Evaluation of control strategies for anaerobic digestion processes
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INTERNATIONAL JOURNAL OF ADAPTIVE CONTROL AND SIGNAL PROCESSING, VOL. 7,309-321 (1993)
EVALUATION OF CONTROL STRATEGIES FOR ANAEROBIC DIGESTION PROCESSES*
M. PERRIER Systems and Control Division, Pulp and Paper Research, Institute of Canada, 570 Boulevard St. Jean,
Poinle Claire H9R 3JR, Canada
D. DOCHAIN Chercheur QuaIijP FNRS, Cesame, Universite' Catholique de Louvain, Place du Levant 3, B- I348
SUMMARY Adaptive non-linear controller designs for various controlled variables are evaluated for the operation of the anaerobic digestion process. The selected controlled variables are the chemical oxygen demand (COD), propionate concentration and dissolved hydrogen concentration. The manipulated variable is the dilution rate. Because the non-linear control algorithms are designed on the basis of the mass balance equations, distinct assumptions have to be made depending on the choice of the controlled variable. For each selected controlled variable the linearizing control design based on a reduced-order model of the process results in a similar controller structure. Although the obtained responses are model-specific, the methodology presented can be readily applied to an actual laboratory- or industrial-scale bioreactor. Study of the time response of the volatile fatty acids and dissolved hydrogen concentrations and of the methane flow rate obtained by controlling one variable at a time reveals the intrinsic behaviour of the balances between the assumed bacterial species and can be useful to investigate for instance the inhibition induced by a given product present in the reactor. The developed methodology demonstrates the usefulness of advanced control techniques to get a better understanding of the process under study even if the controller designs are based on incomplete knowledge.
KEY WORDS Anaerobic digestion Adaptive control Non-linear systems
It is generally recognized that the operation of bioreactors for the anaerobic treatment of waste-water is particularly sensitive to input variations such as organic load, influent flow rate and the presence of toxic material. Procedures for the design, start-up and operation of anaerobic treatment systems to improve their stability are available in the literature (see e.g. References 1-3). The application of process control techniques may provide an additional way to achieve stabilization while maintaining the highest possible conversion of wastes.
Several simulation and experimental studies have appeared to help understand the
* A reduced version of this paper was presented at the ICCAFTS-IFAC B10/2 Conference, Keystone, CO, March 1992.
This paper was recommended for publication by editor M. J . Grimble
0890-6327/93/040309-13$11.50 0 1993 by John Wiley & Sons, Ltd.
Received March 1992 Revised May 1993
310 M. PERRIER AND D. DOCHAIN
consequences of controlled and manipulated variables selection on the operation of anaerobic bioreactors. Only a few of these are reported here to illustrate the diversity of approaches. Dalla Torre and Stephanopo~los~ have proposed several feeding procedures to minimize the start-up time. Denac et al. have suggested a pH control strategy based on the alkaline consumption, which appeared to be the most satisfactory performance indicator for their experiments. In References 6 and 7 the control of the COD and propionate concentration has been studied and applied in a pilot-scale reactor. It was shown that adaptive non-linear controllers could maintain the controlled variables at their setpoint. Various non-linear adaptive control algorithms based on different modelling assumptions are presented and evaluated in Reference 8.
As pointed out in Reference 3, it is difficult to compare the various approaches on the basis of results obtained from different reactors, treating different wastes, with different control strategies and algorithms. It would appear useful to compare the performance of a given control algorithm on the basis of the same model structure for selected controlled and manipulated variables.
This paper presents the design and evaluation of control algorithms for the dissolved hydrogen concentration, propionate concentration and COD by manipulation of the dilution rate. All the controllers use the same information, i.e. measurements of the controlled variable concentration, influent substrate concentration and methane flow rate. A description of the modelling approach is given first. The necessary steps for the synthesis of the non-linear adaptive control laws are then presented. The performance of each controller is finally compared by simulation for load changes in the influent substrate concentration.
2. MODEL DESCRIPTION
The five-bacterial-population model structure proposed in Reference 9 has been used by several researchers (see e.g. References 10 and 11) to adequately represent the dynamic behaviour of anaerobic bioreactors after step changes in influent concentrations and hydraulic residence times. These bacterial populations are the acid formers, propionate-utilizing acetogens, butyrate-utilizing acetogens, acetoclastic methanogens and hydrogen-utilizing methanogens. The model presented here includes only four bacterial populations, the butyrate-utilizing and propionate-utilizing acetogens having been merged into the obligate hydrogen-producing acetogens (OHPAs).
2.1. Reaction path ways
The following reaction pathways are assumed to describe the anaerobic digestion process.
