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Computers and Chemical Engineering 32 (2008) 1877–1891

A Tabu search-based algorithm for mixed-integer nonlinear problemsand its application to integrated process and control system design�

Oliver Exler 1, Luis T. Antelo, Jose A. Egea, Antonio A. Alonso, Julio R. Banga ∗Process Engineering Group, IIM-CSIC, C/ Eduardo Cabello 6, 36208 Vigo, Spain

Received 18 October 2006; received in revised form 20 September 2007; accepted 16 October 2007Available online 25 October 2007

bstract

In this contribution, we consider mixed-integer nonlinear programming problems subject to differential-algebraic constraints. This class ofroblems arises frequently in process design, and the particular case of integrated process and control system design is considered. Since theseroblems are frequently non-convex, local optimization techniques usually fail to locate the global solution. Here, we propose a global optimizationlgorithm, based on extensions of the metaheuristic Tabu Search, in order to solve this challenging class of problems in an efficient and robustay. The ideas of the methodology are explained and, on the basis of two case studies, the performance of the approach is evaluated. The firstenchmark problem is a Wastewater Treatment Plant model [Alex, J., Bteau, J. F., Copp, J. B., Hellinga, C., Jeppsson, U., Marsili-Libelli, S., et al.1999). Benchmark for evaluating control strategies in wastewater treatment plants. In Proceedings of the ECC’99 conference] for nitrogen removalnd the second case study is the well-known Tennessee Eastman Process [Downs, J. J., & Vogel, E. F. (1993). A plant-wide industrial process

ontrol problem. Computers & Chemical Engineering, 17, 245-255]. Numerical experiments with our new method indicate that we can achieve anmproved performance in both cases. Additionally, our method outperforms several other recent competitive solvers for the two challenging casetudies considered.

2007 Elsevier Ltd. All rights reserved.

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eywords: Integrated process and control design; Mixed-integer nonlinear prog

. Introduction

During the last decade, the importance of an integrated pro-ess design approach, considering operability together with theconomic issues, has been widely recognized. The aim is tobtain profitable and operable process and control structures in aystematic way. Both the process design characteristics, controltrategies, control structure and controller’s tuning parametersave to be selected optimally in order to minimize the total

ost of the system while satisfying a large number of feasi-ility constraints in the presence of time-varying disturbances.onsequently, the use of global optimization techniques is there-

� This research was supported by the EU Marie-Curie Research Trainingetwork under project number MRTN-CT-2004-512233.∗ Corresponding author. Tel.: +34 986 214473; fax: +34 986 292762.

E-mail address: julio@iim.csic.es (J.R. Banga).URL: http://www.iim.csic.es/∼julio/ (J.R. Banga).

1 New address: University of Bayreuth, Department of Computer Science,5440 Bayreuth, Germany.

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ing (MINLP); Metaheuristic; Tabu search; Tennessee Eastman plant

ore advisable, although one also should take into account theegree of uncertainty associated with the process at the designevel.

More and more researchers are following the trend towardsonsidering the design and control aspects simultaneously. Asresult, a number of new methodologies has been developed

uring the last years for addressing the solution of processesign and process control problems. Sakizlis, Perkins, andistikopoulos (2004) presented an overview of the state of the art.ollowing this review the developed methods can be classified

nto two categories:

The first category methods use a multi-objective approach.In order to design an economically optimal process that canoperate in an efficient dynamic mode within an envelopearound the nominal point, usually, the objective contains

of two parts. The first part represents the economic issuesof the process, and the second one measures the dynamicperformance, i.e. controllability, of the system. Since thesetwo parts are competing, weights have to be predefined to

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represent the importance of the two objectives. In general,this is a hard task, since the quality of the of the solutiondepends on these weights. A second drawback of these meth-ods is that they also have problems to treat the dynamic ofthe process in a systematic way. As a result, these meth-ods often use a steady state model to determine the processdesign.The second category approaches use dynamic optimizationin order to find the most economic design. Instead of usingsteady state models, the system operation is represented witha dynamic model. In this case a single objective which mea-sures the economical performance can be used. The arisingoptimization problem is a challenging mixed-integer dynamicoptimization (MIDO) problem.

Recently a number of algorithms were suggested for solvinghe MIDO problem. The methods can be also categorized intowo groups. Table 1 shows an overview.

Some methods reformulate the problem and as a result aixed integer nonlinear program (MINLP) has to be solved. TheINLP is then solved by Outer Approximation, General Ben-

ers Decomposition or Branch and Bound frameworks. All theseethods are well-known for the ability to solve MINLPs. Draw-

acks of these methods are the facts that they either need to solveelaxed problems, i.e. the integer constraints are dropped, or theyolve a sequence of nonlinear problems (NLP) with fixed inte-er values. The latter situation implies a waste of computationalffort, if the integer values are far away from optimality. In thisence our developed methodology is different, since we neitherolve relaxed problems nor do we solve NLPs with fixed inte-er values. The integer and the continuous variables are treatedimultaneously.

The multimodal (non-convex) nature of MIDO problems haseen highlighted by, e.g. Schweiger and Floudas (1997) andansal, Perkins, and Pistikopoulos (2000), among others. Con-

equently, the use of global optimization (GO) techniques seemo be promising.

In the domain of deterministic GO methods, Esposito andloudas (2000) have presented approaches to solve dynamicptimization problems. This is indeed a very promising and pow-rful approach, but restrictions may apply for the type of pathonstraints which can be handled. Other groups (Singer, Bok, &arton, 2001; Papamichail & Adjiman, 2002) are also makingood progress in deterministic global optimization of dynamicystems, yet several issues regarding requirements (absence ofoise, discontinuities, etc.) and computational performance aretill present.

Regarding stochastic GO methods, a number of researchesave shown that they can locate the vicinity of global solu-ions for nonlinear dynamic problems with relative efficiencyBanga, Moles, & Alonso, 2003; Moles, Gutierrez, Alonso, &anga, 2003; Sendin, Moles, Alonso, & Banga, 2004), but theost to pay is that global optimality can not be guaranteed.

owever, in many practical situations these methods can be

atisfactory if they provide us with a good enough (often, theest available) solution in modest computation times. Further-ore, stochastic methods do not require transformation of the Ta

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riginal problem, which can be treated as a black box. Thus,hey can handle problems with complicated dynamics (e.g.iscontinuities, non-smoothness, etc.). In summary, stochasticethods do not provide guarantees that attraction basin of the

lobal minimum has been located. On the other hand, deter-inistic methods do provide guarantees, but they can not be

pplied to realistic problems like the ones considered here dueo the characteristics mentioned previously. However, stochas-ic methods have shown, empirically, that they are very goodt locating the vicinity of the global solution in reasonableimes.

In our case we present here a method based on extensions ofhe metaheuristic Tabu Search (TS) – which has been success-ully used in the area of operations research ((Glover & Laguna,997)– for solving the integrated process and control systemesign problem.

