of them focus on models appropriate for state observation and control. A simple mass-balance model involving five variables has been proposed by Bernard, Hadj-Sadok, and Dochain (1999) to design a software sensor, with particular emphasis to alkalinity balance. Simeonov (1999) developed a secondorder non-linear model based on a mere stage to be useful 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 X1 concentration of acidogenic bacteria, g/L S1 concentration of substrate for acidogenic bacteria (mainly glucose), g/L X2 concentration of methane-producing (methanogenic) bacteria, g/L S2 concentration of substrate for methane-producing (methanogenic) bacteria (acetate), g/L S000 concentration of the acetate added in the influent liquid, g/L (a new control input) Q biogas flow rate, L/day S COD Chemical Oxygen Demand
dynamic model (hydrolysis, acidogenesis and methanogenesis) involving seven variables but only two biomass compounds to cope with identifiability problems related to the hydrolysis part. Moreover, because of very restrictive on-line information, the control of such a process is often reduced to the regulation of the biogas production rate (energy supply) or of the concentration of polluting organic matter (depollution control) at a desired value in presence of perturbations (Bastin & Dochain, 1991; Steyer, Buffiere, Rolland, & Molleta, 1999). According to the strongly non-linear input–output characteristics of the process, classical linear controllers have good performances only in a locally linear zone related to small variations of the dilution. More sophisticated robust and variable structure controllers (VSC) may be applied (Simeonov & Stoyanov, 1995; Zlateva & Simeonov, 1995) but even in that case, the performances of the closed loop system may be degraded due to the strongly non-linear dynamics of the process. On the contrary, linearizing algorithms for control of the anaerobic digestion proved to have very good performances (Bastin & Dochain, 1991; Dochain, 1995). Moreover, recent investigations have shown that addition of stimulating substances (acetate or glucose) in appropriate concentrations allow to stabilize the process and to increase the biogas flow rate (Simeonov & Galabova, 2000; Simeonov, Galabova, & Queinnec, 2001). The aim of this paper is then to design and investigate different algorithms for linearizing control of the anaerobic digestion using the addition of acetate as a control action. The control algorithms are based on a relatively simple model developed for this purpose. The outline of the paper is as follows. Section 2 concerns the process modelling. Experimental studies used for mathematical modelling and parameter estimation are presented, so as identifiability properties and identification procedure. In Section 3 steady-state analysis and
specific growth rate of the acidogenic bacteria, day1 m2 specific growth rate of the methanogenic bacteria, day1 k1 ; k2 ; k3 ; k4 ; mmax 1 ; mmax 2 ; kS1 and kS2 coefficients D1 dilution rate for the inlet diluted organics, day1 D2 dilution rate for the acetate added in the influent liquid, day1 D ¼ D1 þ D2 the total dilution rate, day1 xT ¼ ½X 1 S 1 X 2 S 2 the state vector uT ¼ ½D S000 the input vector y ¼ Q the measured output vector pH acidity/alkalinity index
m1
optimal steady states following different criteria on the basis of the developed model are performed. The control problem is formulated in Section 4, and two linearized algorithms for regulation of the biogas flow rate Q are designed on the basis of the reduced model (obtained for the particular case when only acetate is added) for the process. Both approaches are evaluated by simulation in Section 5. Finally, Section 6 contains some concluding remarks.
2. Process modelling and parameter estimation 2.1. Experimental studies Laboratory experiments have been carried out in CSTR with highly concentrated organic pollutants (cattle wastes) at mesophillic temperature and with addition of acetate in low concentrations (Simeonov & Galabova, 2000). The laboratory experimental set-up includes an automated bioreactor of a 3-l glass vessel developed and adapted to fulfil the requirements for anaerobic digestion. It is mechanically stirred by electrical drive and maintained at a constant temperature (34 0:5 C) by computer controller. The monitoring of the methane reactor is carried out by data acquisition computer system of on-line sensors, which provide the following measurements: pH, temperature, redox, speed of agitation and biogas flow rate (Q). A schematic diagram of the experimental laboratory-scale set-up is shown in Fig. 1, where 1 is the bioreactor; 2 the DC drive; 3 the biogas flow-meter; 4 the heating system; 5 the peristaltic pump; 6 the gas holder; 7 the converters unit; 8 the gas chromatograph; 9 the biogas flame; 10 the watt-hour meter; 11 the personal computer and 12 the printer. It is well known that anaerobic digestion is a selfstabilization process as long as disturbance magnitude
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Inlet
5
Table 1 The effect of acetate on the methane fermentation (in steady-state)
Substrate,Acetate,Glucose 11
1
4
nt t pH 2 6
12
CH4
9
8
10
220 V
VFA Outlet
Fig. 1. Experimental set-up.
