Proceedings of the44th IEEE Conference on Decision and Control, andthe European Control Conference 2005Seville, Spain, December 12-15, 2005

Global Optimization for Integrated Design and Control ofComputationally Expensive Process Models

Jose A. Egea, Dirk Vries, Antonio A. Alonso and Julio R. Banga.

Abstract- The problem of integrated design and controloptimization of process plants is discussed in this paper. Weconsider it as a nonlinear programming problem subject todifferential-algebraic constraints. This class of problems isfrequently (i) non-convex and (ii) "costly" (i.e.computationally expensive to evaluate). Thus, on the onehand, local optimization techniques usually fail to locate theglobal solution and, on the second hand, most globaloptimization methods require many simulations of themodel, resulting in unaffordable computation times. As analternative, one may consider global optimization methodswhich employ surrogate-based approaches to reducecomputation times, and which require no knowledge of theproblem structure. A challenging Wastewater TreatmentPlant benchmark model (see [1] and references therein) isused to evaluate the performance of these techniques.Numerical experiments with different optimization solversindicate that the proposed benchmark optimization problemis indeed non-convex, and that we can achieve animprovement of the controller performance compared to thebest tuned controller settings available in the literature.Moreover, these results show that surrogate-based methodsmay indeed reduce computation times while, at the sametime ensuring convergence to the best known solutions.

I. INTRODUCTION

Many optimization problems which arise during thedesign and/or operation of chemical and biochemicalprocesses are frequently non-convex (also referred asmultimodal). Thus, it is not surprising that globaloptimization (GO) has received much attention during thelast decade from many researches in computer aidedprocess engineering. In fact, many advances have beenmade regarding both deterministic and stochastic methodsduring recent years [2]. However, the current state of theart is far from satisfactory, especially when we considerthe global optimization of complex process models. Thesemodels are typically complex due to their dynamic (non-linear) behaviour and high number of states. Thereforethey are typically expensive to evaluate (i.e. each

J. A. Egea, A. A. Alonso and J. R. Banga are with the ProcessEngineering Group, Instituto de Investigaciones Marinas (CSIC)C/Eduardo Cabello 6, 36208 Vigo, Spain (email:j)

D. Vries is with the Systems and Control Group, WageningenUniversity, P.O. Box 17, 6700 AA Wageningen, The Netherlands.

simulation can take minutes, or even hours, in CPUcomputation time of an ordinary PC) and, for a numberof reasons, can only be treated as black-box models.Thus, several authors [3] have recently proposed the useof so-called surrogate models which are(computationally) cheaper to evaluate.In the area of process systems engineering, Moles et al.[4] showed how stochastic global optimization methodscan be applied for simultaneous design and control ofprocess plants of low complexity. In this contribution, ouraim is to apply a similar approach to more complex (andcostly) models. In order to keep the computational burdenacceptable, the capabilities of recent global surrogatemodel based optimization methods have been evaluated.Such evaluation is performed considering a challengingbenchmark case study: the integrated design and controlof a wastewater treatment plant (WWTP) for nitrogenremoval, as developed by the COST 624 work group [5].This paper is structured as follows: in the next section, thegeneral statement of the integrated design problem ispresented. Next, we briefly review the state of the artregarding global optimization for such problems, and wepresent the methods selected for our research. In thefollowing section, the WWTP case is outlined. Finally,sections with results and discussion and conclusions areprovided.