Tabu search has been recently reported to solve MINLProblems in chemical engineering, such as process synthesisLinke & Kokossis, 2003; Lin & Miller, 2004), process designWang, Quan, & Xu, 1999), scheduling (Balasubramanian &rossmann, 2003) and computer-aided molecular design (Lin,havali, Camarda, & Miller, 2005). The main differences of ourpproach with respect to these previous works are that (i) weave improved the TS basic scheme regarding the generationf new decision vectors (see detailed description below), andii) we have added a novel local solver (MISQP) to increasehe overall efficiency of the algorithm. This is particular impor-ant considering the computational cost of the dynamic systemsmbedded in the MINLPs that we are considering.

This paper is structured as follows. The general statement ofhe problem is presented in Section 2. Section 3 is dedicatedo the optimization methodology that we have developed. Inection 4 two challenging case studies are presented. The firstenchmark problem is a Waste Water Treatment Plant model (seelex et al., 1999, and references therein) for nitrogen removal

nd the second case study is the well-known Tennessee Eastmanrocess (see Downs & Vogel, 1993). Finally, we present theumerical results obtained by using our methodology and weompare the performance of our approach with other competitiveodern solvers.

. The integrated process and control system designroblem

In order to represent the interaction of process design andontrol the formulation of the problem has to combine compo-ents that express both design alternatives and the operability ofhe system. In general a superstructure is developed that con-ains possible alternatives that one wants to consider. A keylement that contributes significantly in the calculation of opti-al and sensible design solutions is the good definition of the

esign space and more specifically of the design superstruc-ure. Measuring the controllability of the process is achievedy introducing a system of differential and algebraic equa-ions that simulates the behaviour of the process under dynamicperation.

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Engineering 32 (2008) 1877–1891 1879

Taking this into account the mathematical formulation is asollows:

minv

J(v, tf ),

s.t. f (x, x, p, v) = 0,

x(t0) = x0,

h(x, p, v) = 0,

g(x, p, v) ≥ 0,

vl ≤ v ≤ vu,

(1)

here v is the vector of decision variables, J is the objectiveunction (often representing the costs) to minimize, x is the stateector, p are some parameters, f is the set of differential and alge-raic equality constraints describing the system dynamics and hnd g are possible equality and inequality path and/or point con-traints which express additional requirements for the processerformance. The starting time for the simulation is denoted by

0 and the end, respectively, as tf . Consequently, x0 expresseshe state variables at the beginning of the simulation. The lowernd upper bounds for the decision variables are given with vl

nd vu.A superstructure is developed that contains alternatives for

he process design. The common way to express alternativesn the design is introducing binary variables. In this way thective state of a design alternative can be easily expressed. Aalue equal to one stands for an active alternative whereas inhe case of zero the alternative is inactive, i.e. not used. Thus,ome of the elements of the decision variable vector v can beestricted to integer values. As a result the problem formulated in1) is a mixed-integer dynamic optimization (MIDO) problem.t is quite known that nonlinear problems with a large numberf integer variables are extremely difficult to solve to optimal-ty. Taking this into account the number of integer variables inhe superstructure should be as small as possible. Engineeringnowledge and experience have to be used to limit the searchpace for the optimizer.

Although the resulting problem is very difficult to solve it isorth investigating this kind of problems, since it is the onlyay to obtain both, a process that has low cost and a process

hat can be easily controlled. This approach is superior to allethods that only take one aspect into account. Reducing only

he cost might result in an instable process, while on the otherand the best process from the controllability point of view cane very expensive.

As mentioned before, there are different approaches to solvehis MIDO problem. Dynamic programming suffers from theurse of dimensionality. Complete discretization, when appliedo problems which contain a significant number of dynamictates (like the examples considered here) results in a largeumber of decision variables, complicating the application oflobal optimization methods. As an alternative, we think that the

ontrol parameterization approach, which results in a relativelymall number of decision variables, facilitates the solution of theesulting MINLP using stochastic global optimization methods,ike those based on Tabu search.

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The control parameterization technique parameterizes onlyhe control variables, so the decision vector will contain thatiscretization information plus other selected time invariantarameters. The optimization is carried out in the space of thesearticular decision variables only. In our work we have followedhis technique, i.e. we discretize the control variables and obtains a result a finite dimensional mixed-integer nonlinear pro-ramming (MINLP) problem with a dynamic system embedded,sually as a set of DAEs.

Since this resulting problem is frequently nonconvex, globalptimization (GO) methods are needed. As reviewed in the intro-uction, deterministic GO methods for problems with DAEsmbedded can only handle (i) small problems and (ii) problemshich are differentiable and continuous. This is never met in

ealistic integrated design problems, so this is why these prob-ems are so challenging, and why stochastic GO methods, likehe one presented here, can be useful to overcome these chal-enges.

. Description of the optimization method

Many algorithms for global optimization use a local solvero identify a local minimum by starting from an initial point,nd in order to reach the global minimum, a special strategyor deciding where to start the local solver is applied. Foronvex problems the local solver is able to find the globalinimum. If the problem is not convex the global optimality

f the solution of the local solver cannot be guaranteed. Inhis case one has to guide the local solver to the global opti-

um. This second component of the algorithm has to locatehe attraction basin of the global minimum so that a run ofhe local solver started in this basin will find the global min-mum. The approach developed here uses this strategy. As thelobal component, a procedure based on extensions of the Tabuearch (TS) algorithm is applied. Regarding the local solver,e have used an special adaptation of a sequential quadraticrogramming method for the mixed-integer case. The follow-ng subsection is dedicated to the description of the interactionf the two different components and the basic ideas behind theroposed methodology, whereas the local solver by Exler andchittkowski (2007), called MISQP, is described in a separateubsection.

.1. The hybrid strategy—MITS

TS is a metaheuristic originally developed by Glover (seeGlover & Laguna, 1997). For the optimization of combinato-ial problems, TS has proved to be a very successful strategy.uring recent years, it has also been applied to the optimizationf continuous problems. Here, an adaptation of TS to mixed-nteger nonlinear optimization problems, called Mixed-Integerabu Search (MITS), is presented. This algorithm is an enhance-ent of an approach proposed by Battiti and Tecchiolli (1996).

s Battiti and Tecchiolli, we also use a local solver to intensify

he search if necessary. The aim is to profit from the fast conver-ence of the local method. In this sense, the local solver MISQPs integrated in the TS framework.

rtrr

Engineering 32 (2008) 1877–1891

First, let us summarize the basic idea of a Tabu search algo-ithm. The algorithm starts from an initial solution vk. For thisurrent solution vk a set of neighbors is generated and the bestmong these neighbors is chosen to be the next iteration pointk+1, even if the function value is worse than the one of the cur-ent iterate vk. Allowing an increase in the objective function isecessary to escape from a local minimum. To avoid cycling ando guide the search into unexplored areas, some former visitedoints are set to be Tabu and so prohibited for some time. Thisrocedure is repeated by starting from the new current point,ntil some stopping condition is fulfilled.