Q (L/day), S0" (g/L)
Date (day)
7 3
801
1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0
Feeding D1 (day1 ) D2 (day1 ) S00 (g/L) S000 (g/L) Average value of Q (L/day) at steady state
From 1st to 34th
From 35th to 40th
From 41st to 50th
From 51st to 90th
0.0375 0.0125 68 0 0.35
0.0375 0.0125 68 25 0.5
0.0375 0.0125 68 50 0.9
0.0375 0.0125 68 75 1.2
2.2. Mathematical modelling of the process
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 .
does not exceed the buffer capacity of the medium, which would results in pH breakdown related to accumulation of volatile fatty acids (VFA) produced (Angelidaki et al., 1999; Batstone et al., 2002). In the approach proposed in the paper, addition of acetate acts as a control input. Then biogas production will increase, but pH breakdown may occur. To prevent from such a failure, Simeonov and Galabova (2000) have shown that pH regulation (correction till pH 7.5) has to be done in the inlet mixture ðsubstrate þ acetateÞ rather than in the bioreactor. Experimental design has then been developed consisting in appropriate (by amplitude and time) step and pulse changes of the acetate addition (S000 ) and measurements of the responses of the biogas flow rate ðQÞ and of the acetate concentration in the anaerobic bioreactor (by gas chromatography). Some rather repeatable results are shown in Fig. 2 (for step changes of (S 000 ) from 0 to 25 g/ L at 35th day, from 25 to 50 g/L at 41st and to 75 g/L at 51st day) and the steady state of biogas flow rate after step changes are given in Table 1. It is seen that the settling time for each step response (new steady-state) is about 5–6 days. The reported data offer the suggestion that acetate addition positively affects the methane production and increased levels of acetate as electron donor result in faster rates of methanogenesis (the second important phase of the methane fermentation) (Simeonov & Galabova, 2000).
On the basis of the above-presented experimental investigations and following the so-called two-stage biochemical scheme of the methane fermentation (Bastin & Dochain, 1991), the following simplest realistic non-linear model with two control inputs is proposed: dX 1 ¼ ðm1 DÞX 1 , dt
(1)
dS 1 ¼ k1 m1 X 1 þ D1 S00 DS 1 , dt
(2)
dX 2 ¼ ðm2 DÞX 2 , dt
(3)
dS 2 ¼ k2 m2 X 2 þ k3 m1 X 1 þ D2 S 000 DS 2 , dt
(4)
Q ¼ k4 m2 X 2 .
(5)
In this mass balance model, Eq. (1) describes the growth and changes of the acidogenic bacteria (X 1 ), consuming the appropriate substrate (S1 ), where the first term in the right side reflects the growth of the acidogenic bacteria and the second one reflects the effluent flow rate of liquid. The mass balance for this substrate is described by (2), where the first term reflects the consumption by the acidogenic bacteria, the second term reflects the influent flow rate of liquid with concentration of diluted organics S 00 , and the third one the effluent flow rate of liquid. Eq. (3) describes the growth and changes of the methane-producing (methanogenic) bacteria, with concentration X 2 , consuming acetate, with concentration S2 , where the first term in the right side reflects the growth of the methanogenic bacteria and the second one reflects the effluent flow rate of liquid. The mass balance equation for acetate (4) has four terms in his right side. The first one reflects the consumption of acetate by the methanogenic bacteria, the second one the acetate formed as a result of the
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activity of acidogenic bacteria, the third one the direct addition of acetate, with concentration S000 g/L, (a new control input) and the last one the acetate in the effluent liquid. The algebraic equation (5) describes the formation of biogas with flow rate Q. The specific growth rate of the acidogenic bacteria (m1 ), and the specific growth rate of the methanogenic bacteria (m2 ), are described by Monod type structures as follows: m1 ¼
mmax 1 S 1 kS1 þ S 1
m2 ¼
mmax 2 S2 . kS2 þ S 2
(6)
Generally S00 is an unmeasurable (in real time) perturbation, while S 000 is a known constant or control input. In all cases, the washout of microorganisms is undesirable, that is why changes of the total dilution rate D ¼ D1 þ D2 and the perturbation S00 are possible only in some admissible ranges (for fixed value of S 000 ): 0pDpDsup ;
S00 inf pS 00 pS00sup .