II. INTEGRATED DESIGN: PROBLEM STATEMENT

The general statement of the simultaneous (integrated)design and control problem takes into account the processand control superstructures indicating the different designalternatives [6]. This general approach results in mixedinteger optimal control problems.In this work, we consider a simpler, yet non-trivial,subproblem, where it is assumed that the flowsheet (i.e.plant configuration) is given. It should be noted that thissubproblem is challenging enough to serve as a case studyfor the comparison of surrogate-based global optimizationmethods, which is the main objective of this work.The aim is to simultaneously find the static variables ofthe process design, the operating conditions and thecontrollers' parameters which optimize a combinedmeasure of the plant economics and its controllability,subject to a set of constraints which ensure appropriate

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dynamic behaviour and process specifications. We stateour problem as follows:

Find v to minimize:

C=I TV Xi

subject tof(x, x, p, v) = 0

x(to ) = Xo

h(x, p, v) = 0

g(x, p, v) < 0

V <V <V

where v is the vector of decision variables (e.g. designvariables, operating conditions, parameters of controllers,setpoints, etc.); C is the cost (objective function) tominimize (normally a weighted combination of capital,operation and controllability costs, qi); f is the set ofdifferential and algebraic equality constraints describingthe system dynamics (i.e. the nonlinear process model), xis the vector of the states, to the initial time for theintegration of the ODE's (and, consequently, x0 is thevector of the states at that initial time); p is the vector ofparameters of the dynamic model; h and g are possibleequality and inequality path and/or point constraintswhich express additional requirements for the processperformance; finally, vL and vU are the upper and lowerbounds for the decision variables.The formulation above is that of a non-linearprogramming problem (NLP) with differential-algebraic(DAEs) constraints. Due to the nonlinear and constrainednature of the system dynamics, these problems are veryoften multimodal. Further, it is known that using standardcontrollability measures such as the Integral Square Error(ISE) in the objective function often causes non-convexity [6]. Therefore, if this NLP-DAEs is solved viastandard NLP methods, such as Sequential QuadraticProgramming (SQP), it is very likely that the solutionfound will be of local nature. To avoid this, globaloptimization (GO) should be used.

III. OPTIMIZATION METHODS

A. IntroductionIn principle, model based optimization can besuccessfully used to improve the design and/or operationof single units or full process plants. Typically, most ofthe problems in process engineering applications arehighly constrained and exhibit nonlinear dynamics. Theseproperties often result in non-convexity. Furthermore, inmany complex process models some kind of noise and/or

discontinuities (either due to numerical methods, or tointrinsic properties of the model) is often present.Therefore, there is great need of robust globaloptimization solvers which can (i) locate the vicinity ofthe global solution in a reasonable number of iterations,(ii) handle noise and/or discontinuities, and (iii) use somesort of approximation or reduction of the original processmodel, in order to keep the computational effortacceptable.In general, (iterative) gradient-based local methods forconstrained NLP problems are very efficient, but they canonly handle smooth nonlinear functions subject to smoothconstraints based on a set of continuous variables.Additionally, only convergence to local solutions isguaranteed. Therefore, one must use the so-called globaloptimization (GO) methods.GO methods can be roughly classified as beingdeterministic [7] and stochastic [8]. Deterministic GOmethods assure convergence to the global optimum forcertain problems, although no algorithm can solve generalGO problems with certainty in a finite time [8]. For thesemethods, the computational effort usually increasesexponentially with the problem size. Further, they haverequirements (e.g. smoothness, differentiability) whichare rarely met in realistic applications.Stochastic GO methods can find solutions in the vicinityof the global solution in relatively short computationtimes compared to deterministic GO. Note that with GO,the convergence to global optimality (in finite time) is notguaranteed either. Another advantage of these methods isthat they are easy to implement, and they can treat theobjective function as a black box (i.e. a simple connectionbetween inputs and outputs, with no derivativeinformation needed). This feature is specially appealing inthe case of complex dynamic systems where the objectivefunction is the result of e.g. a simulation provided by athird party software with restricted access for the user.