The efficiency of the above mentioned procedure depends onhe choice of some key parameters, i.e., the length of the period aoint is set to be Tabu, and this is especially important in the casef continuous variables. Since we do not know anything abouthe topology of the objective functions, choosing right valuesor these parameters is non trivial. Ideally, we would like to useprocedure that is independent from such choices. This was

he main reason why we decided to use the basic concepts fromattiti and Tecchiolli (1996). The authors proposed a TS algo-

ithm that is robust for any kind of functions and self-adjusting,o that no parameters have to be set a priori. The componentsill be described now in more detail.

.1.1. NeighborhoodIn the basic Tabu search for discrete optimization the neigh-

orhood of an iteration point is built by all the direct neighborsf this point. Since we have to handle large mixed integer prob-ems, it is impossible to consider all neighbors and it is even aard task to define exactly what a direct neighbor is. Thus, weave to restrict the neighborhood to some generated points. Thedea is to subdivide the search space into regions that representsasins of attraction and to perform the search over these regions.s mentioned before a neighbor generating routine is used that

s self-adjusting and independent from parameters that have toe set a priori.

The initial search region is specified by bounds on each inde-endent variable vi: vl

i ≤ vi ≤ vui , for i = 1, . . . , n. n is the

umber of independent variable and includes both continuousariables and integer variables. Initially, this search region isubdivided into 2n equal-sized regions, obtained by dividing theearch space in half the range on each variable. This first guessor the different regions is very crude but will be updated duringhe search. The regions are stored as a search tree and we performhe search over the different nodes of the tree. In the beginninghe tree contains of 2n leaves representing the 2n regions. Asoon as two different local minima v∗ and v∗∗ are identified inregion, the current region is subdivided into 2n equal-sized

maller regions. If v∗ and v∗∗ belong to two different regionsf the new partition, the splitting is terminated; otherwise theplitting is applied to the region containig v∗ and v∗∗ until theireparation is obtained. Because of the splitting, a region canontain at most one local minimum that was found so far. The

egions will be stored internally as a tree and after some itera-ion of MITS the tree will be of varying depth in the differentegions, with regions of smaller sizes representing regions thatequire an intensification of the search. The leaves of the tree

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artition the initial search region, where the intersection of twoeaves is empty and the union of all leaves represents the initialearch space.

In each iteration MITS generates a set of points, called theeighborhood N(vk) of the current iterate vk. The iterate vk

ies in a unique region that can be identified by a well-definedinary string B = [b11, . . . , b1D, . . . , bn1, . . . , bnD], where Depresents the depth of the region in the tree. Initially, all regionre of depth one. After splitting a box, the new smaller regionsave a depth increased by one compared to their father regions,.e. the higher the value of D, the smaller the size of the region.he length of the region edge along the ith coordinate is there-

ore equal to (vui − vl

i)/2D. The position of the origin BDi of the

egion containing the current iterate along the i-th coordinate is

Di = vl

i + (vui − vl

i)D∑

j=1

bij

2j. (2)

he neighborhood N(vk) is a set of randomly generated pointsn the neighbored regions. Depending on the depth of the cur-ent region up to 3 × n neighbors are created randomly in theorresponding regions. The number can vary according to theize of the current region. The origins of the neighbored regionsan be obtained as follows. In each coordinate up to two neigh-ored regions are considered. Only the regions that do not violatehe box constraints for the variables are evaluated. The originseighBD of these neighbored regions are obtained by changingnly the ith coordinate according to the rule:

eigh BDi = BD

i ± vui − vl

i

2D. (3)

he length of the neighbor region edge once again is (vui −

li)/2D. In this way up to 2 × n neighbors are created. Addi-ionally, we start from the region of depth 1 that contains theurrent iterate that is represented by B1 = [b1, . . . , bn]. Fromhis box we generate n regions where the origins are calculateds follows:

eigh Bli = Bl

i ± vui − vl

i

2. (4)

ig. 1 shows an example for the generated trail points for awo-dimensional problem. x∗ is the current iterate and yi, i =, . . . , 6, are the generated trail points in the neighbored regions.he lower right region has been subdivided into 4 smaller regionsefore. The lower left and the upper right region are of the initialize, but as one can see we generate two new trail points in eachf these regions.

Among these neighbors the best one is chosen and the nextteration starting from this point and the corresponding region.or the unconstrained case that is

k+1 ∈ N(vk) : J(vk+1) = minv ∈ N(vk)

J(v), (5)

here v is not Tabu, i.e. v does not lie in a region that is markeds Tabu.

impm

Fig. 1. Example for generated neighbors.

.1.2. Diversification vs. intensificationAn important question arising in global optimization is when

o intensify the search in a specified region and when to exploreew regions. The risk is to waste a lot of time and evaluationsf the objective function in a region that does not contain thelobal optimum, or to miss it if the search was not intensified.fficient metaheuristics provide a good compromise betweeniversification (exploration by global search) and intensificationlocal search). In our approach the TS component has the purposef diversifying the search whereas the local solver intensifies theearch in a promising area found by the TS component and findshe local minimum with high precision. This section specifiesow this goal is accomplished.

The diversification is done by generating the neighbors inhe whole search space. In this manner every minimum can beeached. To enforce diversification we also prohibit recently vis-ted regions. If the objective function or the constraint violationor the infeasible case of the current iterate vk is less than all val-es of the generated neighbors, the local solver MISQP mighte started. To start MISQP:

(vk) ≤ J(v) for all v ∈ N(vk) (6)

s a necessary condition, but it is not sufficient unless for therst time a point in the region containing vk is locally optimal.n this case MISQP will always be executed. Otherwise, a newun of the local solver has to be justified. Battiti and Tecchiolli1996) apply some Bayesian stopping rules for multi-start globalptimization to determine if a new run should be executed. Wese the same strategy for MITS and so we summarize the idean the following paragraph.

Let r > 1 be the number of times MISQP has been started inhe current region. An additional MISQP run should be started ifnd only if there is a high probability of finding a new local min-

mum in the region. Starting MISQP in some parts of the region

ight lead the solver in a neighbored region. The region can beartitioned into W components, the attraction basins of the localinima contained in the region and the possible basins that lead

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ISQP outside. The value of W is updated during the optimiza-ion depending on the information the local solver returns, i.e.ermination inside a region or outside. It should be highlightedhat every region has its own value for W. MISQP is always exe-uted if r ≤ W + 1. This means that so far every MISQP runas detected a new attraction basin in the corresponding region.n the other case, if r > W + 1 restarts have been performednd W different components have been identified, we have toustify a new run of the solver. Since at least one previous run of

ISQP has led to one minimum that had been located before.n this case MISQP is started only if the following equation isatisfied:

and >(r − W − 1)(r + W)

r(r − 1), (7)

here rand is randomly generated number in the range [0,1]see, Battiti and Tecchiolli (1996)). In that way, the number of

ISQP runs can be kept down if the above estimate predicts amall probability of finding a new local minimum, but a new runs never completely prohibited for reasons of robustness.

The initial starting point for MISQP is the current iterate vk

nd the initial search region is the current region enlarged bywice the range of the region. If v∗ is a local minimum that wasound by MISQP, it is saved in a memory structure associated tohe region. If MISQP converges to a point outside the originalegion the solution will be the next starting point for the next iter-tion; otherwise, MITS continues from the best neighbor foundn this iteration.