(7)
To summarize, the process is characterized by the state xT ¼ ½X 1 S1 X 2 S2 , the input vector uT ¼ ½D S 000 or uT ¼ ½D2 S 000 and the measured output vector y ¼ Q. 2.3. Parameter estimation For parameter estimation, the value of eight parameters has to be determined, so as the initial state variable X 1 ð0Þ; S1 ð0Þ and X 2 ð0Þ. S 2 ð0Þ is directly related to the measured biogas flow rate Qð0Þ. An identifiability test quickly establishes that this whole set of parameters cannot be identified in one step. A sensitivity analysis with 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 , is the most sensitive group, i.e., small variations of these parameters involve strong variation of the simulated behaviour of the process model (1)–(5). The second one, composed of the parameters of Monod expressions, is less sensitive, as much as some standard values may be fixed. Since prior knowledge about initial parameter values is essential in solving non-linear estimation problems (to avoid biased estimates to a large extent), parameter identification has started with initial values known 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 other coefficients using optimization method; estimation of the second group of coefficients with the abovedetermined values of the first group in the following step, etc. The identification procedure has been initiated in the present case with mmax 1 ¼ 0:2 day1 , mmax 2 ¼ 0:25 day1 , kS1 ¼ 0:3 g=L and kS2 ¼ 0:37 g=L and initial value of the state vector corresponding to the initial steady state Simeonov (2000). These values had
been determined from previous experiments without acetate addition. A simplex method has been used for each step of the estimation procedure. Parameter identification has been done with experimental data for Q provided from experiments with known values of D1 ¼ 0:0375 day1 ; the influent (S 00 ¼ 75 g=L; 1 D2 ¼ 0:0125 day ). The experimental data presented in Fig. 2 (with step addition of acetate) served for parameter identification. They involve 44 measurements of the biogas flow rate Q. The first period, from t ¼ 35 to 41 days, is with S 000 ¼ 25 g=L, the second one, from day 42 to day 90, is with S 000 ¼ 50 g=L, and the third one is with S 000 ¼ 75 g=L. The parameter identification step resulted in the estimates given in Table 2. Experimental data and model simulation results for the same case are presented in Fig. 3. Experimental data and model simulation results with pulse addition of acetate (4 pulses with amplitudes of 0.5, 0.75, 1.0 and 1.5 g/L) are presented 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) and simulated Q (solid line) confirms the quality of the modelling step. Comparing the results from Simeonov (2000) and Angelidaki et al. (1999) (without acetate addition) with the new obtained parameter values the conclusion is, as expected that differences exist only for values of kS2 and k4 , related with the methanogenic step of the process.
Table 2 Values obtained for the coefficients of the fourth order model with acetate addition mmax 1 ðday1 Þ
mmax 2 ðday1 Þ
k S1 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
1.6 Qexp
1.4 1.2
X1
1 0.8 Q
0.6
5 S1
0.5 S2
0.4 0.2
0.1 X2 0
5
10
15 20 T i m e (days)
25
30
Fig. 3. Evolution of the main variables in the case of step addition of acetate S000 .
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Qm [L/day], Qexp[L/day]
2.0 Qexp 1.5
1.0 Qm 0.5
0
5
10
15 20 Time (days)
25
30
Fig. 4. Evolution of Q (from experiments and simulations) in the case of step addition of acetate S000 .
3
110
Q[L/day ], S [g/L]
2.5 2
90
1.5 70
1 0.5 0
_ S = C1.S1 + C2.S2 0
0.05
0.1
0.15 D [day-1]
50 0.2
0.25
¯ Fig. 5. Input–output characteristics Q ¼ QðDÞ and S¯ ¼ SðDÞ.
2. Maximal depollution effect. This criterion aims at reducing the concentration of organic matter at the outlet of the process. It then corresponds to an ¯ ecological criterion and may be expressed as min S, ¯S ¼ C 1 S1 þ C 2 S 2 ; C 1 and C 2 being given constants. S¯ is generally associated with the COD (Chemical Oxygen Demands) of the outlet substance. 3. Compromise between energetical and depollution criteria. In this case, the criterion is expressed as a combination of Q and S¯ to maximize, for example ¯ or Q=kS, ¯ k40. This criterion may be very (Q kS) useful 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, for various values of the influent organics S 00 . It brings to the fore the opposite effect of the dilution rate D on the biogas flow rate (which has to be maximized) and on S¯ (which has to be minimized). Indeed, S¯ is minimized as much as D decreases. Then, for a small value of the dilution rate, the retention time related to the inverse of D is very large, which induces poor efficiency of the process and small production of biogas. On the contrary, when high quantity of biogas is wanted it results in poor depollution effect. This justifies the ¯ necessity 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 the derivatives in the model (1)–(4) to zero are solved to determine the analytical steady-state values:
3. Input–output characteristics S1 ¼ In this section, the steady-state analysis and optimality conditions determination are achieved under the hypothesis of identical maximum specific rates for both acidogenic and methanogenic bacteria ðmmax 1 ¼ mmax 2 ¼ mm Þ. This hypothesis could be relaxed and is only considered to simplify the presentation. 3.1. Optimal steady states From industrial point of view, operating conditions have to be searched such that the process runs nearby some static optimal points. However, there exist several optimal operating conditions related to several optimality criteria. The main optimality criteria may be formulated as: 1. Maximal amount of biogas production Q. This is an energetical criterion, in the sense that the control objective rather concerns the production of additional energy source than the reduction of wastes.
