state estimation of large-scale waste water treatment system

8/6/2019 State Estimation of Large-scale Waste Water Treatment System

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Pergamon ooos1098(%)00141-7Autom atic a, Vo l. 32. No. 3. pp . 305-317. 19%Copyright 0 1996 Elsevier Science LtdPrinted in Great Britain. All rights reservedooos-lo98/% $15.00 + 0.00

State Estimation for a Large-scale Wastewater TreatmentSystem*

R. TENNO? and P. URONEN?A filter developed for a large-scale wastewater treatment system permitsestimation of the concentration profile of biomass and organic matter inany area of the treatment basin more precisely than a filter developed fora completely mixed basin.Key Words-Biotechnology; wastewater treatment; modelling; distributed parameters; partiallyobservable stochastic system; state estimation.

Abstract-A new distributed parameter stochastic model isintroduced for modelling a large-scale wastewater treatmentsystem with distributed water feeding, sludge recycling andremoval. The state estimation problem is formulated as abiomass and organic matter profile estimation problem forthe aeration basin. It is solved by approximation of theoriginal model with a finite-dimensional bilinear model andthe filtration distribution with the normal distribution. It isshown that the profile can be estimated online usingdissolved oxygen and gas analysis results measured fromseveral points over the aeration basin. The estimation qualityis higher in the case of a large-scale aeration basin than in thecase of a completely mixed basin. The estimation algorithm isobtained in a computationally effective form. The model andthe estimation algorithm are tested in a simulationexperiment. It is demonstrated that the sludge and waterdistribution has a strong effect on treated water quality. Thedistribution is considered as a new control parameter. It canbe used in practice for improvement of treatment processquality.

NOMENCLATURE

ProcessS f

SXX,(dX),

organic matter concentration inwastewater: readily biodegradablecomponent;organic matter concentration intreated water;activated sludge concentration inthe aeration basin;return sludge concentration;biological growth (growthdifference),

*Received 9 July 1994; revised 18 March 1995; received infinal form 7 August 1995. This paper was not presented atany IFAC meeting. This paper was recommended forpublication in revised form by Associate Editor K. P. Whiteunder the direction of Editor A. P. Sage. Correspondingauthor Professor R. Tenno. Tel. +35804513845; Fax+358 0 462373.fLaboratory of Process Control and Management,Helsinki University of Technology, Kemistintie 1, SF-02150Espoo, Finland.

biomass growth rate;maximum growth rate;Michaelis-Menten saturationconstant;biomass yield;oxygen uptake rate;oxygen yield;mass transfer constant;saturation constant for oxygen;carbon dioxide production rate;carbon dioxide yield;endogenous respiration rates.

Measurementsto dissolved oxygen concentration;oxygen uptake rate;ic carbon dioxide production rate;PO oxygen volumetric concentration;PC carbon dioxide volumetricconcentration;Q&T outlet gas flow rate;Qr wastewater feed rate;::

sludge recycling rate;excess sludge removal rate.

StochasticsW, W, U$:.,W, W,, WC, L, V, and K areWiener processes;L+,ui, uj, aft a, and a, are standard deviations

of errors caused by model inadequacy;rdO, r, and r, are measurement accuracies.

SystemF

state;measurements;W,, W,, W, Wiener processes.305


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R. Termo and P. UronenEstimationm ,Y lLengths2c1SubscriptsI, iJ, jf0, 0c, cDO, do

online estimated current state;covariance of the estimation errors.

length of the aeration basin:moving coordinate:index: I = 1, , L.

biomass:growth substratewastewater:oxygen:carbon dioxide:dissolved oxygen.

A lower-case subscript is used for scalardesignation, and an upper-case one for a vectoror matrix designation.

Ek,, is an elementary matrix: the kth elementin the pth row is a 1, with all the rest zeros: dropthe second index if the indices are equal.

1. INIKODU(TIONBiological wastewater treatment is a relativelyslow process. Therefore a large-scale treatmentbasins must be used for massive treatment ofwater. These basins are completely rear-mixed.The organic matter concentration at a basin inletis about 10 times that at the outlet, as are theload and loading characteristics. Because of poormixing, biological growth is highly disturbed, andthe treated water quality is frequently low.