As mentioned before, as soon as two different local minima v∗nd v∗∗ are identified in a region, the current region is subdividednto 2n equal-sized smaller regions. If v∗ and v∗∗ belong to twoifferent regions of the new partition, the splitting is terminated;therwise the splitting is applied to the region containing v∗nd v∗∗ until their separation is obtained. The local minima aressociated with the new corresponding smaller regions.

.1.3. Tabu tenor updateTo guide the search into unexplored regions some of the pre-

ious visited regions are set to be Tabu, i.e. as soon as a regionontains the current iterate vk it is prohibited for the next iter-tion. Changing the Tabu tenor T– the number of iterations aegion is Tabu and cannot be revisited – the algorithm can man-ge the interaction between intensification and diversification.he following basic rules are used:

As soon as a region is visited again, the Tabu tenor is increasedand the algorithm is forced to start some diversification. Thealgorithm forces the search to choose a different neighbor forbeing the next iterate and new regions can be investigated. Inthis way cycling should be avoided.When repetitions are absent for a sufficiently long period,the Tabu tenor is reduced and the diversification will disap-pear. In order to decide how long this period has to be the

algorithm stores the average cycle length, i.e. the number ofiteration it takes to revisit a region. If more iterations havebeen taken than this average value, the Tabu tenor will bereduced.

••

Engineering 32 (2008) 1877–1891

When the described Tabu tenor changes are not sufficient andcycles still occur an escape mechanism is needed. This mech-anism will force the search in new regions in a very drasticway. This is done by choosing randomly one of the generatedneighbors that does not lie in a Tabu region. This is repeatedn times, where n is the number of variables.

Since the number of generated neighbors depends on the sizef the current region, respectively, the depth of the region inhe tree, our prohibition rule has to take this into account. Forhat reason we introduce the Tabu tenor fraction Tf ∈ [0, 1]. Inther words a specified percentage of the neighbored regions isrohibited. In more detail, all neighbored regions that have beenisited in the last:

={

max(1, min(�Tfn�, n − 2)) if D = 1,

max(1, min(�Tf2n�, 2n − 2)) otherwise(8)

terations are Tabu and cannot be visited. If D > 1 up to 3 ×neighbors can be generated, but since sometimes only 2 ×neighbors can be generated – additional regions would lie

utside the box constraints – we use the formula �Tf2n�, where·� denotes the largest integer value lower than Tf2n. It mightappen that this value is too small but this is feasible because thescape mechanism will be activated, avoiding these cycles. Inhe normal situation, the list is proportional to �Tfn� (or �Tf2n�),he max operator ensures that the list is at least one, the min

perator ensures that it is at most n − 2 (or 2n − 2). In thisay, the last visited region is always prohibited, and at least two

egions are available from a given point, so that the chosen moves influenced by the values of the objective function J– for thenconstrained case – in the neighborhood.

.1.4. Aspiration criterionWhile the search is proceeding, several regions of the search

pace are classified as Tabu. In some cases, the best neighborolution may lie in a Tabu area where its objective functionalue is better than the current best value. In this situation theabu property can be invalidated and the point will be chosen.his feature is necessary to enforce faster convergence to a good

ocal minimum but it also might bias the search. If this happenshe escape mechanism will be started after some time and theearch is forced to leave this region and to explore a new one.

.1.5. Stopping conditionsDefining a stopping criterion for GO algorithms is very dif-

cult. If the algorithm stops too early, the global optimum cane missed. Otherwise, if it stops too late, computational effortill be wasted. Our procedure will stop if one of the following

riteria is fulfilled:

A predefined maximum number of iterations is exceeded.A given number of iterations without any further improvement

on the value of the objective function.A predefined maximum time has elapsed.A predefined maximum number of function evaluations hasbeen reached.

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Since we have to handle computationally expensive models,PU time will be the bottleneck. Hence, in most cases we will

top our algorithm using the third criterion above. In our casetudies we deactived the second criterion since we wanted toompare the performance of the different algorithm subject tohe maximum elapsed time respectively the maximum numberf function evaluations. Fig. 2 shows the flow chart of the MITSlgorithm. The integration of MISQP in the framework MITS isbvious in this figure.

.2. The local solver MISQP

The purpose of this section is to give a short overview ofhe mathematical theory of the local solver that is activated by

ITS. The solver called Mixed-Integer Sequential Quadraticrogramming (MISQP) is a SQP Trust-Region method recentlyeveloped by Exler and Schittkowski (2007). We will only sum-arize the basic ideas here, for further information please read

he mentioned paper. We consider the general optimization prob-em to minimize an objective function J subject to nonlinearquality and inequality constraints:

∈Rnc , y ∈Zni :

min J(x, y)

hj(x, y) = 0, j = 1, . . . , me,

gj(x, y) ≥ 0, j = 1, . . . , mi,

xl ≤ x ≤ xu,

yl ≤ y ≤ yu,

(9)

here x denotes the vector of the continuous and y the vector ofhe integer variables, nc is the number of continuous variables,i the number of integer variables, respectively. Once again, xl

nd yl denote the lower bounds, whereas xu and yu stand for

Fig. 2. MITS—flow chart.

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Engineering 32 (2008) 1877–1891 1883

he upper bounds on the variables. The number of equality con-traints h is expressed by me, the one for inequality constraints

by mi. We define n = nc + ni the number of all variablesnd c = (h1, . . . , hme , g1, . . . , gmi )

T the vector of constraints.urthermore, we use the ‘−’ to declare a vector c− with theollowing properties:

j(x, y)− :={

hj(x, y), j = 1, . . . , me,

min(0, gj(x, y)), j = 1, . . . , mi.. (10)

he Lagrangian function plays a key role in nonlinear optimiza-ion and is defined as follows:

(x, y, λ, μ) : = J(x, y) −me∑j=1

λjhj(x, y)

−mi∑

j=1

μjgj(x, y), (11)

here λj , j = 1, . . . , me, are the Lagrangian multipliers for thequality constraints and μj , j = 1, . . . , mi, are the Lagrangianultipliers for the inequality constraints.MISQP is a Sequential Quadratic Programming method and

o enforce convergence a trust region method is used. We proceedrom the continuous trust region method of Yuan (1995) withecond order corrections. For the rest of this section subscriptsdenote the iteration. As a norm we use the ∞-norm.

In each iteration k we have to solve the mixed integeruadratic problem (MIQP):

∈Rnc × Zni :

min φk(d) = 1

2dTHkd + ∇J(xk, yk)Td

+ σk||(∇c(xk, yk)Td + c(xk, yk))−||∞

||d||∞ ≤ �k,

(12)

here Hk denotes the approximation of the Hessian of the Lagra-ian and dk is the solution to that MIQP in the k th iteration. Thetep size is restricted to the trust region radius Δk. The constraintiolation is measured in the last term of φk and is penalized witharameter σk. Here, ∇ denotes the first derivatives. Thus, ∇J

s the gradient of the objective function, respectively, ∇c is theacobian of the constraints. The Hessian of the Lagrangian func-ion is approximated by a quasi-Newton update formula subjecto the continuous and integer variables. In our implementationhe well-known BFGS-Formula is used.