803
kS1 D , mmax 1 D
kS1 D 1 D1 S 00 X1 ¼ , mmax 1 D k1 D S2 ¼
kS2 D , mmax 2 D
kS1 D 1 k3 D1 S00 X2 ¼ mmax 1 D k2 k1 D kS2 D D2 S 000 þ , mmax 2 D D
(8)
(9)
(10)
ð11Þ
which results for the steady-state values of the biogas flow rate in: kS1 D k4 k3 D1 S 00 Q¼ D mmax 1 D k2 k1 D kS2 D D2 S 000 þ . ð12Þ mmax 2 D D
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Under hypothesis ðmmax 1 ¼ mmax 2 ¼ mm Þ and taking into account that D2 ¼ D D1 (D1 is assumed to be constant), it is shown in Appendix A that DQ opt
¼ mm
pffiffiffiffiffiffi 1 W
(13)
with k3 kS1 þ k1 kS2 W¼ ; k3 kS1 þ k1 kS2 þ k1 S 000
W o1
Qopt
ð15Þ
3.3. Steady-state analysis for combination of energetical and ecological criteria The same procedure as in the previous section may now be applied to determine an optimal Dopt maximiz¯ Let us consider the ing some mixed criterion on Q and S. ¯ k40. As previously, it is shown in criterion J ¼ Q=kS; Appendix B that DJopt
25
50
75
Dsup 1 (day1 ) Dsup 2 (day1 ) 1 DQ opt (day )
0.166 0.358 0.159
0.166 0.5 0.17
0.166 0. 647 0.176
DJopt (day1 )
0.0695
0.075
0.077
In the same way, the condition on the dilution rate such that X 2 is positive expresses as: DoDsup 2 ¼
Moreover, the optimum Qopt is larger that the one which would be obtained in the case without acetate addition.
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðk1 S000 k3 S00 ÞD1 mm ¼ k3 kS1 þ k1 kS2 þ k1 S 000
S000 (g/L)
(14)
It results in an optimal biogas flow rate Qopt is given by: pffiffiffiffiffiffi k3 k4 0 k4 ¼ S 0 D1 þ mm 1 W D1 S 000 k1 k2 k2 pffiffiffiffiffiffi2 mm k4 ðk3 kS1 þ k1 kS2 Þ 1 W p ffiffiffiffiffi ffi . k1 k2 W
Table 3 Upper bounds and optimal values of the dilution rate for various acetate influent conditions
(16)
k3 D1 S 00 þ k1 D2 S000 2k1 ðkS1 þ kS2 Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! 4mm k1 ðkS1 þ kS2 Þ 1 . 1þ k3 D1 S 00 þ k1 D2 S 000
ð20Þ
Finally, Dsup ¼ MinðDsup 1 ; Dsup 2 Þ. Let us consider the kinetic parameters given in Table 2 and the following condition for organic waste addition S 00 ¼ 75 g=L and D1 ¼ 0:0375 day1 . The conditions 1 J (19), (20) and optimal values for DQ opt (day ) and Dopt 1 (day ) are given in Table 3 for various acetate influent conditions. From the table, it may be concluded that the optimal dilution rate which would maximize the production of 00 biogas, DQ opt , is only admissible for S 0 ¼ 25 g=L, but cannot be reached for larger values of the influent acetate concentration. The optimal value of DJopt maximizing a mixed criterion on Q and S¯ is always attainable, and, as it was expected from Fig. 5, is much smaller than for the case of DQ opt .
is an optimum, with existence condition given by k1 S 000 k3 S00 40.
(17)
Remark. It may be checked that, according to the numerical value for k1 and k3 , all the simulated and experimental evaluations respect the existence condition (17).