A special treatment system with distributedwater feeding, sludge recycling and removal canbe used for improvement of quality. The mostgeneral configuration is shown in Fig. 1. Sogeneral a configuration is rare in practice.although distributed water feeding is widelyused, as is, to some extent a distributed sludge

Wastewater feeding

I 111111111111 IAeration basinI * I I I I I I1TlTlTlTlT ISludge feeding and removal

Fig. 1. Large-scale wastewater treatment system withdistributed water feeding. sludge recycling and removal.

recycling system. The latter two are known(Gray. 1990) as the step and tapered feedingsystems respectively. In this paper the generalconfiguration is considered in order to solve thestate estimation problem for any sludge or waterfeeding system.

A state estimation algorithm can be developedmore effectively for a large-scale aeration basinthan for a completely mixed basin. This isbecause of the use of more measurements, thefavourable situation for identification of low andhigh concentrations at the same time at differentpoints of the aeration basin, and the strongcorrelation between local processes in nearneighbourhoods. The estimation algorithm canbc developed even more precisely if there is nodistributed feeding along the aeration basin, forexample if a conventional sludge and waterfeeding system is used.

The main reason to analyse the processes in alarge-scale aeration basin is the particularopportunities they present for stabilisation.Thus, for example, a troublesome waves oforganics in wastewater can be treated moreeffectively (Tenno and Uronen, 1995c, d).

The purpose of this paper is to propose a newmethod for online state estimation for alarge-scale aeration basin. This is a method thatcan be used for estimation of biomass andorganic matter distribution using dissolvedoxygen and gas analysis results measured atseveral points over the aeration basin. Thedevelopment of the method is as follows.

The model of the process is introduced inSection 2 as a system of stochastic first-orderpartial differential equations. It is obtained inthis form by extension of the lumped parametermodel (Tenno and Uronen, 1995a) for thedistributed parameter case. State, observationand control are all considered as distributedparameter processes. The state estimationproblem is formulated as a distribution estima-tion problem. It is solved by approximation ofthe original processes by finite-dimensionalbilinear processes (see Appendices B and C) andthe filtration distribution as the normal distribu-tion (Appendix D). The estimation method isrepresented in Section 3 and tested in thesimulation experiment in Section 4. It isdemonstrated that the sludge and water distribu-tion has a strong effect on treated water quality.The treatment process depends bilinearly on thecontrols. The exact dependence of the model onthe controls is given in Appendices B and C andsummarised in Section 5.

Stochastic distributed parameter processes arerelatively new in control theory, especially in thepartially observable case. Most results have been


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Estimation for large-scale system 307obtained for completely observable processesand for the general linear operator case(Benssousan, 1982; Rozovskii, 1983). They havebeen applied for analysis of the filtrationequation (Rozovskii, 1983) and for modellingsome random field processes, such as populationgenetics (Fleming, 1975). The process is muchsimpler in the pollution control application. Tosome extent it is close to the fixed and fluid bedreactor model (Dochain et al., 1992). Similarly tothat case, the process here is considered in itssimplest form using a system of first-order partialdifferential equations, but it is stochastic and notcompletely observable in this case. Thus, it canbe considered more like a random transportprocess (Funaki, 1979) or a wave formationprocess in a random medium (Sobczyk, 1985).Unfortunately, analysis of these processes is notapplicable in the partially observable case.The state estimation problem has been solvedby Glonti (1985) in the general linear operatorcase. These results are not used in this paper,because the treatment process is simpler than thegeneral case, and therefore the estimation andcontrol algorithm can be developed more easily.It is natural to use a system of ordinarydifferential equations for approximation of thestate and observation processes and for solutionof the estimation and control problems. Thisapproach is used in this paper.

2. MODEL

2.1. Treatment processRemoval of organic substances in a full-scaleaeration basin with distributed water feeding,sludge recycling and removal can be describedby the following system of stochastic functionaldifferential equations (for the basic ideas ofstochastic partial differential equations, seeRozovskii, 1983).(dX), = /_@)X dt + dW,

dX = (dX), + 2 Q&f, - X) - Qo,, $(1)

- (Qf + 8.)X] dt + dW> (2)dS = - YP(dX), + 2 1Qr(Sf - S) - Q,,, $

+ QJ S (t, L) - S 1) dt -t dW. (3)This is the simplest description of the large-scaleaeration basin. It can be obtained (Appendix B)in this form by extension of the lumpedparameter model (Tenno and Uronen, 1995a).