It is not assumed that the MINLP is relaxable, i.e., thatJ(x, y),j(x, y), j = 1,. . ., me and gj(x, y), j = 1, . . . , mi, can be eval-ated at any fractional parts of the integer variables. Thus, therst derivatives at J(x, y) are approximated by the differenceormula:

yJ(x, y) = J(x, y1, . . . , yj + 1, . . . , yni ) − J(x, y1, . . . , yj − 1, . . . , yni )

2(13)

or j = 1, . . . , ni, at neighbored grid points. If either yj + 1 orj − 1 violates a bound, we apply a non-symmetric differenceormula. Similarly, dyhj(x, y) and dygj(x, y) denote a difference

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884 O. Exler et al. / Computers and Che

ormula for first derivatives for hj(x, y) and gj(x, y) computedt neighbored grid points. For the continuous variables the gra-ients are numerically approximated by a forward differenceormula.

After computing a step dk, one has to decide if the step isccepted or not. Otherwise, convergence cannot be guaranteed.e have to measure the improvement compared to a merit func-

ion. In our implementation we use the exact penalty function:

σ(x, y) := J(x, y) + σ||c(x, y)−||∞ (14)

ith a penalty parameter σ > 0.The solution dk of the quadratic subproblem (12) is used to

ompute the next iteration point:

(xk+1, yk+1)

={

(xk, yk) + dk if Pσk((xk, yk) + dk) ≤ Pσk

(xk, yk)

(xk, yk) otherwise.(15)

key role in the trust region algorithm is played by the predictionf a new trust region radius for the next iteration. The trust regionadius is adjusted according to the ratio of the actual and theredicted reduction of the merit function:

k := Pσk(xk, yk) − Pσk

((xk, yk) + dk)

φk(0) − φk(dk). (16)

he trust region radius Δk has to be updated in order to enforceonvergence. If rk is close to one or even greater than one, thenk is enlarged and if rk is very small, Δk is decreased. If rk

emains in the intermediate range, Δk is not changed at all. Moreormally, we use the same constants proposed by Yuan (1995),nd set:

k+1 =

⎧⎪⎨⎪⎩

max[2Δk, 4||dk||∞] if rk > 0.9,

Δk if 0.1 ≤ rk ≤ 0.9,

min[Δk/4, ||dk||∞/2] if 0 < rk < 0.1.

(17)

f, on the other hand, rk < 0, then Δk is decreased and we solveubproblem (12) again.

To increase the convergence rate a second quadratic problems used:

∈Rnc × Zni :min φk(d)

||d + dk||∞ ≤ Δk,(18)

here

¯k(d) : = 1

2(d + dk)THk(d + dk) + ∇J(xk, yk)T(d + dk)

+ σk||(∇c(xk, yk)Td + c((xk, yk) + dk))−||∞. (19)

et the solution be dk. In order to state the algorithm we definehe following two equations:

¯k := rk + φk(0) − φk(dk)

φk(0) − φk(dk), (20)

nd

ˆk := Pk(xk, yk) − Pk((xk, yk) + dk + dk)

φk(0) − φk(dk). (21)

Tpoh

Engineering 32 (2008) 1877–1891

n Algorithm 1 the formulation of the local solver MISQP isresented. As a termination tolerance the parameter ε is used.

lgorithm 1 (MISQP).

: Let (x1, y1) ∈Rnc × Zni , Δ1 > 0, H1 ∈Rn×n positive defi-nite, σ1 > 0, ε > 0, and let k := 1.

: Solve subproblem (12) to get dk and the multiplier λk and μk.If φk(0) − φk(dk) < ε and c(xk, yk)− < ε, then stop. Updatepenalty parameter σk.

: Compute rk by (16). If rk > 0.75, goto Step 5. Solve sub-problem (18) to get dk and the corresponding multiplier λk

and μk, and compute rk by (20). Let λk := λk and μ := μk. Ifrk < 0.25, goto Step 3. If 0.9 < rk < 1.1, let Δk+1 := 2Δk,else Δk+1 := Δk and goto Step 6.

: If rk < 0.75, goto Step 4. Otherwise, compute J((xk, yk) +dk + dk) and c((xk, yk) + dk + dk). If Pk((xk, yk) + dk +dk) ≥ Pk((xk, yk) + dk), goto Step 4. Calculate rk by (21)and let dk := dk + dk, rk := rk. If rk ≥ 0.75, goto Step 5. Ifrk ≥ 0.25, goto Step 6.

: Let Δk+1 := ||dk||∞/2 and goto Step 6.: If ‖dk‖∞ < Δk, then Δk+1 := Δk and goto Step 6. If rk >

0.9, then Δk+1 := 4Δk, else Δk+1 := 2Δk.: If rk > 0, goto Step 7. Otherwise, let (xk+1, yk+1) :=

(xk, yk), Hk+1 := Hk, increment k, and goto Step 1.: Define a new iterate (xk+1, yk+1) := (xk, yk) + dk, com-

pute J(xk+1, yk+1) and c(xk+1, yk+1), update Hk+1 by theBFGS formula applied to dk and ∇L(xk+1, yk+1, λk, μk) −∇L(xk, yk, λk, μk). Increment k and goto Step 1.

. Case studies

Our algorithm, MITS, was implemented in Matlab. An inter-ace (software gateway) was also developed in order to call theocal solver MISQP, which was implemented in FORTRAN (andompiled as a dynamic link library). We have tested MITS on twoell-known and challenging benchmark problems: a wastewater

reatment plant (WWTP), and the Tennessee Eastman processTEP).

In optimal design problems, it is usually useful to performensitivity analysis of the obtained optimal solutions. However,iven the complexity of the models considered here, plus the facthat they have noisy dynamics, standard codes for computingensitivities of dynamic systems (like, e.g. ODESSA) could note applied. Finite differences estimates could be used, althoughor noisy systems their reliability would be questionable unlessome sort of filtering was used, but this was out of the scope ofhe present paper.

herefore, questions regarding alternative formulations for theseroblems (e.g. like considering other dynamic scenarios) wereut of the scope of this paper and have not been investigatedere.