Acid- forming bacteria (XI)
Obligato hydrogen-producing acetogens (X2)
Acetoclastic methanogenic bacteria (X3)
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Hydrogen-utilizing methanogenic bacteria (X,)
k7S4 + kioSs + X4 + kisPi (4) Here S1, SZ, S3, S4, SS and P1 represent glucose, propionate, acetate, dissolved hydrogen,
inorganic carbon and methane respectively. The yield coefficients are represented by ki, i = 1, ..., 15.
2.2. Dynamic model
The dynamic mass balance equations for each bacterial population, substrate and product are readily obtained from equations (1)-(4) and can be formulated in the general matrix form'
( 5 ) dt/dt =K(P - D[ - Q + F where
E ' = [XI S! xz sz x3 s3 x4 s4 ss Pll, (PT= [ 9 i (oz (P3 +u]
0 0 0 0 1 -k4 0 0 k13 k14 0 0 0 0 0 0 1 -k7 -kio kis :I 1 -k1 0 k3 0 k5 0 ks kll 0 0 1 -kz 0 k6 0 k9 k12 K'= [
FT= [0 DSin 0 0 0 0 0 0 0 01, Q T = 10 0 0 0 0 Q H ~ Qco, QCHJ D is the dilution rate (h-'), Sin is the influent organic matter concentration (gl-I), (PI, cpz, cp3 and (04 are the reaction rates (gl-'h-') of reactions (1)-(4) respectively and QH2, Qco2 and Q C H ~ are the gaseous outflow rates (g1-Ih-l) of hydrogen, carbon dioxide and methane. Note also that in the dynamical model ( 5 ) S1, SZ, S3, S4, SS and PI hold for the concentrations (g 1-I) of glucose, propionate, acetate, dissolved hydrogen, inorganic carbon and methane respectively.
3. ADAPTIVE NON-LINEAR CONTROL ALGORITHMS
The controller algorithm for each variable (dissolved hydrogen, propionate and chemical oxygen demand) is obtained in three steps: model reduction, feedback linearization and parameter adaptation.
3.1. Model reduction
The dynamical model of the anaerobic digestion process is quite complex (the system order is equal to lo!), but especially for control design purposes, the model order can be reduced by using singular perturbation techniques. A systematic approach to model order reduction via singular perturbation for (bio)reaction systems is described in detail in References 8 and 12. Its key ideas can be summarized here as follows. Consider that there are low-solubility products (such as methane for instance) and/or substrates which intervene only in fast reactions. Each of them is denoted t i in the general dynamical model ( 5 ) and its corresponding dynamical mass balance equation is
where K i is the row of K corresponding to ti. The application of singular perturbation then dEi/dt = Kip - Dti - Qi + Fi
312 M . PERRIER AND D. DOCHAIN
results in the following model order reduction rule: set the concentration of the low-solubility product and/or the fast substrate and its time derivative to zero, i.e.
i = 0, dtildt = 0 (8)
(9) However, in the application of model order reduction there may be some difficulties in
deciding which reactions are fast and which are slow, especially in bioprocess applications where the notion of fast and slow reactions may depend on the operating conditions: one reaction may be fast under some operating conditions and slow (or limiting) under other operating conditions. This question can be addressed here as follows. The choice of a controlled variable depends on some u priori process knowledge. In anaerobic digestion a key control objective is to avoid instability in the form of substrate accumulation and inhibition. The choice of the controlled variable (e.g. COD, propionate or hydrogen) is at least partially based on the idea that the operator wishes to avoid its accumulation and this also means that the underlying consumption reaction is limiting, i.e. slow with respect to the other reactions. This idea has been followed here for the model reduction and control design of each of the proposed controlled variables (dissolved hydrogen, COD and propionate).
and replace the dynamical equation (7) by the algebraic equation
K i p = Qi - Fi
3.1.1. Dissolved hydrogen control &). Let us see how to apply the above model order reduction rule (8) to a specific example, namely the control of the dissolved hydrogen concentration S4. First of all, it is well known that methane is a low solubility product. Furthermore, assume that the second methanization path (hydrogen consumption) is limiting, i.e. that the first three reactions (1)-(3) are fast and the fourth one (4) is slow. We can then apply the model order reduction rule (8) to the the glucose concentration SI, the propionate concentration SZ, the acetate concentration S3 and the dissolved methane concentration PI. By setting their values and their time derivatives to zero, i.e.