Here the biomass growth and substrate oxidationprocesses are considered as the coordinate-dependent processes, unlike the situation in acompletely mixed aeration basin.The following notation is used in the model:XS

srx,(dxh,/A(S) = /dI(Kj + S)

:IY

biomass (activated sludgeconcentration);growth substrate (organicmatter concentration inthe aeration basin):wastewater organic matterconcentration;return sludge concentration;biological growth (versusthe mechanical growth);specific growth rate;maximum growth rate;Michaelis-Menten satura-tion constant;biomass yield;

{X, = Xg(t, l)}, {X = X(t , f)} and {S = S(t, e)}are stochastic processes with values in aseparable Hilbert space, {X, = X,(t)} and {S, =$(t)} are stochastic processes in a Euclideanspace, and {W = W(t , f)}, {W = W(t , f?)} and{Wj = Wj(t, e)} are Wiener processes with valuesin a Hilbert space* and with smooth covarianceoperators CT, T;, and oj respectively;u, u;, a, standard deviations of errorscaused by model inadequacy,e moving coordinate;2 length of the aeration basin;Qr = Qf(t, e) wastewater feeding rate,(distribution);Qr = Qrk f> sludge recycling rate:Qe = Q& 4 excess sludge removal rate.The latter three flow rates are local. Totalfeeding Qf(t), recycling Qr(t) and removal QJt)rates for the whole system can be calculated assums of local flows by integration over the lengthof basin, e.g.

Q,r(t) = kyQ,N 4 de.Q,,Jt, f) is the sludge and water flow inside theaeration basin. It can be calculated as themoving coordinate sum

Q,& 0 = 1 W, u> du (4)0of the feeding, recycling and removal flows:

*For any orthogonal basis e E H the scalar product{(W(t, j), e(C))} = {W(r)} is a standard Wiener process,with covariance M{(W(C 4, e(e))(W(s. 4, e(R))} =min (t, s)(cr(e, R)e(e), e(R)), where u*(e, R) is the positiveself-adjoint nuclear operator.


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308 R. Tenno and P. UronenU = Qr + Qr - Q,. For interpretation of theseflows, see Fig. 2.2.2. Observation processThe dissolved oxygen concentration &, KJ, K>, W1 and {WJ are Wienerprocess in a Hilbert space; it is assumed that thecovariance operator is the identity for measuredprocesses {V.} and smooth (a,, a,) for otherprocesses {W.};

standard deviations of errorscaused by model inadequacy,rdcjt r,, rc measurement accuracies.The perturbations W , W ,, W ,, W ,, W C of a realtreatment process are correlated with nearestneighbours. In the simplest case this correlationcan be accounted for in the model using theexponential function

v([, R) = a*R-I, R < 1.The model (l)-(B) can be expressed indiscrete form as a system of ordinary stochasticdifferential equations (see (B.l)-(B.5) in App-endix B). Each equation of the system can beconsidered as a local description of the treatmentprocesses in a small aeration tank connected tothe others as a unit in a sequence (Fig. 3). Thetreatment and observation processes in the

large-scale aeration basin are very similar to theprocess in a chain of small treatment units. Thechain can be considered as the simplest systemfor description of the processes in the large-scaleaeration basin.Remark. The model depends on units ofmeasurement. It is assumed that the flows aremeasured in dilution rate units and calculatedper aeration basin volume. The physicalinterpretation of local flows is special. Theycannot be measured in the same units as thetotal flows. In this paper the following units areused: h- for total and h- m- for local flowrates. The flow inside the aeration basin QO,, isconsidered as total flow, also in units of h-.Standard units (m h-, m* h-) can also be used,but in this case the model has to be modifiedwith the replacement Q + Q/V, 2?+ l/A, whereV is the volume and A is the cross sectional areaof the aeration basin.2.3. ProblemThe problem is to estimate the currentdistribution of the state {X(t, e), .S(t, e), S,(t)} bydiscrete observations of the random field{td&? l>t t& 81, k($ e), 0 5s s th optim allyin the mean-square sense.


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Estimation for large-scale system 309

Fig. 3. Decomposition of a large-scale treatment system into a sequence of small treatment units.