O. Exler et al. / Computers and Chemical Engineering 32 (2008) 1877–1891 1885

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.1. A wastewater treatment plant

The first benchmark problem that we consider is a wastewa-er treatment plant (WWTP) for nitrogen removal. The WWTPas developed by the COST 624 work group (1998–2004)

nd additional information can be found in Copp (2002) andlex et al. (1999). The layout of this benchmark plant – as

hown in Fig. 3– combines nitrification with predenitrifica-ion by a five-compartment reactor with an anoxic zone. Aecondary settler separates the microbial culture from the liq-id being treated. A basic control strategy consisting of 2 PIontrollers is proposed to test the benchmark. Its aim is to con-rol the dissolved oxygen level in the final compartment of theeactor (AS Unit 5) by manipulation of the oxygen transfer,nd to control the nitrate level in the last anoxic compart-ent (AS Unit 2) by manipulating the internal recycle flow

ate.A Simulink implementation of the benchmark model by

eppsson and Pons (2004) was used for the simulations. Eachunction evaluation consists of an initialization period of 100ays to achieve steady state, followed by a period of 14ays of dry weather and a third period of 14 days of rainyeather. Calculations of the controller performance criterion

re based on data from the last seven rain days. Each sim-lation of this benchmark model takes a significant time onstandard PC (about 60 s in a PC with Intel Pentium IV

,2 GHz).The control performance was evaluated using the Inte-

ral Square Error (ISE) as a criterion. Both the nitrate levelnd oxygen level controllers (further referred as N- and O-ontroller, respectively) are optimized with respect to theirontroller parameters, that is, the gain K, integral time con-tant τi and anti-windup time constant τt . Besides the controllererformance, the objective function also considers the follow-ng:

EQ. Effluent quality in kg pollution units/day includes number

of violations and percentage of time in violation. Limits foreach component are available.PE. Average pumping energy, in kWh/dayAE. Average aeration energy, in kWh/day.PSludge. Sludge production, in m3/day.

toptv

treatment plant.

The problem is formulated as follows:

minv

J(v, tf ) = μ1EQ + μ2PE + μ3AE + μ4PSludge

+μ5 · control

s.t. x = f (x, x, p, v) = 0,

x(t0) = x0,

vl ≤ v ≤ vu,

(22)

here f ∈R150 denotes the system dynamics and v ∈R12 × Zs the vector of decision variables. The system dynamics areescribed by algebraic mass balance equations, ordinary differ-ntial equations for the biological processes in the bioreactors asefined by the ASM1-model (see Henze, Grady, Gujer, Marais,

Matsuo, 1986), and the double-exponential settling velocityunction presented in Takacs, Patry, and Nolasco (1991) as a fairresentation of the settling process, with x ∈R13 the state vector,∈Rp the system parameters and d the influent disturbance.In order to compare the controllability of different configu-

ations, we define control as follows:

ontrol = ω1 · ISE(N)(vk) + ω2 · ISE(0)(vk)

ω1 · ISE(N)(v0) + ω2 · ISE(0)(v0), (23)

here

SE(·) =∫ tf

t0

ε(τ)2(.)dτ. (24)

he weights ω1 and ω2 are chosen such that the ISE(O) equalso the ISE(N) part when using the benchmark default set-ings v0 (i.e. the tuned PI-parameters) provided by the COST24 (1998–2004). The values are ω1 = 1/1001 and ω2 =000/1001. For the initial configuration v0 we obtain control =. If the controllability of a configuration is better than the onet v0 the value of control will be smaller than one; otherwise, itill be greater than one.The objective function is based on this formulated by

anrolleghem and Gillot (2002) which takes into account justhe economics of the process. We have added a controllabilityerm to the cost function since previous tests revealed that just

ptimizing the criterion chosen by Vanrolleghem results in aoor controllability and high values for the pollutants concen-ration. The weight for the control term is chosen so that itsalue has the same order of magnitude as the economic term.

1886 O. Exler et al. / Computers and Chemical Engineering 32 (2008) 1877–1891

Table 2Weights in the objective function

Weight μ1 μ2 μ3 μ4 μ5

Value 2 1 1 3 1000

Table 3Decision variables for the COST benchmark

Variable Description [vlvu]

v (1) N-controller – gain [100 1000]v (2) N-controller – integral time constant [0.0007 0.7]v (3) N-controller – anti-windup time constant [0.0001 0.7]v (4) O-controller – gain [100 50,000]v (5) O-controller – integral time constant [0.01 1]v (6) O-controller – anti-windup time constant [0.0001 0.07]v (7) Aeration factor in reactor 1 [0 360] day−1

v (8) Aeration factor in reactor 2 [0 360] day−1

v (9) Aeration factor in reactor 3 [0 360] day−1

v (10) Aeration factor in reactor 4 [0 360] day−1

v (11) Sludge purge [0 1844.6] m3/dayv

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Table 5Changes in the plant performance

Criterion Initial MITS Units

Objective functionvalue (J)

35226 33537

Effluent qualityindex (EQ)

9032 8356 kg poll units/day

Average aerationenergy (AE)

7173 6787.6 kWh/day

Average pumpingenergy (PE)

1919 2045.7 kWh/day

Sludge production(PSludge)

2357 2393.8 m3/day

ISE(O) 1.2232 × 10−5 2.1779 × 10−4 (mg COD/l)2 day2

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(12) Sludge recycle from settler [0 36,892] m3/day(13) Feed layer in settler v(13) ∈Z [1 10]

he exact values are listed in Table 2. It should be highlightedhat the values can be varied in order to change the weighting ofifferent components, e.g., increasing the weight for the pumpnergy the optimization will lead to a smaller value in the pumpnergy.

Table 3 lists all decision variables with a short descriptionnd the upper and lower bounds. Boundaries on the decisionariables (vl and vu) are chosen such that the process dynamicsould not show (exceptional) unstable behavior.As mentioned before, the weights in the objective function

re chosen with respect to the performance obtained with defaultalues for the decision variables (i.e. the tuned PI-parameters)s provided by the COST 624 (1998–2004). The optimizationas started from these default values for the variables. Since the

valuation of the objective function is costly (in the sense of time

onsuming), we stopped MITS after exceeding a CPU time limitf 48 h using a standard PC. Table 4 shows the default valuesinitial point for the optimizer) and the optimal values obtainedy MITS.

able 4esult for the WWTP

ariable Initial point v0 Result MITS vMITS

(1) 500 486.3(2) 0.001 0.000708(3) 0.0002 0.0001(4) 15000 16369.3(5) 0.05 0.0221(6) 0.03 0.0275(7) 0 0(8) 0 0(9) 10 h−1 (240 day−1) 224.8 day−1

(10) 10 h−1 (240 day−1) 224.8 day−1

(11) 385 m3/day 333.1 m3/day(12) 18446 m3/day 16921 m3/day(13) 5 7

Mnvo

ISE(N) 0.83345 0.46733 (mg N/l) dayControl 1 0.81

The new configuration obtained by our solver leads to sev-ral changes, which are shown in Table 5. The performance hasmproved for all the criteria except the average pumping energynd the sludge production. This behavior can be explained takingnto account the multi-objective nature of the objective functionnd the typical conflicting nature of the different criteria. Aslready stated before, this behavior can be influenced by settingifferent weights. It is worth mentioning the change in the con-rollability: a significant improvement with respect to the defaultalues was obtained, as shown in Table 5.

The performance of MITS was compared with the per-ormance of two other competitive modern solvers, namelyQNLP and MINLPbb. OQNLP, based on the Scatter Searchetaheuristic, has been recently reported as one of the best

lobal optimization solvers for black box problems (Neumaier,hcherbina, Huyer, & Vinko, 2005). The second solver,INLPbb, has been developed by Leyffer (2001) and uses a

ranch and bound strategy to solve the MINLP. Fig. 4 showshe convergence curves of the three different solvers. MITS andQNLP were stopped after the time limit of two days exceeded.INLPbb is a deterministic approach and stopped after the inter-

al criterion was satisfied. In both categories – objective functionalue and convergence speed – MITS clearly outperformed thether solvers.