s1 = s2 = s3 = PI (10) dSl/dt = dSz/dt = dSJ/dt = dPi/dt = 0 (1 1)
we reduce their differential equations to the set of algebraic equations
By inverting the submatrix of the yield coefficients on the left-hand side of (12), we can express the reaction rates CPI, CPZ, (03 and (04 as functions of the feed rate DSin and the gaseous methane outflow rate Q c H ~ :
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Let us replace the reaction rates (01, $92 and $94 by their expressions (13), (14) and (16) respectively in the dynamical equation of the hydrogen concentration S4, which is then rewritten as
(17) dS4/dt = -DS4 - Q H ~ - ~ I ~ Q c H ~ + 824DSin
3.1.2. Propionate control (SZ). The output dynamical equation for propionate is obtained by assuming that pathways (l) , (3) and (4) are faster than pathway (2) and that methane is a low-solubility product and then following the approach described above:
(20) dSz/dt = - DS2 - ~ I ~ Q c H ~ + 82JlSin where
(21) k h k 7 k4k9kis + ki4k7ka 8 1 2 =
3.1.3. Chemical oxygen demand (COD). Let us define the chemical oxygen demand as
COD = C i S i + ~2S2 + ~3S3 where c1, c2 and c3 are conversion factors from g 1-' to (g COD) 1-'. Multiplying the mass balance equations for S1, S2 and S3 by cI, c2 and c3 respectively and adding the three equations, we obtain
d(COD)/dt = (-Cikl + C2k3 + C3kS)$91 + (-~2k2 + Cjk3)$92 - ~3k3$93 + D(C1Sin - COD) (23) The output dynamical equation for COD is obtained by assuming that pathways (2), (3) and
(4) are faster than pathway (1) and that here again methane is a low-solubility product:
d(COD)/dt = -BIlQCH4 + D(C1Sin - COD) where
(25) ciklkzk4k7 + ~3k7(kzk3ks + k:k6 - k2k4k~ - k:k4)
k7ki4(k2ks + k h ) + k4kis(kzke + k3k9) 811 =
314 M. PERRIER AND D. DOCHAIN
3.2. Feedback-linearizing controllers
general expression The output dynamical equation for each controlled variable can be represented by the
dyjldt = -Dyj - 01jQcH4 + OzjDSin (26)
dyj/dt = C1 (Yj,p -Yj) (Ci > 0) (27) If we want to impose first-order stable linear closed-loop dynamics
the following linearizing control law is obtained:
The first term on the right-hand side of equation (28) can be viewed as a proportional control term where the gain depends on the desired closed-loop time constant, the influent substrate concentration and the actual level of the controlled output. The second term represents a non- linear feedforward component of the actual methane gas production. It is shown in Appendix I that equation (28)) coupled with the adaptation mechanism described in the next section, includes the key features (proportional and integral action) of a classical PI controller.
3.3. Adaptation mechanism
The parameters 011 for COD control and 02j) j = 2.4, for propionate and hydrogen control respectively were updated using a Lyapunov approach* where the parameter rate of change is proportional to the control error:
d&j/dt = - Cz(t)DSin(~j,~ - yj), j = 2,4 (C2 > 0) (30) dei i/dt = c 2 ( ~ ) Q c H ~ (Yi,, - yi )
Defining the system formed by the control error j j = ysp - y and the parameter error 6, its dynamics is given by
The last two sets of equations can be used to select values of the tuning parameters CI and C2(t).
3.3.1. Choice of the design parameter values. The values of the design parameters are chosen so as to assign the closed-loop dynamics of the process. This is done by choosing Cz(t) to be inversely proportional to the square of the regressor:
c; > 0 (35) Indeed, with this choice the characteristic polynomial P(A) of the state matrix of the closed-
loop dynamical system is independent of time:
P(A) = A 2 + CIA + c;
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Also, if we consider the reasonable assumptions that QCH, and D S i n are strictly positive and bounded (the boundedness can be shown by using BIBS stability arguments for the process model8), it is shown in Appendix I1 that the (non-stationary) closed-loop dynamics (31) and (32) are equivalent to a stable stationary system via a Lyapunov transformation.
A possible choice for the design parameters CI and Ci is to consider a double real pole. This is achieved by setting Ci to the value
ci = c:/4 (37) This choice has been considered in our simulations. Furthermore, in order to avoid division
by zero when D = 0 in the on-line adaptation of 022 and 824, the value of Cz(t) in (34) has been modified as
where &in is a small positive constant. Note that a similar approach has been considered in a Xanthan gum production appli~ation.'~
4. RESULTS AND DISCUSSION The performance of each controller has been individually evaluated by simulation for step changes in the organic load (Sin). The anaerobic process has been simulated by integration of the mass balance equations with the following yield coefficients, specific growth rate expressions and initial conditions.
kl = 3.2, kz = 1.5, k3 = 0.7, k4 = 12, ks = 0.27, k6 = 0.53, k14 = 0.3, kls = 0.08 (39) k7= 1.15, kg = 0.6, k9 = 0.1,
Specific growth rate expressions
pmaxl= 0.2, KI, = 10 pmaxl= 0.5, Ks, = 0.4, KI, = 0.5 ~cmax, = 0.4, Ks, = 0.5, KI, = 4 pmau = 0.5, Ks, = 4, KI, = 3
316 M. PERRIER AND D. DOCHAIN
The performance of each controller was evaluated for an influent substrate concentration (Sin,) change from 25 to 35 g I - applied at t = 2 days. The closed-loop time constant for each case was chosen to be equal to 2 days (Cl = 1 and Ci = f ) . Dmin was set to 0.01 day-.