2.4. SolutionThe solution of the problem is summarised inSection 3. It is obtained in this form byapproximation of the infinite-dimensional proc-esses as finite-dimensional (Appendices A-C)and of the filtration distribution as the normaldistribution (Appendix D). Approximation ofthe nonlinear distributed parameter processes(l)-(8) in a simpler form is an essential step inthe solution of the problem. It is shown that themodel in final form can be expressed as thefollowing system of bilinear stochasticdifferential equations:

de = [a, + a, 0, + $, a:S(t, fS)X(t, Is)] dt+ b, dW, + bz dW,, (9)

d,$ = [A + A,X, + & A:S(t, lS)X(t, [s)] dt+ B, dW, + B2 dW,, (10)

with the following notation:

For detailsAppendix C.

state, 8 = [X ST S,];measurements. &=vectors, e.g. X =[X(t, 6) . . . X(t, L6)lT;discrete space interval,with L an integer;Wiener processes for modellinguncertainties;vector and matrix functions,depending on sludge and waterflows and the dissolved oxygenconcentration, as well as on thebiological constants and thestandard deviations of measu-rement errors and model in-adequacy errors.

of the system (9)-(lo), see

3. STATE ESTIMATIONThe problem is to estimate the currentlarge-scale state 0, using available observations

&, 0 5 s zz t, optimally in the mean-square sense.An exact solution of the problem can beobtained, but is hardly compatible with real-timeapplication. A relatively simple and at the sametime effective calculation rule can be obtained byapproximation of the infinite-dimensional filtra-tion distribution by the normal distribution. Thealgorithm obtained by this method has a formquite similar to the bilinear filter (Tenno andUronen, 1995a), but the dimension of the filter islarger and it contains a sum over the coordinatesof state and observation processes. The algo-rithm is in the following form:dm = [ U, + u,m + i u:(mimmf rfj)1t + u dw,I=1 (11)

dy = (DuT + uD= + b, b: + b2b: - acr) dt+ i @(% dQ (12)I=1

where m, the estimation of the current state, -y, sthe covariance of is the estimation errors, rij isan element and y.j a column of the matrix yI,(w;) is an innovation process with newmeasurements,dIV=dt- Ao+A,mi+5Ai(mfmj+$j) dt,[ I=1 I

(+is the gain for correction of the state,u = (b, BT + DAT)(BBT)-,

BBT = B, By + B,B;,92t! is the gain for correction of the covarianceof the estimation errors,

@ = r!,ry + y!,yT, %!? (BB=)-A;,I@, d,?V) , the scalar yduct,L a =al u2 . . . u2) and A=[A, A2 . . . AZ) areblock matrices composed of elements of the


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310 R. Tenno and P. Uronensystem (9), (lo), D = [do d, . . . dL] is a block sludge and water feeding was concentrated atmatrix composed of estimation-dependent the beginning of the aeration basin and theelements sludge removal at the end.

d,, = ^J/, d, = mf y!, + mj y!,, I= 1, , L.With the proposed algorithm, one canestimate the current distribution of the state inany area of the aeration basin using commer-cially available equipment: a dissolved oxygenanalyser (multielectrode modification), the outletgas flow rate, O2 and CO2 analysers, and valvesfor gas probe collection over the aeration basin.The derivation of the algorithm can be foundin Appendix D, and its testing in the simulation

experiment in Section 4.

2. In distributed feeding the sludge and waterfeeding was distributed uniformly along theaeration basin as well as the sludge removal.3. In rotated feeding the location of the feedingpoint was changed every hour: rotated to theend and started again from the beginning ofthe aeration basin. Location of the recyclingflow was changed similarly, but with sift backfrom the location of feeding point in aone-hour period. The removal flow wasconcentrated at the end of the aeration basin.

Remark. The dimension of the filter can bereduced if it is used on the conventional aerationbasin, because the sludge concentration there isnearly constant along the basin. In this case theestimation accuracy is higher, since the numberof observations per unknown states is nearly twotimes higher.

The following was observed in the simulationexperiment. Variation of the profile is ratherdrastic for organic matter (Figs 4-6) and nearlyconstant for biomass. Both these characteristicsare strongly affected if the feeding or recyclingdistribution is rotated (Figs 6 and 7). The sludgeand water distribution has a strong effect ontreated water quality (Figs 4-6). The distributionis an important parameter for process control.

4. SIMULATIONA special simulation experiment was used fortesting the model and the state estimationquality.

4.2. Estimation accuracy

4.1. Model qualityThe profiles of biomass and organic matter areshown in Figs. 4-7. They are shown along theaeration basin for a three-day period for threedifferent feeding modes.1. In the conventi onal poi nt-f eeding mode the

The simulated and estimated processes arerelatively close to each other. To see a smalldifference, compare these processes in Figs 8 and9. The estimation error characteristics aresummarised in Table 1. The estimation errordepends on the feeding mode. It is smallest inthe case of uniform feeding, because thevariation of sludge is minimal and the modelinadequacy errors are strongly correlated withnearest neighbours in this case. Variation of the

1 6 I, l6 * 26 3 , & I4 q46 51. .Ime,n 56 6; 7 l671

1350-400moo-3508250-300mzoo-250I150-200E9100-150mo-100m-50

mgth, m

Fig. 4. Organic matter profile along the aeration basin when the conventional point-feeding mode was used.