B. Surrogate-based Global OptimizationIn general, all these GO approaches require a significantnumber of evaluations of the objective function and theconstraints. In case of realistic problems, these models arecostly to evaluate, posing a major challenge to theapplication of GO methods. In recent years, a number ofapproaches have been proposed to obtain surrogatemodels which are cheaper to evaluate than the originalones and which imitate the original model based on areduced number of sampled points (simulations). Hence,these so called surrogate model based solvers try toapproximate the original model over a region by a modelthat is cheaper to evaluate. Provided the surrogate modelis accurate enough, the computation times can still bedramatically reduced. In addition, surrogate models go

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beyond than simply reducing computation time (e.g.Kriging methods provide statistical information aboutdecision variables).In [3], surrogate based methods are classified into twogroups: not interpolating (e.g. quadratic polynomials andother regression models) and interpolating methods (e.g.basis functions and Kriging). At the same time, bothmethods can be one-stage or two stage methods. Two-stage methods fit first a response surface using samplepoints from the real model and then optimize an auxiliaryfunction based on the fitted surface. A potentialdisadvantage of these methods is that the initial surfacemay not accurately fit the real model which can provokethe optimization to stop prematurely or search too locally.On the other hand, one-stage methods evaluatehypotheses about the location of the optimum. This isdone by examining the best-fitting response surfacepassing through the observed data and another point inwhich the optimum is presumed to be located. Thecredibility of each hypothesis is evaluated and the surfaceis being constructed by evaluating the new points wherethis credibility is maximum.The taxonomy of these methods by Jones [3] presents anoverview of the different approaches. Currently, the mostpromising techniques seem to be Radial Basis Functionsand Kriging. In this work, these two strategies are testedand evaluated.

C. Selected Optimization MethodsRegarding surrogate-based GO, we have considered tworecent solvers which are MatlabR implementations of thetwo types of strategies mentioned above:

rbfSolve: this solver, included in the TomlabR toolbox[9], solves costly global optimization problems using aRadial Basis Function (RBF) interpolation algorithm. Itfits a response surface (based on splines) to data collectedby evaluating the objective function at some points andthen applies an optimization algorithm over that surrogatemodel. The initial points to create the response surfacemay be given by the user or selected by the algorithmbased on different strategies.ego: this solver, also included in the TomlabR toolbox [9],solves costly global optimization problems using theEfficient Global Optimization (EGO) algorithm. The ideaof the EGO algorithm is to first fit a response surface todata collected by evaluating the objective function at afew points. Then, EGO balances between finding theminimum of the surface and improving the approximationby sampling where the prediction error may be high.

In order to critically evaluate the performance of thesetwo surrogate-based strategies, we have also considered

selected local and global solvers which only rely onevaluations of the original (costly) model:

fminsearch: a local method implemented in the MatlaboOptimization Toolbox [10] that uses the simplex methodinstead of using gradient information. Although generallyless efficient than gradient based methods, the simplexmethod may be more robust if the problem is highlydiscontinuous or presents noise.fmincon: also part of the MatlabR Optimization Toolbox[10], this solver finds a local minimum of a constrainedmultivariable function by means of a SQP (SequentialQuadratic Programming) algorithm. The method usesnumerical or, if available, analytical gradients.NOMADm: Nonlinear Optimization for Mixed variablesAnd Derivatives-Matlab, abbreviated as NOMADm, is aMatlabo code that runs various Generalized PatternSearch (GPS) algorithms to solve nonlinear and mixedvariable optimization problems [11].Differential Evolution (DE): This method, presented in[12], is a heuristic population based stochastic approachto global optimization for minimizing possibly nonlinearand non-differentiable continuous space functions.

IV. CASE STUDY: OPTIMIZATION OF CONTROLLERSSETTINGS IN AWWT PLANT

A number of control strategies have been proposed tomeet the strict standards that Wastewater TreatmentPlants (WWTP) must comply with, while also trying toreduce costs [13]. Relevant examples from the recentliterature of attempts to optimize the controllers of theseplants are:

1. ad hoc extensive simulation studies [14],(strictly speaking these may not be calledoptimizations, because there is no evidence thata locally or globally best solution is found).

