e Pergamon

PH: S0273-1223(98)00485-S

Wat. ScL Tech. Vol. 38. No.3. pp. 219-226. 1998.IAWQ

~ 1998Publishedby ElsevierScienceLtd.Printedin Great Britain.All rights reserved

0273-1223198 $19-00+ 0'00

219

RESPIROMETRY BASED OPTIMALCONTROL OF AN AEROBICBIOREACTOR FOR THE INDUSTRIALWASTE WATER TREATMENTJaimeMoreno* and German Buitr6n**,t

• AutomaticControlDepartment. Institute ofEngineering. NationalUniversity ofMexico. Ap. Postal 70-472. 045/0 Mexico. D.F.•Mexico•• Environmental Bioprocesses Department. Institute ofEngineering,NationalUniversity ofMexico. Ap. Postal70-472, 04510Mexico. D.F.•Mexico

ABSTRACT

A methodology and mathematical simulation for the time optimal control of an activated sludge sequencingbatch reactor (SBR) for toxic waste water degradation is presented. An optimal strategy for the input flowwas obtained using techniques of optimal control theory. such that the reaction time is as small as possible.The degradation time was controlled through the substrate concentrstion estimated from the liquid oxygenconcentration, which is much simpler to measure. The simulation of a toxic effluent containing phenolshowed that the degradation time, obtained with the optimal strategy, was a half of that obtained when thereactor is stopped at the end of the reaction period, but filled without the optimal strategy. C 1998 Publishedby Elsevier Science Ltd. All rights reserved

KEYWORDS

Biodegradation; Kalman filter; optimalcontrol; respirometry; toxiccompounds; SBR process.

INTRODUCTION

Activated sludge is an aerobic biological process in which wastewater is mixed with a suspension ofmicroorganisms to assimilate pollutants and is then settled to separate the treated effluent. It has beentraditionally applied in continuous flow processes with fixed volume tanks. The treatment of industrialwastewater by the activated sludge process is common, but the nature of many industrial discharges oftencause operationalproblems in continuous flow systems.Sequencing batch reactors (SBR) offer a number ofadvantages over continuous flow systems (Irvine and Ketchum, 1989). They offer for example a greatflexibility in control strategyand are effectivefor fully automated computercontrols.

In general the SBR process is distinguished by three major characteristics: periodicrepetitionof a sequenceof well defined phases; planneddurationofeach process phase in accordance with the treatmentresult to bemet;progressof the variousbiologicaland physicalreactionsin time rather than in space.

In the SBR system all treatment takes place in a single reactorwith differentphases separatedin time. Thecycle in a typical SBR is divided into five discrete time periods: Fill, React,Settle,Draw, and Idle. At thebeginning of each cycle, the SBR contains a certain volume of water, and activated sludge settled at thebottom of the reactor. The cycle starts with a fill phase of distinct duration. The fill phase may be short orlongdependingon the effectswhich are desiredto be achieved.

A reductionof the cycle time of the SBR increasesthe quantityof water that can be treated by the tank. Asthe settle,draw and idle phases are usuallyof fixedtime or not controllable by the operator,the cycle time ofthe SBR can be onJy reduced if the fill and reaction times can be diminished. A simplified model of theset Corresponding author

220 J. MORENO and G. BUlTR6N

phases (see eq. (I», with four state (X, S, V, and 0) and one control (Q,.) variables. is taken in (Moreno,submitted; Moreno and Buitr6n, 1996) as the basis to find the best strategy for the control variable in thesense that the total time of the fill and reaction phases will reach a minimum. Such problems are usuallysolved using the Maximum Principle ofPontryagin of the Optimal Control Theory (Pontryagin et 01., 1964;Lee and Markus, 1967). In order to implement such time optimal strategy in the reactor operation, biomassor substrate concentration have to be known on line. Since this measurement is not feasible. respirometrictechniques can be used to control the biodegradation process (Spanjers et al., 1996).