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Estimation for Iarge-scale system 311

Fig. 5. Organic matter profile hen the distributed feeding mode was used.sludge is drastic in the case of rotated feeding.The estimation errors are largest in this case.They are moderate in the case of theconventional point-feeding mode.The estimated state is about 1% biased for theconventional and distributed feeding modes andl-10% biased for the rotated feeding mode. Thebias is arises from errors due to approximationof the original model with a suite-dimensionaibilinear model and the exact ~~tratjo~ d~str~bution with the normal dist~but~on.The estimation quality is much better for thelarge-scale aeration basin than for the compl-etely mixed aeration tank. This can be concludedfrom comparison of the estimation accuracy

obtained here and by Tennu and Uronen,(199Sa). A similar simulation experiment wastested there. Xn ocal area the aeration tank wasloaded s~m~lar~iyn both cases, The geometry ofthe large-scale treatment system was choseneight times larger in volume and length than thegeometry of tank considered earlier. For moredetails of the simulation experiment, see Tennoand Uronen, (1995a).

Sludge and water distribution is an importantparameter for process control. ~~furtunate~y, itis not clear froqn the model (l)-(8) or from the

Fig. 6. Organic matter profile when tie rotated feeding mode was used.


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31 2 R. Tenno and P. Uronen

1 6 1776 2, & m3 36 4, &--

Time, h 5 56 ;,Fig. 7. Biomass profile when the rotated feeding mode was used.

system (9), (10) how the treatment processdepends on the controls. Therefore it is shown inAppendices B and C that the only controllableelements of the system (9) (10) are the vector a,,and the matrix a,. a,, depends on the distributionin the following linear manner:

al has the following quasi-linear dependence(strictly, the matrix a, depends nonlinearly onthe controls):a, 0 = a, + b,QF- + &QK + &Q,...

Here 8 is the state, a,) is a state independentvector, b0 is a state-independent matrix, a,, b,

164

and b2 are state-dependent vectors, b3 is astate-dependent matrix, uK is the recyclingdistribution, with normalised values uR = QR/Q,,Qx is the (unnormalised) recycling distribution,Q,: is the wastewater feeding distribution, QE isthe excess sludge removal distribution, and U,,and QE are vectors, e.g.i;:;. o)Q:. . u(t, L6 - S)].

U=The system is linear in the control variables. It

is bilinear in the state variables and alsomutually in the state and control variables.The observation process depends on the waterdistribution through the washout effect ofoxygen. This dependence is rather weak. It canbe ignored as an optimisable parameter in acontrol problem statement.

a0

60

501 '00 1 2 3 4Time, daysFig. 8. Estimation quality for organic matter when the distributed feeding mode was used.


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Estimation for large-scale system 3137.2 ,

5.6

1=16m

5.2 m0 2 4 6 6 10 12 14 16 16 20 22 2Time, h

6

Fig. 9. Estimation quality for biomass when the distributed feeding mode was used.

6. CONCLUSIONSA relatively simple model can be applied for

description of the processes in a large-scaleaeration basin. Using the model and a simulationexperiment, it can be verified that the sludge andwater distribution has a strong effect on treatedwater quality. The distribution can be used as anew parameter for wastewater treatment processcontrol. Using the model, a new state estimation

problem has been solved. The estimationalgorithm is the main result of the paper. It isobtained in a computationally effective form. Itcan be applied to real processes for biomass andorganic matter concentration estimation in anyarea of the aeration basin using dissolved oxygenand gas analysis measurements. The estimationaccuracy is higher for a large-scale aeration basinthan for a completely mixed basin.