4.1. Formulation of the control problem The problem of optimal control of anaerobic digestion may be decomposed in three subproblems: (a) static optimization; (b) optimal start-up; (c) dynamic optimization.
3.4. Physical admissibility of optimal dilution rates According to model (1)–(5) and definition of the dilution rate, the first existence condition is classically 0oDommax .
4. Linearizing control
(18)
Moreover, according to (8), it may be verified that the biomass concentration X 1 is positive implies that sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! mm kS1 D1 S 00 sup 1 1þ 1 . (19) DoD ¼ 2kS1 D1 S00
The static optimization of the process was presented in the previous section. The problem for optimal startup of the process with the new defined control input (D2 ) is a very promising one. The problem of the dynamic optimization is reduced to regulation of: (1) the biogas production rate Q (energy supply), or (2) the organics concentration S¯ (depollution control),
ARTICLE IN PRESS I. Simeonov, I. Queinnec / Control Engineering Practice 14 (2006) 799–810 at a prescribed value (Q and S¯ , respectively) by acting upon the dilution rate D ¼ D1 þ D2 . 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 of unmeasured variations of the inlet soluble organics S 00 using the addition of stimulating substance (dilution rate D2 or influent acetate concentration S000 ) as the control input.
805
Manipulation of time-derivative of m2 may cause many computation problems, and it is much more careful to consider an algebraic expression of dm2 =dt derived from the non-linear expression for m2 (6) and the timederivative of S2 , i.e., kS2 1 2 dS2 kS2 1 2 dm2 k2 m m Q ¼ ¼ 2 2 2 2 dt mmax 2 S 2 dt mmax 2 S2 k4 þ k3 m1 X 1 þ D2 S 000 DS 2 . ð25Þ
4.2. Control algorithm for regulation of Q The model (1)–(5) may be decomposed into two parts following 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 is needed. Proposition. The regulation of the biogas flow rate Q may be achieved through linearizing control, where the control input is given as: (1) D2 ðtÞ ¼ 1 l ðQ QÞ 1 ðm2 DÞ þ DS 2 00 S0 y Q y k2 ð21Þ k3 m1 X 1 þ Q ; ðS000 ¼ const:Þ k4 with 0oD2 ðtÞoDsup
Then after substitutions of (24) and (25) in (23), one obtains: D2 S 000 ¼
l Q Q 1 k2 ðm2 DÞ þ DS 2 k3 m1 X 1 þ Q, y Q y k4 (26)
where y¼
kS2 m2 mmax 2 S 22
.
From expression (26), two input variables may be considered. Either the dilution rate D2 (S 000 being constant) related to the addition of influent acetate or the concentration of the influent acetate S 000 (D2 being constant) may be used as control input, which results in the two proposed control laws (21) and (22). Remark. The laws (21) and (22) do not directly depend on the influent diluted organics S00 . They depend, however, on the acidogenic reaction rate m1 X 1 which has to be estimated on-line. This can be done by using an observer-based estimator (Lubenova, Simeonov, & Queinnec, 2002).
or (2) S00 ðtÞ ¼ 1 l ðQ QÞ 1 ðm2 DÞ þ DS 2 0 D2 y Q y k2 k3 m1 X 1 þ Q ; ðD2 ¼ const:Þ k4
5. Simulation studies and discussion ð22Þ
with 0oS000 ðtÞoS 00sup 0 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 firstorder closed-loop dynamics: dQ lðQ QÞ ¼ 0, (23) dt where the first time-derivative of Q is given by differentiation of (5) as dQ dm ¼ k4 2 X 2 þ Qðm2 DÞ. dt dt
(24)
The designed algorithms are evaluated by simulation. Some results of the simulations with the control algorithm (21) are shown in Fig. 6 (for l ¼ 0:4) and Fig. 7 (for l ¼ 0:1). In both figures the simulation conditions are as follows: step changes of the set point Q (L/day) (0.6 from 0 to 30th day, 1.2 from 30th to 60th day, then 0.6 after 60th day); step and sinusoidal changes of the disturbance S 00 (7.5 g/L between day 0 and day 20, then again between day 40 and day 60, 15 g/ L between day 20 and day 40, then again between day 60 and day 80 and a sinusoidal signal of 20% of amplitude with period of 8 h is added on the step disturbance); D1 ¼ 0:0325 day1 ¼ const:, S000 ¼ 25 g=L ¼ const:; 10% of noise under Q (in L/day). Biomasses X 1 and X 2 are plotted in subplot (a), S1 (in g/L), S 2 (in g/L), Q, S 00 and D2 are plotted in subplots (b), (c), (d), (e) and (f), respectively.