Fig. 4. WWTP—convergence curves.

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.2. The tennessee eastman process

Since the publication of the Tennessee Eastman processTEP) example by Downs and Vogel (1993), it has been widelysed in the literature as a benchmark due to its challengingroperties from a control engineering point of view: it is highlyonlinear, open-loop unstable and it presents a large number ofeasured and manipulated variables which offer a wide set of

andidates for possible control strategies. The flowsheet for theEP is depicted in Fig. 5. Two products (G and H) are produced

rom four reactants (A, C, D and E). A further inert trace com-onent (B) and one byproduct (F) are present. The process unitsonsist of a continuous stirred tank reactor, a condenser, a flashrum and a stripper. The gaseous reactants are fed to the reactorhere they are transformed into liquid products. The following

eactions take place in gas phase:

A(g) + C(g) + D(g) → G(l),

A(g) + C(g) + E(g) → H(l),

A(g) + E(g) → H(l),

3D(g) → 2F(l).

(25)

hese reactions are irreversible and exothermic with rates thatepend on temperature through Arrhenius expressions and onhe reactor gas phase concentration of the reactants. The reac-ion heat is removed from the reactor by a cooling bundle. Theroducts and the unreacted feeds pass through a cooler and,nce condensed, they enter a vapour-liquid separator. The non-ondensed components recycle back to the reactor feed and theondensed ones go to a product stripper in order to remove theemaining reactants by stripping with feed stream. Products Gnd H are obtained in bottoms. The inert (B) and the byproduct

F) are mainly purged from the system as a vapour from theapour-liquid separator.

Recently, Antelo, Otero-Muras, Banga, and Alonso (2007)pplied their systematic approach to a plant-wide control design

Ti

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Fig. 5. The tennessee eastm

Engineering 32 (2008) 1877–1891 1887

eveloped in a previous work (Antelo, Otero-Muras, Banga, &lonso, 2005) to derive robust decentralized controllers for theennessee Eastman Process. In this framework, the TEP is repre-ented as a process network. Then, conceptual mass and energynventory control loops for each node are designed first to guar-ntee that the states of the plant will remain on a convex invariantegion, where the system will be passive and therefore input-utput stability can be stated (Antelo et al., 2005). The next steps to realize the proposed conceptual inventory control loopssing the physical inputs-outputs of the process. Some extra con-rol loops are needed to achieve the convergence of the intensiveariables since the inventory control by itself does not ensurehe convergence of these variables to a desired operation point.n some cases, the available degrees of freedom are not enougho implement the complete control structure that ensures bothxtensive and intensive variables convergence to the referencealues. As a consequence, the setpoints of the inventory con-rollers can be used as new manipulated variables to completehe decentralized control design.

We explain the alternatives we introduced to extend the origi-al hierarchical control design proposed by Antelo et al. (2007).oncerning the reactor level control loop in the original design,

ts set point modifies the reference for the flow controller actingver E feed. As an alternative for closing this loop, D feed isroposed as manipulated variable.

For the reactor pressure case, the original proposal by Antelot al. (2007) considers the condenser cooling water flow as theanipulated variable. By using this, the reactor pressure can be

aried since the separator pressure and, as a consequence, theecycle rate can be modified. It is at this point where the controlver the vapor mass inventory in the separator by using the purgeate is established in order to ensure that all inventories in the

EP will remain bounded and, therefore, input–output stability

s guaranteed (Antelo et al., 2007).As an alternative, we are considering here a manipulated vari-

ble widely used in the literature for the reactor pressure control

an process flowsheet.

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888 O. Exler et al. / Computers and Che

oop: the purge flow. By modifying this, it is possible to regulatehe separator pressure as well as the recycle flow, and there-ore the reactor pressure. When this alternative is considered, anxtra loop controlling the separator temperature (energy inven-ory) by acting over the condenser coolant flow is defined. Notehat we are not considering other alternatives to control the reac-or pressure as the A feed, that is a disturbance in the model, or

feed.In order to determine the best control alternative among

he proposed ones, a new binary vector b is added to ourystem dynamics. These 0–1 variables express which of theour control strategies is being used, and they are defined asollows:

b1 ∈ {0, 1} (E feed),

b2 ∈ {0, 1} (D feed),

b3 ∈ {0, 1} (condenser coolant),

b4 ∈ {0, 1} (purge flow).

(26)

herefore, the original control design proposal by Antelo et al.2007) will be characterized by the vector b = (1, 0, 1, 0)T sincet uses E Feed to control the reactor level and the condenseroolant flow to control the reactor pressure.

From all the exposed, the optimization problem consists nown solving the following MINLP of the form:

minv,b

J(x, v, b),

s.t.

f (x, x, p, v, b, t) = 0,

h(x, p, v, b) = 0,

g(x, p, v, b) ≥ 0,

b1 + b2 = 1,

b3 + b4 − 1 ≥ 0,

vl ≤ v ≤ vu,

bl ≤ b ≤ bu,

(27)

here b ∈ {0, 1}4 is the vector of binary variables (0–1 vari-bles) and v ∈R36 are the continuous variables (the controllerarameters). The lower and upper bounds for the binary vari-bles will be of the form bl = (0, . . . , 0)T and bu = (1, . . . , 1)T.t must be pointed out that we are considering that only onef the two alternatives for each loop can be active at one time,eing necessary to introduce the additional linear constraints1 + b2 = 1. The linear constraint b3 + b4 − 1 ≥ 0 ensure thatt least one of the alternatives b3 or b4 is active. The rest ofhe decision variables are connected to the tuning of the PIontrollers.

Note that the MINLP is also subject to the dynamics (DAEs)

f the system which are expressed by f in Eq. (27). The TEP has71 DAEs (30 ODEs and 141 algebraic equations). The MINLPs also made up of the following constraints which are relatedith the reactor pressure, temperature and volume, and with the

ov

tv

Engineering 32 (2008) 1877–1891

eparator and the stripper volumes:

Preactor ≤ 3000 kPa,

2 m3 ≤ Vreactor ≤ 24 m3,

Treactor ≤ 175 ◦C,

1 m3 ≤ Vseparator ≤ 12 m3,

1 m3 ≤ Vstripper ≤ 6 m3.