Table 1. Mean and standard deviation of estimation errorsDistributed feeding

I=1 2 3 4 5 6 7 8BiomassTreatedwaterWastewater

MeanStand.MeanStand.Mean

-0.062 -0.079 -0.087 -0.073 -0.072 -0.058 -0.057 -0.0540.312 0.314 0.296 0.286 0.266 0.262 0.251 0.2490.096 0.068 0.054 0.096 0.109 0.119 0.121 0.1081.527 1.668 1.527 1.504 1.488 1.485 1.4% 1.5152.21 Stand. 14.8

Conventional feedingI=1 2 3 4 5 6 7 8

Biomass Mean 0.010 0.000 -0.011 -0.011 -0.015 -0.001 0.009 -0.020Stand. 0.149 0.169 0.191 0.206 0.212 0.221 0.231 0.232Treated Means -1.142 -1.026 -0.889 -0.706 -0.526 -0.362 -0.247 -0.169water Stand. 8.672 6.870 5.197 3.728 2.578 1.747 1.173 0.804Wastewater Mean 6.51 Stand. 17.8Rotated feeding

I=1 2 3 4 5 6 7 8Biomass Mean -0.402 -0.020 -0.153 -0.055 0.002 0.078 0.011 0.006Stand. 0.922 0.539 0.507 0.494 0.434 0.474 0.427 0.431Treated Mean -2.305 -1.166 -0.093 0.469 0.827 0.516 -0.559 -0.730water Stand. 15.67 14.21 14.06 14.42 16.06 17.38 18.00 16.58Wastewater Means 9.41 Stand. 24.2


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314 R. Tenno and P. LJronenREFERENCES

Benssousan. A. (I 982). StochusticControl hj, bitr~rronulAnalysis Methods. North-Holland. Amsterdam.

Dochain, D., _I. P. Babary and N. Tali-Maamar (1992).Modelling and adaptive control of nonlinear distributedparameter bioreactors via orthogonal collocation.Automatica, 28, 873-883.

Fleming, W. H. (1975). Distributed parameter stochasticsystems in population biology. In A. Benssousan and J. L.Lions (Eds). Control Theory, Numerical Methods andComputer Systems Modelling, pp. 179-191. Lecture Notesin Economic and Mathematical Systems. Vol. 107.Springer-Verlag. Berlin.

Funaki, T. (1979). Construction of a solution ol randomtransport equation with boundary condition. ./. Moth. Sot.Jpn, 31. 719-744.

Glonti. 0. A. (198s). lnuestigations in the 17frorv ofConditional Gaussian Processes. Metsniereba. Tbilisi (inRussian).

Gray, N. F. (1990). Actfvated Sludge. lleory and fracfrw.Oxford University Press.Pugachev, V. S. and I. N. Sinitsyn (1987). Stochtr.\frc,

Differential Systems. Springer-Verlag. New York.Rozovskii, B. L. (1983). Stocha.stic Evolution .S~.sfcm.~.Nauka, Moscow (in Russian).Sobczyk. K. (1985). Stochastic Wave Propagatum. Else&r.

Amsterdam.Tenno. R. and P. Uronen (199Sa). State and paramctci

estimation for wastewater treatment processes using astochastic model. Conrroi Engineering Practice, 3, 79%+Ik!.Tenno. R. and P. Uronen (1995b). Stock and concentrationdynamics of activated sludge processes. In Proc. 6thInternational Conjl on Computer Applicutiom inBiotechnology, Garmisch-Partenkirchen, pp. 3 10-3 14.

Tenno. R. and P. llronen (199%). Optimal fcedinpdistribution for large scale aeration basin. In Proc,. 7thIFAC/IFORS/IMACS Symp. on Large Scule Systems :Theory and Applications. London. Vol. 2. pp. 603-608.

Tenno. R. and P. Uronen (1995d). Stochastic control for alarge scale wastewater treatment system. Automafictr.submitted.

APPENDIX A-SPECIFICATION OF THE MODELThe following steps are necessary for transformation of the

model in explicit form and for its approximation in a formsuitable for control application. Here the same approach canbe used as in Tenno and Uronen (199Sa). but withmodifications due to distributed parameter processes.Return sludge fluxThe return sludge concentration can be eliminated fromthe model (2) using the following relationship between thesludge concentrations in the aeration and settling basins(Tenno and Uronen, 199Sb):Q,(t, P) X,(t) = Q,(t, i) X(f. Y) + {pa,cu + larQ,(f) pa,(1 ~ Z)]}X(f. Y)}v(f)u,(f. (). (A. 11Here (Y s the specific settling velocity and c the reduction 01the velocity due to thickening, p is the volumetric ratio:settling per aeration basin, k,, is the decay rate and 174, thewashout rate, n is the mass ratio of nonseparable perseparable part of the total biomass, 2 is the stock per settlingbasin capacity, v(t) is the recycling efficiency. (,, and ~2~arclinear approximation coefficients.The biomass concentration in the aeration basin is affectedby the total recycling rate through the recycling efficiency

Q,(r)

The latter depends on the sludge concentration according tothe relationshipu(X) z X(1.W)X(f, Y) + p U, a 2 0. (A.2)

lhe lmear approximation coetbcients also depend onconcentration through the function (A.2). An average valueof the stock : = MZ(t) and concentration .Y = MX(t, 2) canhe used for simplification of the model, i.e. forapproximation of the function (A.2) and specification of thelinear approximation coefficients a,, = u,,(x), p = 1, 2..Appro_~m~utiorf ol the growth rate fimction

The nonlinear model (l)-(8) can be simplifed using apiece wise-linear approximation of the growth rate function

(A.3)where CL;) nd CL: are linear approximation coehicients.