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(a)
4
0.4
3
substrate S1
biomasses X1, X2
806
X2 : dashdot line
2 X1 : solid line
1 0
0
20
40
60
80
0.3 0.2 0.1 0
(b)
S’0
(c)
40
60
80
0
20
40
60
80
0
20
40 time (d)
60
80
1.5
0.3 Q
1
0.2
0.5
0.1 0
20
0
20
40
60
80
0
(d)
20
0.06
15
0.04 D2
substrate S2
0.4
0
0.02
10 5
0
20
(e)
40 time (d)
60
0
80
(f)
(a)
4
0.4 X2 : dashdot line
3
substrate S1
biomasses X1, X2
Fig. 6. Simulations with control action D2 for l ¼ 0:4.
2 X1 : solid line 1 0
0
20
40
60
80
(b)
0
0
20
40
60
80
0
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60
80
0
20
40 time (d)
60
80
1
0.15
0.5
0.1 0
20
40
60
80
0
(d) 0.06
15
0.04 D2
S0’
20
10 5
(e)
0.1
Q
substrate S2
0.2
0.05
0.2
1.5
0.25
(c)
0.3
0.02
0
20
40 time (d)
60
0
80
(f)
Fig. 7. Simulations with control action D2 for l ¼ 0:1.
Some results of the simulations with the control algorithm (22) are shown in Fig. 8 for l ¼ 0:4. The simulation conditions are the same as in the previous case (except D2 ¼ 0:0125 day1 ¼ const:, S 000 ¼ var.). Biomasses X 1 and X 2 are plotted in subplot (a),
S 1 ; S2 ; Q; S 00 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 S 000 ), however
ARTICLE IN PRESS
(a)
6 X2 : dashdot line
4 2 0
X1 : solid line 0
20
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0.3 0.2 0.1 0
(b)
20
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60
80
0
20
40
60
80
0
20
40 time (d)
60
80
1.5
0.3
1
0.2
0.5
0.1 0
0
Q
substrate S2
0.4
(c)
807
0.4 substrate S1
biomasses X1, X2
I. Simeonov, I. Queinnec / Control Engineering Practice 14 (2006) 799–810
0
20
40
60
80
0
(d)
20
100
S0"
,
S0
15 50
10 5
(e)
0
20
40 time (d)
60
0
80
(f)
Fig. 8. Simulations with control action S000 for l ¼ 0:4.
1.6 nominal case 0.8 µ max1 1.2 µ max1
1.4 1.2 Q (L/day)
differences exist between evolutions of X 1 ; X 2 and S 1 due to the fact that changes of D2 result in changes of D. Simulation studies with the linearizing control algorithms show that they present very good performances of regulation with different values of the tuning parameter l. Some problems, however, arise when l is too much increased. It may result into some vanishing oscillations (for l ¼ 1) or even instability due to saturations on the actuators. This problem may be overcome with more precise determination of Dsup and 0 S 0inf (in each particular case it is possible to measure 0 S 0inf 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 are estimated by suitable observers (Lubenova et al., 2002). Moreover, Figs. 9–16 show the influence on the control of the biogas production Q of model errors. Simulations compare the biogas production Q in the nominal case (control law D2 ðtÞ is computed by using the process parameters) with respect to the biogas production Q obtained when the control law is computed with a model error of þ or 20% on each parameter. It may be checked from these figures that k1 has no influence on the quality of the control and that mmax 1 ; mmax 2 ; kS1 ; kS2 and k3 have a small influence on the quality of the control. On the other hand, the efficiency of the control is closely related to the quality of estimates k2 and k4 . This is an expected result since the steady-state value of D2 is closely related to the factor k2 =k4 .
1 0.8 0.6 0.4 0.2 0
10
20
30
40 50 time (day)
60
70
80
Fig. 9. Influence of model error—error of þ or 20% on mmax 1 .
The algorithm (22) is slightly more difficult to implement than algorithm (21) since it is generally easier to act on a pump, i.e., on a dilution rate than on a concentration. This can, however, be done by using a system of two pumps relied to two tanks allowing to control (with constant flow rate) variations of the concentration. But in spite of the technical difficulty, the algorithm (22) is theoretically more correct since D is kept constant. Then the control only acts on the methanogenic phase of the process.