(28)

he objective function proposed by Downs and Vogel (1993) inhe TEP definitions is based on the operating costs and can beefined as follows:

TC = PC · PR + PrC · PrR + CC · CW + SC · SR,

TC = 7.5973 $/kmol · PC + 0.1434 $/kmol · PrR

+ 0.0536 $/kWh · CW + 0.0318 $/kg · SC,

(29)

here TC are the total operating costs at the base case, PC andR are the purge costs and purge flowrate, respectively. Analo-ously, PrC, CC and SC are the costs associated to the producttream, compressor and steam, and PrR, CW and SR are theroduct rate, the compressor work and the steam rate, respec-ively. Operating costs for this process are primarily determinedy the loss of raw materials (in the purge, in the product streamnd by means of the two side reactions). Economic costs for therocess are determined by summing the costs of the raw materi-ls and the products leaving in the purge stream and the producttream, and using an assigned cost to the amount of F formed.he costs concerning the compressor work and the steam to thetripper are also included. Note that the objective function usedn the MINLP formulation will be the mean of these operatingosts along the whole simulation time horizon. For this work,his simulation time horizon was set to t = 10 h. This is enoughime for stabilization of the TEP.

After all these considerations concerning the objective func-ion, the problem can be represented as an MINLP of the form27):

v ∈R36, b ∈ {0, 1}4 :

min J(x, v, b) = total operating costs at base case

v0 − 0.5v0 ≤ v ≤ v0 + 0.5v0.(30)

he lower and upper bound for the decision variables have beenet to be the ±50% of the initial value for the decision vector.he reason for this selection is to avoid as much as possible theaturation problems that can be exhibited by the valves. Theseituations have been detected in preliminary dynamic simula-ions when considering a value of ±100% of v0 as bounds forhe decision vector.

Note that changes in the decision variables (v) will be trans-ated into variations in the states x that can even drive the systemo shutdown due to the fact that one or more of the constraintsefined in (28) have been violated. More precisely, the reac-or volume, pressure and temperature are related with the pairs

f decisions variables (gain-time constant) v(11)– v(30), v(12)–(31) and v(16)– v(35), respectively. Finally, the separator andhe stripper volumes are linked with the pairs v(9)– v(28) and(10)– v(29), respectively.

O. Exler et al. / Computers and Chemical Engineering 32 (2008) 1877–1891 1889

Table 6Result for the TEP

Decision variable Control loop Default values Result MITS

v(1) A Feed flow 0.001 1.4029e −3v(2) D Feed flow 0.003 0.004043v(3) E Feed flow 1.8e −10 9.6882e −11v(4) C Feed flow 20 30.0v(5) Condenser coolant 7e −7 9.8858e −7v(6) Separator flow 4e −4 4.5727e −4v(7) Stripper flow 0.004 0.004242v(8) Production rate 3.2 4.8v(9) Stripper level −0.02 −0.021864v(10) Separator level −0.05 −0.048376v(11) Reactor level 10 8.5024v(12) Reactor pressure −0.0001 −5e −5v(13) %G in product −0.032 −0.02811v(14) %A in purge 0.0009 5.664e −4v(15) Recycle rate 0.00125 0.001271v(16) Reactor temperature −8 −8.1636v(17) Separator temperature 100 90.219v(18) G /H Product ratio 32 21.762v(19) G /H Product ratio 46 30.891v(20) A Feed flow 1.6667e −5 1.584e −5v(21) D Feed flow 1.6667e −5 1.4535e −5v(22) E Feed flow 4.1667 3.1266v(23) C Feed flow 0.1667 0.83334v(24) Condenser coolant 4.1667 2.0834v(25) Separator flow 1.6667e −5 1.42116e −5v(26) Stripper flow 1.6667e −5 1.302e −5v(27) Production rate 2 1.2995v(28) Stripper level 0.3333 0.19213v(29) Separator level 3.3333 1.6704v(30) Reactor level 0.01667 0.011874v(31) Reactor pressure 0.3333 0.18142v(32) %G in product 1.6667 1.0927v(33) %A in purge 9.3667 10.284v(34) Recycle rate 25 30.338v(35) Reactor temperature 0.125 0.06475v(36) Separator temperature 8.3333 6.4701b(1) Reactor level (E feed) 1 0b(2) Reactor level (D feed) 0 1b nt) 1 0b 0 1C 156.843 84.289

stmapeao

cPTtwoap

(3) Reactor pressure (condenser coola(4) Reactor pressure (purge rate)ost value ($/h)

In order to solve the MINLP (27) problem we used theolvers MITS and OQNLP. The dynamic model for the proposedhermodynamic-based control design has been also imple-

ented as a SIMULINK code by the Process Engineering Groupt IIM-CSIC. Both solvers were started from the same initialoint. As stopping criterion, we set the maximum of functionvaluations equal to 10000. Table 6 lists the default values (useds initial point) and the best point located by MITS. The pairsf

Note that the decision vector pairs v (1)– v (20), v (2)– v (21),orresponds to the gain and time constants, respectively, of theI controllers employed to close the different loops presented inable 5. In other words, variables v (1)– v (19) are the gains of

he PI controllers used to close the corresponding control loop,

hile variables v (20)– v (16) are the related time constantsf these controllers. Note that for the G/H Product Ratio therere no time constants, since the associated controllers are onlyroportional. Fig. 6. TEP—convergence curves.

1 mical

mpbrtFcf

5

thfeSsmocfprt

mstppvt

A

pnPfStSt

R

A

A

A

A

A

A

A

B

B

B

B

B

B

B

B

C

C

D

D

E

E

F

GH

J

890 O. Exler et al. / Computers and Che

Note that for the case of the binary variables in Table 6, theanipulated variable that has been chosen to close the loop is

resented in brackets. The solution obtained by MITS for theinary vector is b = (0, 1, 0, 1)T. This vector defines the newealization of the control loops for the pressure and the level inhe reactor by acting over the purge and the D feed, respectively.ig. 6 shows the convergence curves for both solvers. MITSlearly outperformed OQNLP, both regarding the final objectiveunction value and the convergence speed.

. Conclusions

We have developed a hybrid algorithm for the optimization ofhe integrated process and control system design problem. Weave considered a mixed integer nonlinear program (MINLP)ormulation for this class of problems. Our novel hybrid strat-gy, MITS, uses a combinatorial component, based on Tabuearch, to guide the search into promising areas, and a localolver, MISQP, which is activated to precisely approximate localinima. A Matlab implementation of this technique was devel-

ped, and its performance and robustness was tested on twohallenging benchmark problems: a wastewater treatment plantor nitrogen removal and the Tennessee Eastman Process. MITSresented very good performance, and it was able to obtainemarkable results on both case studies, clearly outperformingwo selected modern MINLP solvers.

The purpose of this paper was not to illustrate the perfor-ance of MITS for large-scale MINLP, but to show how it can

olve complex integrated design problems where other state ofhe art solvers failed. The application of MITS to larger scaleroblems is currently under research in our group. Initial testserformed with problems from standard collections have beenery promising so far, and they will be reported in future con-ributions.

cknowledgements

We acknowledge the support of the EU Marie-Curie Actionsrogram. In particular, we thank the EU PRISM project – projectumber MRTN-CT-2004-512233 – “Towards Knowledge-basedrocessing Systems” for the support. Author Jose A. Egea grate-ully acknowledges financial support (FPU fellowship) from thepanish Ministry of Education and Science. The authors wish

o thank Dr. Ulf Jeppsson (IEA, Lund University of Technology,weden) for providing his MATLAB/SIMULINK implementa-

ion of the COST Benchmark.

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