*Corresponding author. Tel.: 00 33 3 83 17 52 34; fax: 00 33 3 83 1753 26; e-mail: lati"@ensic.u.-nancy.fr.

Chemical Engineering Science 54 (1999) 2707}2714

Methodology of dynamic optimization and optimal controlof batch electrochemical reactors

F. Fournier, M.A. Lati"*, G. Valentin

Laboratories des Sciences du GeHnie Chimique, CNRS-ENSIC, B.P.451, 1- rue Grandville, 54001 Nancy Cedex, France

Abstract

This paper presents a methodology of dynamic optimization and optimal control of a batch electrochemical reactor where a seriesof two reactions AHBHD takes place. The optimization problem is stated as to determine the optimal pro"les of the electrodepotential which maximize the selectivity of product B, in a given batch period and under speci"ed constraints. The main importantsteps which are worth to be analyzed prior to any on-line implementation are presented and discussed. Thus, after the formulation ofthe optimization problem, di!erent dynamic optimization methods have been used for the determination of optimal time-varyingpro"les of electrode potential. These pro"les were then compared to the best uniform operations through the relative improvements ofthe performance index. The sensitivity analysis of the optimization criterion and control variables to the model parameters wascarried out before state observation and implementation of optimal control pro"les. ( 1999 Elsevier Science Ltd. All rights reserved.

Keywords: Dynamic optimization; Sensitivity analysis; Optimal control; Electrochemical reactor; Series of reactions; Selectivity

1. Introduction

Recent studies, based on dynamic optimizationmethods, have shown that the traditional operatingmodes of batch electrochemical reactors, i.e. constantvoltage or constant current, are not always the best(Lati" et al., 1991; Bakshi and Fedkiw, 1993; Fournierand Lati", 1998). This has been proved by determiningthe best transient operating pro"les of control variableswhich optimize a de"ned performance index (cost func-tional or optimization criterion) subject to speci"ed con-straints.

The search for the best transient pro"les may be per-formed either in open loop or in closed loop control. Thelatter is referred to as optimal control. Its experimentalimplementation, i.e. on-line optimization, constitutes thelast stage of a series of steps which may be achieved whenone deals with optimal control of processes. These stepswill be presented and constitute a methodology ofdynamic optimization and optimal control of a batchelectrochemical reactor. The objective here is toshow how to handle a dynamic optimization problem

from the speci"ed optimization objective to experimentalimplementation.

Thus, considering that a process model is available,hence avoiding the modeling stage, the aforementionedsteps are: (i) determination of the best transient pro"les;(ii) search for the best uniform operation; (iii) comparisonof uniform and transient operations; (iv) sensitivity analy-sis; (v) state and parameter estimation; (vi) on-line optim-ization.

In this paper, these steps will be discussed and illus-trated on a batch electrochemical reactor where a seriesof two reactions AHB HD takes place. The optimalcontrol problem is stated as to determine the optimalpro"les of the electrode potential which maximize theselectivity of B, in a given batch period and under speci-"ed constraints. All the optimization methods used arebased on the Maximum Principle.

2. Model description

In this study, we consider a batch reactor where thefollowing series of two electrochemical reactions occurs:

A#l1eN kf1H

kb1B#l

2eN kf2H

kb2D.

0009-2509/99/$} see front matter ( 1999 Elsevier Science Ltd. All rights reserved.PII: S 0 0 0 9 - 2 5 0 9 ( 9 8 ) 0 0 3 7 4 - 1

The kinetics of the two reactions are assumed to be "rstorder with respect to the reactants and both forward andbackward reaction rates are given by Arrhenius-typeexpressions where the reaction potential is the electrodepotential:

kfi"k

fi0expA!b

i

liF

R¹

EB and kbi"k

bi0expAai

liF

R¹

EB.(1)

Under these conditions, the mass balance equations maybe written as

CQA"!

1

<

I1

l1F"a

11C

A#a

12C

B#b

1, C

A(0)"C

A0,

CQB"

1

< AI1

l1F!

I2

l2FB

"a21

CA#a

22C

B#b

2, C

B(0)"C

B0. (2)

For this system, the electrode potential E is the controlvariable (u), (C

A, C

B)T is the state vector (x) and the

current intensity I is the only measured output given by

I"I1#I

2"c

1C

A#c

2C

B#d, (3)

where aij, b

i, c

i(i, j"1, 2) and d, are complex non-linear

functions (Scott, 1985; Fournier and Lati", 1998) of theelectrode potential, rate constants and liquid/solid masstransfer coe$cients.