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808
1.6
1.6 nominal case 0.8 µ max2 1.2 µ
1.4
max2
1.2
1.2
1
1
Q (L/day)
Q (L/day)
1.4
0.8
0.6
0.4
0.4
0
10
20
30
40 50 time (day)
60
70
0.2
80
Fig. 10. Influence of model error—error of þ or 20% on mmax 2 .
30
40 50 time (day)
60
70
80
1.2 Q (L/day)
Q (L/day)
20
nominal case 0.8k2 1.2k2
1.4
1.2 1
1
0.8
0.8
0.6
0.6
0.4
0.4 0.2
0
10
20
30 40 50 time (day)
60
70
80
1.6
10
20
30
40 50 time (day)
60
70
80
1.6
nominal case 0.8kS2 1.2kS2
1.4
0
Fig. 14. Influence of model error—error of þ or 20% on k2 .
Fig. 11. Influence of model error—error of þ or 20% on kS1 .
nominal case 0.8k3 1.2k3
1.4
1.2
Q (L/day)
1.2
1 0.8
1 0.8
0.6
0.6
0.4
0.4
0.2
10
1.6
nominal case 0.8kS 1.2kS11
1.4
0.2
0
Fig. 13. Influence of model error—error of þ or 20% on k1 .
1.6
Q (L/day)
0.8
0.6
0.2
nominal case 0.8k1 1.2k1
0
10
20
30
40 50 time (day)
60
70
80
Fig. 12. Influence of model error—error of þ or 20% on kS2 .
0.2
0
10
20
30
40 50 time (day)
60
70
Fig. 15. Influence of model error—error of þ or 20% on k3 .
80
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may be expressed in the form:
1.6 nominal case 0.8k4 1.2k4
Q (L/day)
1.4
Q ¼ AD1 B
1.2
with
1
A¼
D2 þ CD mm D
k4 k3 0 S 0 S000 ; k2 k1 k4 C ¼ S000 . k2
0.8 0.6
B¼
(27)
ðk3 kS1 þ k1 kS2 Þk4 ; k1 k2
It is then possible to evaluate an optimal biogas flow rate Qopt from the determination of an extremum of (27), obtained when the derivative of Q with respect to D is equal to zero:
0.4 0.2
809
0
10
20
30
40 50 time (day)
60
70
80
Fig. 16. Influence of model error—error of þ or 20% on k4 .
dQ 2Dðmm DÞ þ D2 ¼CB dD ðmm DÞ2 ¼ 0 ! D2 2mm D þ
6. Conclusion Experimental and analytical studies have shown that addition of acetate (with pH correction of the added substrate) allows to stabilize the process with respect to load and activity disturbances and to increase the amount of biogas obtained from the anaerobic digestion of organic wastes. This fact is very promising for stabilization of the biogas plants in the case of strong variations of the influent organic matter. Theoretical studies and simulation results have proven that the linearizing control design on the basis of an appropriate model of the anaerobic digestion with a new control input, the addition of acetate, may be very useful for the regulation of the amount of biogas in the realistic case of strong variations of the influent organic matter. From practical point of view both linearizing control algorithms ((21) and (22)) are easy to implement. However, even if it is easier to control variations of a flow rate (law (21)), the control of the influent acetate concentration presents the advantage to keep the process dilution rate constant.
Acknowledgements This work was supported by Contract no TH-1004/00 of The Bulgarian National Found ‘‘Scientific researches’’ and by a CNRS-BAS exchange program.
Only one root of this second-order equation is admissible, i.e., pffiffiffiffiffiffi W ; with DQ opt ¼ mm 1 W¼
k3 kS1 þ k1 kS2 ; k3 kS1 þ k1 kS2 þ k1 S000
W o1.
pffiffiffiffiffiffi The other root, D ¼ mm ð1 þ W Þ, is not admissible since D4mm is not physically admissible (it would results in the washout of the process). From evaluation of the second derivative d2 Q=dD2 , it may be checked that: pffiffiffiffiffiffi k3 k4 0 k4 Qopt ¼ S 0 D1 þ mm 1 W D1 S000 k1 k2 k2 pffiffiffiffiffiffi2 mm k4 ðk3 kS1 þ k1 kS2 Þ 1 W pffiffiffiffiffiffi k1 k2 W is a maximum.