This paper presents a methodology and mathematical simulations for the time optimal control of an activatedsludge SBR used for toxic wastewater degradation. The degradation time is controlled through the substrateconcentration deducted from the oxygen concentration. This estimation is realized through an extendedKalman filter based on the nonlinear model of the process (I). In order to optimize (minimize) thedegradation time the influent flow rate Qoo is manipulated.

MATHEMATICAL MODEL OF THE PROCESS

The fill and reaction phases ofan SBR can be mathematically described by the following system ofordinarydifferential equations, which are obtained by mass balance in the reaction tank (Henze et 01., 1986; Schugerl,1987):

d(v.¥)-;jf=f.JVX

d(VS) =_f.JVX +n Sdt Y ~ Of

dVdi"=Q..

d(VO) =_f.JVX

+Q..O.. +K,aV(O_-O)-rVXdt Yxo

where: X = [MV] Biomass concentration in the tank ;S =[ML·3] Substrate (toxic substance) concentration in the tank ;V =[L3

] Volume ofwater in the tank;o =[ML·3] DO (Dissolved Oxygen) concentration in water.Y Biomass Yield coefficient;Qoo =[L3T I

] Waste water input flow to the tank;Soo =[MV] Substrate concentration in the input flow;f.J = [TI

] Specific biomass growth rate;Yxo = Oxygen yield coefficient;0,. = [ML·3] DO concentration in the input flow;0... = [ML·3] Saturation DO concentration in water;Ka =[TI

] Oxygen mass transfer coefficient;r '" [TI

] Specific endogenous respiration rate of the biomass in the water.

(I)

Naturally the state variables are constrained to be positive, i.e., X,S,V,O ~ O. Furthermore, the volume V is

restricted by the dimension of the tank, i.e. V S V..... ' and the input flow, the control variable, can only takevalues in the interval 0 S Q.. S Q......In (I) the decay rate of the biomass will be neglected.

In model (I) the first two equations describe the substratelbiomass dynamics; the third one represents thefiIIing process of the tank and the fourth describes the DO mass balance over the liquid phase. In this lastequation the second term on the right hand side represents mass flow of DO in the liquid phase. The thirdterm describes the mass transfer of oxygen from the gas phase to the liquid phase. First and last terms

Respirometry based optimal control 221

contain the respiration rate to be derived from the mass balance. The first one describes the exogenousrespiration and the last one the endogenous respiration process. It has been demonstrated that biomassgrowth and substrate removal are proportional to the oxygen consumption. thus an oxygen uptake curve canprovide the same information as either a substrate removal or a biomass growth curve (Gaudy et ai.• 1988;Grady et al.• 1989). Hence in this study. the dynamic of the DO will enable us to determine the biomass andsubstrate concentrations without having to measure them.

The waste of interest in this work are toxic organic substances present in industrial sewage. These substancesinhibit the growth ofthe biomass. This effect is usually described by the Haldane equation

I'tmxS(2)

where : Pnm = [il]Ks = [ML')]

K, =[ML')]

Maximum specific biomass growth rate;

Michaelis-Menten constant;

Inhibition constant.

This function is illustrated in Fig. I, where the maximum value of the specific growth rate J.1. and itscorresponding substrate concentration S· are shown.

vs

II.

S·

Figure I. Haldane equation

S

Q=Ov...t----__+---,

P,

Q=O

s'

Figure 2. Filling strategy

s

TIME OPTIMAL CONTROL

The operation of a SBR in the waste water treatment can be described as follows: at the beginning of thefilling phase the substrate concentration is So and the volume of water in the tank is small Yo' As waste wateris filled in the quantity of toxic in the tank and the volume of water will increase. The reaction begins, andthe microorganisms will degrade the toxic substance. When the substrate concentration reaches a desiredvalue S... and the tank is full, i.e, V=V_, the reaction is stopped, taking place the other phases of the SBR:settling, decant and idle.