3. Dynamic optimization

3.1. Problem statement

For such a simple but typical electrochemical system,the key product is often the intermediate component B.Product quality or selectivity optimization is thereforea frequently encountered problem in industry. Moreover,in order to avoid reactant rejection or separation processor to better use of high cost reactant it is sometimesnecessary to achieve a speci"ed conversion rate of thereactant A.

Thus, it is possible to state the problem as to "nd theelectrode potential}time pro"les which satisfy the lowerand upper bounds:

E.*/

)E(t))E.!9

(4)

such that the cost functional

J0"C

B(tf) (5)

is maximized while subjected to the constraints (2) (massbalance equations) and the following terminal-point con-straint ("nal conversion rate of A):

XAf

!

(CA0

!CA(tf))

CA0

"0. (6)

The reactions considered here are the electroreduction ofoxalic acid (A"COOHCOOH) to glyoxylic acid (B"

CHOCOOH) followed by the reduction of glyoxylic acidto glycollic acid (D"CH

2OHCOOH) (Pickett and Yap,

1974; Scott, 1985).The physical data used are the following:

Ae"0.022m2, <"0.157]10~3m3, ¹"293.15 K,

CA0"680 molm~3, C

B0"C

D0"0, k

d"k

dA"10 k

dB"

10~4 ms~1, kf10

"10~13m s~1, kf20

"kf10

/3, kb10

"

kb20

"0, b1"0.162, b

2"0.157, a

1"a

2"0, E

.*/"

!1.7 V, E.!9

"!1.0 V, tf"3600 s, X

Af"0.8.

3.2. Optimization method

The general formulation of optimal control problemfor batch processes is as follows:Maximize the performance index J

0:

J0"G

0[x(t

f), t

f]#P

tf

t0

F0(x, u) dt (7)

subject to the di!erential constraints (state equations):

xR "f (x, u), x(0)"x0, (8)

where x is the state vector and u the control vector.The additional constraints may be of di!erent types:

lower and upper bounds of control, in"nite dimensional,interior-point and terminal-point equality and inequalityconstraints. All these constraints, generally involved inan optimization problem, can be written in the followingcanonical form (Chen and Hwang, 1990):

Ji"G

i[x(t

f), t

f]#P

ti

0

Fi(x, u) dt, i"1, 2,2, q. (9)

The constrained optimization problem (7)}(9) can betransformed into a non-constrained optimization prob-lem by de"ning an augmented performance index:

J"J0#

q+i/1

liJi"G[x(t

f),t

f]#P

tf

0

F(x, u) dt, (10)

where G"G0#+q

i/1liG

iand F"F

0#+q

i/1liFi. l

i's

are the Lagrange multipliers; they are determined opti-mally by the computational method used (Secion 3.3).

The resulting problem (8), (10) is then a standard non-constrained optimization problem whose solution, if itexists, satis"es the following di!erential-algebraic system(Pontryagin et al., 1964):

xR "LH

Lj, x(0)"x

0,

jQ "!

LH

Lx, j(t

f)"C

LG

LxDtf

0"LH

Lu,

(11)

2708 F. Fournier et al./Chemical Engineering Science 54 (1999) 2707}2714

Fig. 1. Optimal time-varying pro"les of electrode potential obtainedby three di!erent dynamic optimization methods.

where j is the co-state vector and H the Hamiltoniande"ned by

H"F(x, u)#jTf (x, u)"H0#

q+i/1

liH

i, (12)

where

Hi"F

i#jT

if, jQ

i"!LH

i/Lx and j

i(tf)

"LGi/LxD

tf, i"0, 1, 2,

2, q.

In system (11), the "rst di!erential equation is an alterna-tive expression of the state Eq. (8), the second is theadjoint equation and the algebraic equation is the opti-mality condition which enables the determination ofoptimal control u. It can be seen that the boundaryconditions of the state and adjoint variables are split.This is called a two-point boundary value problem(TPBVP) and is not easy to solve analytically. Computa-tional techniques are therefore needed.

3.3. Computational techniques

There are numerous computational techniques tosolve dynamic optimization problems. These techniquesmay be divided into three categories depending on thedegree of approximation of either the state or the controlpro"les. We therefore distinguish:

(i) The continuous}continuous techniques where boththe state and the control solutions are continuous.The two-point boundary value problem (TPBVP)resulting from the application of the maximum prin-ciple is then solved. The best known techniquesbelonging to this category are the boundary condi-tion iteration (BCI) (Ray and Szekely, 1973), thecontrol vector iteration (CVI) (Bryson and Ho, 1975)and the quasi-linearization (Lee, 1968).