Appendix B. Mixed energetical and ecological criterion ¯ k40. Let us consider the mixed criterion J ¼ Q=kS; Similarly to Appendix A, we only present the case mmax 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: J¼
a1 þ a2 D a3 D 2 a4 D
with Appendix A. Optimal biogas flow rate criterion a1 ¼ AD1 mm ¼ A ¼ For sake of simplicity, we only consider the hypothesis mmax 1 ¼ mmax 2 ¼ mm . Taking into account that D2 ¼ D D1 (D1 is assumed to be constant), Eq. (12)
m2m C ¼ 0. BþC
a2 ¼
k4 k3 0 S 0 S 000 D1 mm ; k2 k1
k4 ðk1 S 000 ðmm þ D1 Þ k3 S00 D1 Þ; k1 k2
(28)
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810
k4 ðk3 kS1 þ k1 kS2 Þ þ k1 k4 S000 40; k1 k2 þ c2 kS2 Þ40.
a3 ¼ B þ C ¼ a4 ¼ kðc1 kS1
It is then possible to evaluate an optimal value for the criterion J from the determination of an extremum of (28), obtained when the derivative of J with respect to D is equal to zero: dJ a1 þ a3 D2 ðk3 S 00 k1 S 000 ÞD1 mm 2 ¼ ¼ 0 ! D ¼ . dD k1 S 000 þ k3 kS1 þ k1 kS2 a4 D2 Only one root of this second-order equation is admissible, i.e., sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðk1 S000 k3 S00 ÞD1 mm J Dopt ¼ , k3 kS1 þ k1 kS2 þ k1 S 000 where the existence condition is k1 S000 k3 S 00 40. From evaluation of the second derivative d2 J=dD2 , it may be checked that Dopt is a maximum. References Angelidaki, I., Ellegaard, L., & Ahring, B. (1999). A comprehensive model of anaerobic bioconversion of complex substrates to biogas. Biotechnology and Bioengineering, 63, 363–372. Bastin, G., & Dochain, D. (1991). On-line estimation and adaptive control of bioreactors. Amsterdam: Elsevier Science Publication. Batstone, D. J., Keller, J., Angelidaki, R. I., Kalyuzhny, S. V., Pavlostathis, S. G., Rozzi, A., et al. (2002). The IWA anaerobic digestion model No 1 (ADM1). Water Science and Technology, 45(10), 65–73. Bernard, O., Hadj-Sadok, Z., & Dochain, D. (1999). Dynamical modelling and state estimation of anaerobic wastewater treatment plants. European control conference, Karslru¨he, Germany.
Dochain, D. (1995). Recent approaches for the modelling, monitoring and control of anaerobic digestion processes. In Proceedings of the international workshop on monitoring and control of anaerobic digestion processes (pp. 23–29). Narbonne, France. Haag, J., Vande Wouwer, A., & Queinnec, I. (2003). Macroscopic modelling and identification of a biogas production process. Chemical Engineering Sciences, 58(9), 4307–4316. Lubenova, V., Simeonov, I., Queinnec, I. (2002). Two-step parameter and state estimation of the anaerobic digestion. In Proceedings of the 15th IFAC world congress. Barcelona, Spain. Simeonov, I. (1999). Mathematical modelling and parameters estimation of anaerobic fermentation processes. Bioprocess Engineering, 21, 377–381. Simeonov, I. (2000). Methodology for parameter estimation of nonlinear models of anaerobic wastewaters treatment processes in stirred tank bioreactors. Proceedings of the fifth international symposium on systems analysis and computing in water quality management— WATERMATEX 2000 (pp. 8.40–8.47). Gent, Belgium. Simeonov, I., & Galabova, D. (2000). Investigations and mathematical modelling of the anaerobic digestion of organic wastes. Proceedings of the fifth international conference on environmental pollution (pp. 285–295). Thessaloniki, Greece. Simeonov, I., Galabova, D., & Queinnec, I. (2001). Investigations and mathematical modelling of the anaerobic digestion of organic wastes with addition of electron acceptors. Proceedings of the ninth world congress on anaerobic digestion 2001 (pp. 381–383). Antwerpen, Belgium. Simeonov, I., & Stoyanov, S. (1995). Dynamic output compensator control of methane fermentation. Proceedings of the workshop on monitoring and control of anaerobic digesters (pp. 47–51). Narbonne, France. Steyer, J.-Ph., Buffiere, P., Rolland, D., & Molleta, R. (1999). Advanced control of anaerobic digestion processes through disturbances monitoring. Water Research, 33, 2059–2068. Van Impe, J. F. M., Vanrolleghem, P. A. V., & Iserentant, D. M. (1998). Advanced instrumentation, date interpretation and control of biotechnological processes. Dordrecht: Kluwer Academic Publication. Zlateva, P., & Simeonov, I. (1995). Variable structure control of methane fermentation. System, Modelling, Control, 2, 433–436.