4.1. Dissolved hydrogen control
The concentrations of each substrate and bacterial population are shown in Figures l(a) and l(b) respectively for dissolved hydrogen control. The controller proves to be able to maintain the hydrogen concentration at the desired setpoint. The concentration of bacterial populations slowly reaches new levels, while the substrates go back to their initial values.
4.2. COD control
Good regulatory performance is also achieved by using the COD controller as shown in Figures 2a and 2b. However, it can be seen in Figure2(a) that the dissolved hydrogen concentration reaches a new level (S4 = 4-2 pM > ( K s , K I ~ ) ~ = 3.5 pM) which corresponds to an inhibitory value in the Haldane model of c ~ q .
4.3. Propionate control
controller (see Figures 3(a) and 3(b)). Results obtained using the propionate controller are very similar to those for the COD
Figure 1. Adaptive linearizing control of hydrogen
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Figure 3. Adaptive linearizing control of propionate
318 M. PERRIER AND D. DOCHAIN
4.4. Absence of feedforward action It is also interesting to check how the controllers will perform when the step change in inlet
substrate concentration is not monitored. As shown in Figures 4(a) and 4(b), the substrate and biomass concentration behaviour is only slightly worse than for the case where the change is included in the hydrogen controller. Adaptation of the parameter 024 (Figure 4(c)) is able to account for the change.
The influence of the feedforward term on the closed-loop dynamics appears more drastically in the control of COD and propionate. The COD controller is not able to reject the disturbance as efficiently, because the dilution rate is not reduced sufficiently rapidly as can be seen from Figure 5(a). As a consequence, hydrogen will tend to accumulate and will inhibit the methane production (Figure 5(b)) . Again, similar results are obtained using the propionate controller.
- 9 ' COD Prop. 2 - 3 - - Acet. 4 0 - - H,
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0 10 20 30 40
Figure 5. Comparison of COD control with and without including ASin
5 . CONCLUSIONS
A general non-linear controller structure has been proposed for dissolved hydrogen, propionate and COD. The control law has been derived by assuming a model structure based on four bacterial populations. In addition to the measurement of the chosen controlled variable, the controller requires knowledge of the methane flow rate and influent substrate concentration.
Simulation results have shown that each controller is able to maintain the controlled variable at the desired setpoint after a step change in influent substrate concentration. At this point it has been found out that after an unmeasured organic load change, only the hydrogen controller is able to recover quickly.
The authors would like to thank Iven Mareels for his insightful comments about Lyapunov transformations.
APPENDIX I: CONDITIONS FOR EQUIVALENCE BETWEEN ADAPTIVE LINEARIZING CONTROL AND PI CONTROL
The equations of the adaptive linearizing controller were found to be ~ I ( Y ~ ~ - Y ) + Q I Q c H ,
&Sin - Y D =
320 M. PERRIER AND D. DOCHAIN
ddl/dt = CZQCH~(YSP -Y)
Integration of equation (45) yields
81 = 1' CZQCH~(YSP-Y) df 0
Then cz = C ~ / ~ Q $ H .
8, = 1' (ysp-y) dr o ~ Q c H ~
Substituting equation (48) into equation (44) gives cl(Ysp-Y) + [ji (C~/~QCH~)(YSP-Y) ~ ~ I Q C H I
VzSin - Y D =
Assuming Q C H ~ = Q C H ~ z. constant near an operating point, then
The expression for a PI controller may be written as
By equating equations (50) and (51). we finally obtain the expressions for Kc and 7-1 as Cl K c = -
W i n - Y TI = 4 c l / c i
Equations (52) and (53) may be useful to obtain the tuning parameters of a PI controller.
APPENDIX 11: STABILITY ANALYSIS
Theorem Let us consider the second-order linear time-varying system
dx(f)/df = A(f)x(f ) + h(f) with
0 C Zmin 6 Z ( t ) 6 Zmax V f
Then the free system ( u ( f ) = 0) is exponentially stable and the forced system is BIBS stable.
Proof. Let us consider the state transformation = TX
By applying the transformation T. the above time-varying system becomes d[(f)/df = A [ ( f ) + & ( f )
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A = ( -c, I ), B=(&) -c2 0
Since A is stable and z ( t ) is strictly positive and bounded, the transformation T is a Lyapunov 0 transformation and the rest of the proof follows.
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