2. dynamic optimizations of control or designstrategies using local gradient-basedoptimization methods [15], often based onsimplified or linearized models.

3. global optimization methods for simultaneouslyoptimizing operation and design [4].

4. an integrated approach for the optimization ofcontrol strategies, where a small selection ofglobal and local optimization methods was used[16].

Evaluation of these and similar strategies, either inpractice or by simulation, is a real problem due to the lackof a standard with respect to evaluation criteria, processcomplexity and large variations in plant configuration. Inorder to enhance the development and acceptance of newcontrol strategies, the International Water Association

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(IWA) Task Group on Respirometry, together with theEuropean COST work group, proposed a standardsimulation benchmarking methodology for evaluating theperformance of activated sludge plants. The COST 624work group defines the benchmark as "a protocol toobtain a measure ofperformance of control strategies foractivated sludge plants based on numerical, realisticsimulations ofthe controlledplant" [5]. According to thisdefinition, the benchmark consists of a description of theplant layout, a simulation model and definitions of(controller) performance criteria.The layout of this benchmark plant combines nitrificationwith predenitrification by a five compartment reactor withan anoxic zone (see Figure 1). A secondary settlerseparates the microbial culture from the liquid beingtreated. A basic control strategy consisting of 2 PIcontrollers is proposed to test the benchmark. Its aim is tocontrol the dissolved oxygen level in the finalcompartment of the reactor (AS Unit 5) by manipulationof the oxygen transfer, and to control the nitrate level inthe last anoxic compartment (AS Unit 2) by manipulatingthe internal recycle flow rate. A detailed description of thebenchmark can be found in [5]. In this work, a SimulinkRimplementation of the benchmark model by Jeppsson wasused for the simulations [17]. Each function evaluationconsists of an initialization period of 100 days to achievesteady state, followed by a period of 14 days of dryweather and a third period of 14 days of rainy weather.Calculations of the controller performance criterion arebased on data from the last 7 rain days.

respectively) are optimized with respect to their controllerparameters, that is, the gain K, integral time constant zaand anti-windup time constant za. The problem isformulated as follows:

min J(v,tf)=W*ISEst.

x= f(x,v,p,d)

x(to) = Xo

h(x, v, p) =0g(x, v, p) < 0

V < V <V

(7)

(8)

(9)(10)

(1 1)(12)

where W E Jjtlx2 contains the weighting coefficients; ISEE 9jj2xl contains the integral squared errors of the two PIcontrollers;f E 91i5Oxl denotes the system dynamics; h andg are possible equality and inequality trajectory and/orendpoint constraints which express additionalrequirements for the process performance. The weightingvector W, the integral square error ISE and the decisionparameter vector are as follows:

(13)FIS 10 ]=w=[Iw2]=F 1 l°

1 1001 -1001]ISE = ISEO

LISEN J

ISE() = e(r)2(.,dr

F K(o) 1

V[vo]

ri(O)Tt(O)K(N)Ti(N)

Tt(N)

(14)

(15)

(16)

Figure 1. Benchmark Layout

As far as we know, global optimization methods have notbeen used for the integrated design and control ofWWTPs models of this complexity. Further, since eachsimulation of this benchmark model takes a significanttime on a standard PC (about 3 minutes in a PC-PIll 800MHz), it is an illustrating example to evaluate thesurrogate-based strategies mentioned previously.The control performance is tested by using the ISE(Integral Square Error) as a controller performancecriterion. Both the nitrate level and oxygen levelcontrollers (further referred as N- and 0-controller

The weighting vector is chosen such that the ISE(0) equalsto the ISE%N) part when using the benchmark defaultsettings (i.e. the tuned PI-parameters see(13)) provided bythe COST project [5].The system dynamics are described by algebraic massbalance equations, ordinary differential equations for thebiological processes in the bioreactors as defined by theASMI-model [18], and the double-exponential settlingvelocity function presented in [1] as a fair presentation ofthe settling process, with x E 9inxl the state vectors, p E9Vpxl the system parameters and d the influent disturbance.Due to the complexity of the system dynamics, theproblem cannot be solved analytically, and the optimalvalues for the decision variables v E 9j6X1 (i.e. the PIcontroller parameters) have to be retrieved by