(ii) The continuous}parameterized techniques where theoptimal control pro"les are approximated by a poly-nomial form and the state is treated as continuous.This technique is known as control vector parametr-ization (CVP) (Goh and Teo, 1988). The most fre-quently reported approximation consists on usingzeroth order spline functions (piece-wise constantfunction) to approximate the control pro"les.

(iii) The parameterized}parameterized techniques whereboth state and control variables are approximatedby polynomial forms. Accurate approximation canbe obtained by the use of weighted residual methodssuch as orthogonal collocation (Villadsen andMichelsen, 1978).

The continuous}continuous techniques obviously lead tothe exact optimal solution of the problem. This solutionis, however, obtained at the expense of a long computa-tional time. Moreover, the maximum principle is notappropriate to easily handle neither equality nor inequal-

ity constraints even when they are accounted for informulation (10). This is an important drawback sincephysical problems are often subjected to constraints.

For the two other categories, the dynamic optimiza-tion problem is transformed into a more classical non-linear programming problem. It thus can be solved bysequential quadratic programming methods like NLPQLtechnique (Schittkowski, 1985). These two kinds of tech-niques are therefore more adapted to handle any kindof constraints.

3.4. Results

Fig. 1 presents the optimal electrode potential}timepro"les obtained by three di!erent methods. Each ofthese methods belongs to one of the three aforemen-tioned categories.

It can be seen that all the methods lead to the sameoptimal pro"les. As mentioned above, the continuous}continuous techniques require between tens of second(CVI) and hours (BCI) to yield a solution. Moreover, theinability to handle constraints leads, in practice, to amuch longer computational e!ort when the control isrestricted to its domain. As far as parameterized tech-niques (CVP, collocation) are concerned, the solutioncould be obtained within tens of second with no majordi!erences when constraints are involved. It is worth-while noticing that the accuracy of the optimal perfor-mance index for all these methods was equivalent.

The performance indices resulting from optimized op-erations are then compared to those obtained with thebest static (uniform potential) operations. The relativeimprovements of the optimal performance index withrespect to the static performance index for di!erent "nalconversion rate of product A and for three values of themass transfer coe$cient are presented in Fig. 2. Theseimprovements point out two major results. First, forsome operating conditions (X

Af)60%), the same

F. Fournier et al./Chemical Engineering Science 54 (1999) 2707}2714 2709

Fig. 2. Relative improvements of the performance index.

performances could be achieved with both operatingmodes. It is then not necessary to operate at optimaltransient control pro"les in that case. Under some otheroperating conditions, however (X

Af*60%), the im-

provements rise up to 30% when the potential boundsare not accounted for and 25% when the electrodepotential is restricted to its domain, i.e. !1.7 V)

E(t))!1.0 V. These operating conditions have an in-dustrial interest since they correspond to short batchperiod, limiting mass transfer conditions or high conver-sion rate constraints. These improvements thus empha-size the bene"t of optimal operating mode even whencompared to the best industrial operating modes.

It is interesting to notice that the computation ofoptimal pro"les of electrode potential is based on a nom-inal process model (1)} (3) where the parameters, i.e.kinetic parameters and liquid-to-electrode mass transfercoe$cients, are supposed to be measured or determinedwith no errors. Actually, the real process may deviatefrom the assumed model due to model parameter vari-ations. It is therefore important to analyze how theattained performance index and computed potential pro-"les remain optimal when the model parameters aremore or less accurately known. This analysis will becarried in the next section.

4. Sensitivity analysis

The e!ects of possibly parameter variations dp on thesystem performance J may be evaluated by means ofsensitivity analysis. They can be quanti"ed by either anon-normalized sensitivity index which is given by thepartial derivative p(LJ/Lp) or by a normalized sensitivityindex L lnJ/L ln p"(p/J) (LJ/Lp) (Eslami, 1994). These in-dices allow the determination of, respectively, absoluteand relative variations of the performance index withrespect to a parameter relative variation. The non-nor-malized sensitivity index is mainly used in this study. The

parameters considered are the mass transfer coe$cients(k

dAand k

dB) and the kinetic parameters (k

fi0and b

i,

i"1, 2).The variables concerned by the sensitivity analysis are

the performance index and the control.

4.1. Performance index

Here, the interest is to evaluate how far it is worthoperating at optimized electrode potential in case ofmodel mismatch. The performance indices obtainedwhen the best uniform and the nominal optimal electrodepotentials are, respectively, applied to the nominal pro-cess model and to the real system are then compared. Thereal system is simulated by the process model where theparameters are perturbed. The comparison criterion isde"ned by the following relative di!erence of the result-ing performance indices:

In"

Jp(u

n)!J

n(v

n)

Jn(v

n)

, (13)

where the subscripts n and p refer, respectively, to nom-inal and perturbed model, v

nis the best uniform electrode

potential.The sensitivity analysis of the performance index will

be mainly used to determine In. In fact, J

p(u

n) is deduced

from Jn(u

n) as follows:

LJ"Jp(u

n)!J

n(u

n)"P

tf

0CLH

LpT(x

n, j

n, u

n, p

n)D

TLp(t) dt.