In this case the model can be approximated as a bilinearmodel hy state variables.Organic- mutter fktuation vi wastewater

The lluctuation can be accounted for in the model (3)using the relationships, -- /*, t ab&(f ~ 24) ~~p, ] + s,

d.S,, .- Cp,S,, dr + f ) ~ (dX):. + [Q,(r. ( ~ 6)X,(t) ~ Q&r, ( ~ 6) AX(r, l)

(Q, + Q, + Q,.,(r. 4 ~ s)X(t. C)] dti CT:W,(r. I), (B.1)

d.Y(i. 0 = _~Y (dX);+ [Q,(r, ( 6)&(r)~~Q,,,,,(r. / 6)AS(t. 0 t Q,(r, & S)S(f. L6)~~ Q, + Q,)(f. C fi)S(r, t,] dr+ v; dW,(t. 0: (B.2)

rhwrurit~orr pr0ces.r(dam,):, = Y,, (dX),; t rrr,,X(r. f) dt + tr:,dW,,(r, /).

dt,,,,(l, 0 = {RQ,(f. ( ~ S)[P ~~ ,,,(f, ( 6) &xf* (1(cr),+ Vr)(f. i ~~ K,,,(r. t))dt(WX, + r,,,,dV,,,,(f.0. (8.3)


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Estimation for large-scale system 315.&At,) = (4X, + Qoutk 1 8) &,o(t, e)

+ (Q, + Qr)(t, f - %,A e) + roi,(t, e), 03.4)&.(t, e) = Y,(X);+ m,X(t, e) + a,PJt, 6) + r,V,(t, e),

(W

ko, &, W, W, W,, WC, r,,, ib, vc nd Qoor, e.g.x = [X(t, 6) . x(t, L6)lT, and for elements of thematrices o,, u,, uo and tro, e.g.

where AX(t, f) = X(t, e) - X(t, 8- 6) etc. and Q&t, e) isthe flow rate inside the aeration basin,Qo& e) = QourO, - 6) + W, e - 61, QwtO,0) = 0.

In this form the system has a simple physical interpretation.It can be considered as a system of small treatment tanks insequence (Fig. 3). Here the local flows and flow inside theaeration basin are defined as individual flows for each tank,in units of hh. They are defined using the small aerationtank volume (compare with the units in the distributedparameter model). The finite-element model (B.l)-(B.5) canbe obtained on the basis of the distributed parameter model(l)-(S) using the following replacements and renominationsof the specific flows:

Numbers from 0 to L6 - 6 are used for coordinates of thevectors U, QF, QR and Q, and for elements of the matricesOouT, C7, 0+ and b. Here F and OouT are band-typematrices,0I . . . 0 0F= . . . 0 7.0 . i 01

Qdh-I = ,,2,,Jm3 h-9, C?outb3h-l~ QK lh-,lout 7zQ,,h-, m-,l = @Wh-l+ @.b*h-I ~ b,Ih-,l

V KHere the natural (m hh, m2 h-) and specific (hh) units areused, depending on the model, as shown in square brackets.The specific flows are calculated on the basis of wholeaeration basin volume V (distributed parameter model) andsmall aeration tank volume V, finite-element model).