Asthe efficiency of the treatment plant depends heavily on the time needed by the filling and reaction phases(because the time of the other phases is mostly fixed), it is of interest to reduce this period to a minimum.One important factor that influences this reaction time is the acclimation grade of the microorganisms to thetoxic. Therefore for the optimal function of the plant the biomass has to be adapted to the toxic substance(Buitr6n et al., 1994). Another factor is the filling strategy of the tank, i.e, the form of Qin' It is very

222 J. MORENO andG. BUITR6N

interesting because this variable can be manipulated by a controller to obtain optimal results. Usingtechniques of the theory of optimal control, it is possible to show that the optimal form for Q,. such that thetotal time for the filling and reaction phases of the SBR is as small as possible, is given by equation 3(Moreno. submitted; Moreno and Buitron, 1996):

10 if (v=v..... )or(s>s·),

Q", = Q.", if (v<V_)and(S= S·).Q..... if (V <V..... )and(S<S·).

where Q;., is the time function that satisfies

0=- JJ' X + Q". (S - S·)Y V '" '

i.e.

JJ'VXQ.", = r(S,.-s·)·

(3)

(4)

This filling function QIl. is such that the specific growth rate J! is maintained in its maximum value !!. andthe substrate concentration is therefore S·. This function constitutes a so called singular arc in the optimalcontrol theory.

The optimal filling strategy (3) can be given an easy physical interpretation (Fig. 2 ): first the initial substrateconcentration will be brought to the critical one S·. This can be achieved by filling with maximum inputflow Q....or without input flow. i.e. Q•• = o. When the critical concentration S· has been reached the inputflow will be QIl. (4), so that S· will be maintained in the tank until it is full. During this time the specificgrowth rate is at its maximum value J!•• so that the velocity of biodegradation of the substrate is as high aspossible. and that is the reason for the reaction time to be as low as possible. The end phase after the tank isfull is just to wait until the substrate concentration goes under the desired value S..,•.

The filling strategy (3) is given in terms of the state variables of the reactor and builds therefore e feedbackcontrol law. If X, S, and V could be measured on line and the parameters of the model (I) were known, itwould be easy to implement (3) and to minimize the filling and reaction time of the SBR.

RESPIROMETRY BASED TIME OPTIMAL CONTROL

The on-line measurement of the state variables biomass (X) and substrate (S) concentration is not easy. Incontrast the monitoring of the DO through respirometers is easy and reliable. as well as the volume V. Thelast equation of model (I) gives the relationship between the change in the DO concentration in the liquidphase and the evolution of the other state variables (X, S, and V).

Therefore the time optimal control (3) can not be directly implemented. because of the lack of on lineinformation about the state variables X and S. This problem can be solved through the use of an observer. i.e.a mechanism to estimate the actual values of the unmeasured variables X. and S using the measuredvariables 0 and V and the model (3). If the parameters of the mathematical model (3) are known, then theuse of an Extended Kalman Filter (EKF) is a possible solution (Anderson and Moore. 1979). As the model isnonlinear, the design ofan observer is not straightforward and the first approach is to use a linearized tangentmodel to construct the observer. The model (3) can be written in compact form taking

Respirometry basedoptimalcontrol

i =r(I. , u) , 1.(0)=1.0 •

Y=h(l.),

where lor= [X S V 0] and v' = [V 0]. then the EKF has the form

223

(5)

i =f(i. u)+ L(I)[y- h(i)]. i(O) = i o

iJbl8(1) =-iJ . .I. it,)

~f

F(t)=iJ •I. iltl

(6)

where P = pI is the covariance matrix. and i is the estimation of the state vector I.. The elements of thesymmetricand positive definite matrices Q. R and Po, the initial condition of the matrix Riccati equation,are free design parameters. Because the design is based on the linearizationof the model. the convergenceofthe estimated states i of the observer to the actual states I. of the reactor is assured only if the initial

conditions i o and 1.0 are not very distant. The use of the estimated values [X S] and the measuredvalues

v' = [V 0] in the time optimal feedbackcontrol (3) allows the implementationof the optimal control.