(14)

Jn(u

n) is the performance index obtained when the nom-

inal optimal electrode potential is applied to nominalprocess model.

As an illustration, Fig. 3 presents the ratio In

plottedversus the relative deviations of the mass transfer coe$-cients (Fig. 3a), frequency factors (Fig. 3b) and energeticcoe$cients (Fig. 3c) near their respective nominal values(k

dA"1.0]10~4ms~1, k

dB"1.0]10~5ms~1, k

f10"

1.0]10~13ms~1, kf20

"3.3]10~13ms~1, b1"0.162,

b2"0.157).It can be seen that the ratios I

nare more sensitive to

kdB

than to kdA

since the performance index is the "nalconcentration of product B, C

B(tf). These ratios remain

signi"cant when the error on the mass transfer coe$-cients ranges from !10 to 50%. This is an interestingresult since it shows the importance of optimized operat-ing mode with respect to the best static mode. Concern-ing the kinetic parameters k

f10and k

f20, the sensitivity

pro"les are di!erent but in the same order of magnitudeas the mass transfer coe$cients. The energetic coe$-cients b

1and b

2must, however, be known with an

extreme accuracy since a 1% deviation from their nom-inal values can vanish the improvement of dynamic op-timization with respect to static optimization.

2710 F. Fournier et al./Chemical Engineering Science 54 (1999) 2707}2714

Fig. 3. Sensitivity analysis of the performance index with respect to theparameters: (a) mass transfer coe$cients; (b) frequency factors; (c)energetic coe$cients.

4.2. Optimal control proxles

The sensitivity index of the optimal control pro"le=

u"Lu/Lp is evaluated by partially di!erentiating the

di!erential-algebraic system (11) with respect to p andsolving the resulting linear TPBVP in which=

uas well as

the sensitivity of the state=x"Lx/Lp and adjoint vari-

ables=j"Lj/Lp, appear.

=Qx"

L2H

LjLp, =

x(0)"0,

=Q j"!

L2HLxLp

, =j(tf)"0,

0"L2HLuLp

.

(15)

Fig. 4 presents, in the form of temporal variations of p=u,

the resulting sensitivity of the optimal control pro"lewith respect to mass transfer and kinetic parameters. Theanalysis of these curves shows, here again, that the opti-mal pro"les are more sensitive to kinetic parameters thanto mass transfer coe$cients. The energetic coe$cientsbiof the forward reactions were found to be the most

in#uencing parameters.On the other hand, it can be seen that the optimal

pro"les are more sensitive at the end of the operationthan at the beginning with respect to all parameters. Thismeans that in the time interval [0.7t

f, t

f] the control

must be accurately tuned in order to reach the de"nedobjective.

It is interesting to notice that the sensitivity analysis ofthe optimal pro"les allows

(i) to identify the parameters which have the largestin#uence on these pro"les during the whole operatingtime,

(ii) to indicate precisely at what time or time interval ofthe batch period the control must be more or lessaccurate in order to reach the speci"ed performances.

The optimal control pro"les determined do not accountfor external disturbances. In order to practically imple-ment these pro"les, a closed-loop control scheme isnecessary. The next section is devoted to on-line optim-ization problems.

5. On-line optimization

In this section a closed-loop control is then carried outin order to account for ubiquitous non-idealities as statedisturbances, time varying dynamics and process modelmismatch. As opposed to usual control schemes, thecontroller used for the tracking and regulation is optimalin the sense of the same performance index (10). This isimportant since the trajectories remain optimal all thetime even when the system is perturbed.

The original state equation is therefore linearized andthe variations of the criterion are developed up to thesecond order. Indeed, according to the maximum prin-ciple, the "rst-order terms vanish at the optimum. The

F. Fournier et al./Chemical Engineering Science 54 (1999) 2707}2714 2711

Fig. 4. Sensitivity analysis of optimal control pro"les with respect tothe parameters: (a) mass transfer coe$cients; (b) frequency factors; (c)energetic coe$cients.

variations of the criterion are then given by

d2J"1

2CdxTAL2G

Lx2BdxDtf

#

1

2 Ptf

t0

[dxTduT]

]

L2H

Lx2

L2H

LxTLu

L2H

LuLxT

L2H

Lu2

Cdx

duDdt. (16)

As a matter of fact, the optimization of this quadraticcriterion subject to linear state constraints leads exactlyto the linearized version of the original TPBVP (Brysonand Ho, 1975). The solution of the following linear two-point boundary value problem is hence needed.

dQ x"L2HLjLx

dx#L2H

Lj2dj#

L2HLjLu

du ,

dQ j"!