0o = Qoutk 6) -Qout(t, 6)out

b .00

Qou-dt. L6 - 6)6 and 6+ are diagonal matrices with elements {Q(t, e)} and{J$rs ~,~~(;~x4] respectively, Qk e) = Qdt, e) + Q,k 4,OQ=

Remark. Since the total feeding rate Q,(t) is defined on thebasis of the whole aeration basin volume and the localQ,(t, h) feeding rate on the basis of the small aeration tankvolume, the sum over the feeding distribution is equal to L:

5 u,(t, h) = L.h=l

00..6 Q(t, L6-6) -Q,(t, t8-8)

Here u,(t, h) is the feeding distribution: u,(t, h) =Qf(t, h)/Qf(t). A similar sum (equal to L) is valid for therecycling and removal flows.

and rDo, ro and r, are diagonal matrices, e.g. r,o = r,,l,with r,, a scalar and I the identity matrix.Control-dependent elements

Vector presentationThe state and observation processes can be expressed in avector form very similar to the scalar case:

The return sludge flux can be eliminated from the model(B.6) using the relationship (Al) between sludge concentra-tion in the aeration and settling basins. It can be expressed asa linear function either in terms of sludge concentration,treatment process QRX, = g + f-X 03.8)

dX = (dX), + [QRX, + (0our - 0+)X] dt + a, dW,, (B.6)dS = - Y-(dX)G + [Q& + (0our - @)S] dt + u, dW,;

observation processes

or in terms of distributed controls uR = QR/Q,,

HereQRX~ = go+ 8 UR. (B.9)

(do,), = Y,~(dy)~ + m,X dt + uo dWo,d5,, = [RQ&* - &A + (GOUT 015001 t - (dOJu +rD0dV,o,

9 = Q&R, G= [ 0 0 QR + (pvuR],a, = WI av, g1 = (Q, + w)X(t, La),

cp= azQr - w,c - dl - dl- z)l& + vQ,h0 is the zero vector, and QR is the local and Qr the totalrecycling flow.

flow rate inside the aeration basinQow = FQc ur + u, u = QF + QR _ QE. (B.7)

Here lower-case subscripts are used for scalars and uppercase for vectors or matrices. Coordinates of vectors andmatrices are enumerated in two ways. Numbers from S to LSare used for coordinates of the vectors X, S, (dX),, (do,),,

It simple to prove that both vectors (0&r- 0+*-)Xand(ObT - @)S can be expressed in terms of distributedcontrols as linear functions. They can be written in thefollowing forms:

(Oou, - c)X = MQ F + QR) + %QE~ (B.lO)(Gxrr - @)S = &QF + (4 + %~,)QR, (B.ll)


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316 R. Tenno and P. Uronen

d, = 93, + i X(f. B)(E,,, , - E,)(I ~ F) .I- 2

%, = ~ i S(f, /S)E 1.1..I-I

.oi; and !A, are defined similarly to &, and &, but for S,X(t. t8) is replaced with s(t. 16).

Proof: Obviously,

C* :=C + i Q,(r. 16 - 6)E,,, ,I- I

where

o= 2 Q,(f, 16 - 6) + Q,(t, 16 - 6)]E,.I I

El,,, is an elementary matrix (drop the second subscript if thetwo are equal), Q&r, IS - 6) is the coordinate of the vectorQooI-. The latter can be expressed as a solution of (B.7) inthe form

Qou.r = (I - F) U. where L = Qt + QH Q,.Here (I - F)- is a triangular matrix: all of its elements are Ion or under the diagonal, with zeros above the diagonal.For both Z = X or Z = S the following relationship is valid:

%orfTZ= f: Z(t, WE,,, I - E,)Qo,, .I-2Obviously.

PX = i X(r. IS)E,[Q, + QR + Q,-1.I I

PFS = i [.S(t.M)E,(Q,. + QR)+ S(r, 6L)E:,,,Q,yJI= I

Therefore (B.lO) and (9.1 I) are also valid.

APPENDIX-SYSTEMThe model in its final form can be expressed as the

following system of bilinear stochastic differential equations:

de= i u,+a,e,+ia$e{ei dt+h,dW+hldWz.I ~-dt = (A,, + A, X, + i A;O$;) dr + B, dW, + Bz dW,.I- I

and W, are Wiener processes, and a,,, al,. ,& are vectorand matrix functions!

I Y,, 4 - wlA, = Y,;Mo+m, 1A:=L Y, M,+w Iw, = [w;;

d = @z4dm(r - 24)dt - @,b, + @dmf(t - 24)

4 = i d,G is a diagonal matrix,IL,M, = p(E, is a vector,

- I+11

E, is an elementary matrix whose Ith diagonal element is 1,with all the rest zeros, and /.L{, and CL{ are constantparameters (see the piecewise-linear approximation (A.3) ofthe growth rate function.

Dependence on controlThe following decomposition of the system in controllable

and uncontrollable components can be expressed from the(B.9)-(B.11):

For any 0 (or estimated state), the following decompositionis valid:

Here 0 is the state, t represents the measurements. W,, WI


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state estimation of large-scale waste water treatment system

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