As discussed previously, acclimated microorganisms must be present to biodegrade effectively the toxiceffluents. To assure the prevalence of acclimated microorganisms and the proper biomass concentration inthe tank (from 2.5 to 5 gil) an adequate Solid Retention Time, SRT. should be maintained. Once theestimated value of X is known. a suitable sludge wasting strategy can be implemented in order to maintainthis adequate SRT. It must be take in mind that the optimal strategy is based on the maximal activity ofmicroorganisms and thus the minimizationof degradation time is not obtained only by making X very large.

DIGITAL SIMULATIONS

To test the effectivenessof the proposed control methodologysome digital simulations were run. For this theplant the model (I) was used with the parameters shown in Table I. The substrate to be degraded is phenoland the desiredoutput concentration is Sm,,=1 mgll.

Often in practice the control strategy of a SBR for the industrial waste water treatment is very simple : thetank will be filled as fast as possible and the complete cycle of the SBR is fixed to be 24 hours. In this caseinformationabout the state of the process is not used for its control.

Table I. Parametersof the plant used in the simulations.

Vm.. = 50 mJ Qm.. - 50 mJ!h K, = 50 mg/l Y - 0.5

S.. =200 mg/l Ks = 2 mg/l 11m.. = 0.072 l!h r = 0.0097 h·1

Yxo= 1 Kia =40h" 0 10, = 20 mgll 0,. = 0 mgll

Xu = 7000 mgll So = 50 mgll Vo = 5 m' 00 =8mgll

Fig. 3 shows the simulation of the fill and reaction phases with the optimal control strategy (3) if all statevariables were measurable. The time behavior of S. V and Q,. are traced. Under this conditions the optimaltime for both phases is 2.43 hours. If the other phases (settling. draw and idle) last 2 hours. the completecycle would last 4.43 hours instead of 24 hours for the usual cycles! Another strategy can be obtained if a

224 J. MORENO and G. BUITRON

sensor for determining the end of the reaction phase is used, and the reactor is operated in batch with a veryshort fill period. In this case the time for the reaction phase would be 4.88 hours. i.e. two times that of theoptimalstrategy.

50S [mgJIJ

40

30

20

10

o 0\

U.:> Time [h) 2 2.5

250.. [m'lh)

20

15

10

5

0u50

50

40

30

20

10

0.5 Time [h) 2

Figure3. Simulationof the fill and reactionphasesof the SBR usingthe time optimal strategy

Figure 4 shows the behavior of the observer (EKF) used in open loop. i.e. used for estimation but not forcontrol.The initialvalues i o of the observerare 200A! higher than that of the plant and the design parametersare set to: Q=Diag[500 ;0.1 ;1000 ;10000] ; R=Diag[IO· ;10' ] ; and Po=Diag[l0·2 ;10.2]. The estimatedvalue

S converges to the true value S fast.

Fig. 5 shows the behavior of S and its estimated value S if the estimations of the observer are used toimplement the time optimal control (3). In this case the time for the fill and reaction phases is 2.44 hours.This valueis only a little bit bigger thanthe optimal time (2.43 h.).

These results show the advantage of using the time optimal strategy for the operation of a SBR. Theefficiency is greatly increasedcompared to the usual operationmode.-The use of an EKF for the estimationof the not measurable biomassand substrateconcentrations (X and S) is also possible.

..

Figure4. EKF in open loop. Substrateconcentration in the reactorS (-) and its estimatedvalue S(-.-.).

eo11"91J

Respirometry basedoptimalcontrol 225

Figure S. EKF in the time optimal control loop.

Substrate concentration in the tank(-) and its estimated value S (....).

In order to study the influence of the decay rate on the estimated values of S, digital simulations wereobtained for the degradation of 4-chlorophenol. The results show (figure 6) that the influence of thisparameter is negligible since for both cases the evolution of the estimated substrate concentration is verysimilar. It is interesting to note that during the first 2 hours of the first cycle, estimated values of S are higherthan the actual values of S, but for final concentrations the deviation is not important Next cycles of thesimulation agree very well.