L2HLx2

dx!L2HLxLj

dj!L2H

LxLudu ,

0"L2HLuLx

dx#L2H

LuLjdj#

L2HLu2

du

(17)

with the boundary conditions dx(0)"0 and dj(tf)"0.

Such a linear quadratic problem can be solved bya Sweep method which consists of expressing the adjointvariations as a function of state variations as follows:dj"Sdx, where S is the solution of the resulting Riccatiequation. Since a linear quadratic problem is involved, itcan be shown that the optimal control solution du(t) isalso related to the state variables by the following linearexpression: du"Kdx where K is the gain of the optimalcontroller, depending on S and model parameters(Bryson and Ho, 1975). du(t) represents the variations ofthe control variable in the neighborhood of the optimaltrajectory.

In order to implement this control law, the values ofthe state variables are required. A state observer, basedon Kalman Filter (Kalman, 1960) is therefore designed.

The closed-loop optimal control of the electrodepotential is illustrated in Fig. 5. The open and closed-loop optimal control pro"les E are presented in Fig. 5a.Similarly, the open and closed-loop output pro"les Iare given in Fig. 5b. These pro"les result from the ap-plication of the optimal state-feedback law, in the pres-ence of a large and constant output perturbation DIstarting at t/t

f"0.25 and ending at t/t

f"0.50. From

Fig. 5, it can be checked, a posteriori, that the lineariz-ation, on which the developments of the controller arebased, is acceptable since the control deviation remain inthe neighborhood of the open loop trajectory. It can beseen that the resulting output (current) does not necessar-ly go back to the set point trajectory after a perturbation.The corresponding input (electrode potential) hencedeviates from its optimal pro"le obtained in open-loopcontrol.

It is also important to notice that the constraint on theconversion rate is perfectly satis"ed since it was includedin the optimal controller formulation.

2712 F. Fournier et al./Chemical Engineering Science 54 (1999) 2707}2714

Fig. 5. Open and closed loop optimal pro"les: (a) electrode potential(input); (b) current (output).

6. Conclusions

Several computational techniques to solve dynamicoptimization problems and their speci"c features havebeen used to optimize the dynamic behavior of a batchelectrochemical reactor. Optimal pro"les of the electrodepotential are thus determined. They are realistic since themost important physical constraints are considered. Theperformances of the optimized operating modes are com-pared to those of the best static modes. The resultingimprovements rise up to several tens of percent henceshowing the economical interest of dynamic operatingmode. Sensitivity analysis of the criterion and of thecontrol pro"les points out the validity of the optimalconditions even when operating conditions deviate fromtheir ideal values used in the model. The optimal controlpro"les have been implemented in a closed-loop controlscheme, including a Kalman Filter based state observer.The control law is an optimal state feedback law de-signed to optimize the criterion variations in the neigh-borhood of the open-loop optimal trajectory.

It is clear that in on-line optimization, all these impor-tant steps are required prior to any experimental valida-

tion. As an important e!ort has been paid to physicalfeasibility of the numerical results, the optimization of thedynamic operating mode of the electrochemical reactorseems to be very encouraging. The results need now to becompleted by an experimental validation.

Acknowledgements

The authors are grateful to Ministere de l'EducationNationale, de la Recherche et la Technologie for its"nancial support.

Notation

Ae

electrode surface area, m2

C concentration, molm~3

E electrode potential, VF Faraday constant, 96 500C mol~1

I current, AJ performance indexkd

mass transfer coe$cient, m s~1

kfi0

frequency factor of the ith forward reaction,m s~1

kbi0

frequency factor of the ith backward reaction,m s~1

p model parameterq number of constraintsR perfect gas constant, 8.32 J mol~1 K~1

tf

batch period, s¹ temperature, Ku, v control variable< reactor volume, m3

= sensitivity indexx state variableX conversion rate

Greek lettersai

energetic coe$cient of the backward ith reac-tion

bi

energetic coe$cient of the forward ith reac-tion

j adjoint variableli

number of electrons

Subscripts0 initial1 "rst reaction2 second reactionA product AB product BD product Df "naln nominalp perturbed

F. Fournier et al./Chemical Engineering Science 54 (1999) 2707}2714 2713

Superscriptsopt optimal operationstat best static operation

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