458 8IrrBItI

EI<F .. closed k>~ K~o.0111/tt

~20(II

8

Figure 6. Influence of the decay rate, Kd, on the estimated substrate concentration.Degradation of4-chlorophenol. Parameters utilized: Vmi' = 7.5 I; Qm.. = 99 l/h; K, =10.2 mgll; Y = 0.8;S.. =133 mgll; Ks =lOSmgll; J.Lm.. =0.5 l/h; r =0.0097 h"; Yxo=2.1; K.a =30 hoi; 0 ...=6.7 mgll;

Oil = 0 mgll; X, - 2542 mgll; So=IS mgll; Vo= 1.875 I; 0 0 = 6.5 mgll.

226

CONCLUSIONS

I. MORENO andG. BUITR6N

A methodology for the time optimal control of an activated sludge sequencing batch reactor was presented.The control of the reactor was obtained measuring the oxygen concentration in liquid phase and estimatingthe other state variables with an Extended Kalman Filter. The optimal strategy showed that the reaction timeis reduced if the substrate concentration is maintained at a critical value, which corresponds to the maximalspecific growth rate. This concentration was held constant controlling the input flow. It was demonstratedthat the efficiency of the reactor was greatly increased using such optimal strategy.

ACKNOWLEDGMENT

This research was supported by Direcci6n General de Apoyo aI Personal Academico through the grantPAPIIT-1N500396.

REFERENCES

Anderson, B.D.O. and Moore. J.B. (1979). Optimal Filtering. Prentice-Hall. Englewood Cliffs. N.J.Buitron, G.• Capdeville, B. and Horny. P. (1994). Improvement and control of the microbial activity of a

mixed population for degradation of xenobiotic compounds. Wat. Sci. Tech, 29 (7). 317-326.Gaudy A.F.• Rozich A.F., Garniewski S.• Moran N.R. and Ekambarum A. (1988) Methodology for utilizing

respirometric data to assess biodegradation kinetics. Proc. of the 42nd. Purdue Industrial WasteConference. 573-584.

Grady C.P.L. Jr.• Dang J.S.• Harvey D.H.• Jobbagy A. and Wang X.L. (1989) Determination ofbiodegradation kinetics through use of electrolytic respirometry. Wat. Sci. Tech., 21, (8/9). 957-968.

Henze M., Grady C.P.L. Jr., Gujer W., Marais G. v. R. and Matsuo T. (1986) Activated sludge model No. I,IAWPRC Scientific and Technical Reports. No.1. Task Group on Mathematical Modelling for Designand Operation of Biological Wastewater treatment.

Irvine, R.L. and Ketchum, L.H. Jr. (1989). Sequencing batch reactors for biological wastewater treatment.Critical Reviews in Environmental Control, 18 (4).255-294.

Lee, E. B. and Markus L. (1967). Foundations ofoptimal control theory. John Wiley & Sons. New York.Moreno, J. and Buitr6n, G. (1996). Optimaci6n de la biodegradaci6n de efluentes contaminados con

compuestos t6xicos. Memorias del XXV Congreso Interamericano de Ingenieria Sanitaria yAmbiental, Mexico, D.F.•Noviembre 3-7 de 1996. Pp. 816-819.

Moreno, J. A. (Submitted). Optimal time control of sequencing batch reactors for industrial wastewatertreatment. Conference on Decision and Control (CDC) 1997. San Diego, California.

Pontryagin, L. S., Boltyanskii V. G., Gamkrelidze R. V. and Mishchenko E. F. (1964). The mathematicaltheory ofoptimal processes. The Macmillan Company, New York.

SchUgerl, K. (1987). Bioreaction Engineering. Vol. I. J. Wiley. Chichester.Spanjers H.,Vanrolleghem P., Olsson G. and Dold P. (1996). Respirometry in control of the activated sludge

process. Wat. Sci. Tech., 34 (3/4),117-126.