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optimization techniques or open loop controller tuning.However, the problem remains that most tuningtechniques are intended for linear systems so that modelapproximation by linearization around different operatingpoints would be inevitable.In the particular case considered here, no further dynamicrestrictions are laid on the problem, i.e. h and g belong toempty sets. Boundaries on the decision variables arechosen such that the process behavior would not show(exceptional) unstable behavior:

vL = [100 7.010 1.010 100 1.0.10-2 1.010 ] (17)

vu= [1000 7.010-1 7.010-1 50000 1.0 7.010 2] (18)

The objective function values are normalized with respectto the performance obtained with the tuned controllersettings provided by the COST project [5], which are:

v' = [500 1.0 10 2.0 10 15000 5.0.10 23.0.10 2] (19)

surrogate models based on RBFs perform better thanKriging.

11

10 _

9_

8_

7_

6_

LL 5

4_

3_

2_

01 1.5 22.Objective Function value[-

Figure 2. Histogram of solutions obtained by a local method(fmincon) using 14 different initial decision vectors

1 03

V. RESULTS AND DISCUSSION

All runs were carried out on a PC Pentium-Ill 800 MHzcomputer. To illustrate the non-convexity of the problem,we investigated the performance and robustness of thelocal solvers fminsearch and fmincon by applying amultistart procedure. Hence, different initial values for thedecision variables are randomly selected within theirrange. Both solvers converged to local solutions in all theruns, confirming the need of using global optimizationmethods. The histogram of the solutions obtained byfmincon in 14 runs is presented in Figure 2. It isinteresting to note that, despite the large computationaleffort (about 16 h of CPU time per run), none of the runswere able to improve the controllers default settings(which we have normalized to J= 1. 0).As expected, the other methods performed much betterdue to their capabilities to escape from local solutions.Typical convergence curves (i.e. objective function valueversus CPU time) from the same initial point arepresented in Figure 3 (note the log scale for J). The bestfinal solutions were consistently found by rbJSolve, whichalso showed the best convergence rate. Both DE andNOMADm finally reached good values of theperformance index, although their convergence rate wasclearly worse than that of rbJSolve. It is somewhatsurprisingly to the authors why the other surrogate-basedmethod (ego) showed a rather good convergence rate butstopped prematurely in all runs. We are currentlyinvestigating if this behaviour can be avoided. In themeantime, as a preliminary conclusion, it seems that

c,10'

0 1 2 3 4 5 6 7CPU Time [s]

Figure 3. Convergence curves for the different optimizationalgorithm.

8 9x 104

The best solutions found by each solver are presented inTable 1, where it can be seen that several solvers wereable to improve the controllers default settings.Simulations using the best solution obtained by rbJSolveand denoted by v, were performed to compare othercriteria like effluent quality and pollutant violations withthose using v'ef Results are presented in Table 2,indicating that v* is also better than vref for those criteria(except in the case of pumping energy). Hence, althoughwe only minimized an ISE-based objective function inthis work, the optimized plant also performed better inalmost all relevant issues.

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Table 1. Best optimization results obtained by the different optimization algorithmsSolver PI Controller Parameter J

K(O) Ti(O) Td(O) K(N) ti(N) Td(N)fmincon 661.39 3.98 10-2 6.59 10-1 28478 3.74 10-1 5.85 10-2 8.6229

fminsearch 819.08 4.87 10-1 451 10-1 26508 9.47 10-2 2.55 10-2 30.166NOMADm 750.62 6.91 10-3 6.30.10-1 21175 3.07 10-2 2.9710-3 0.8170

DE 357.51 1.73.10-3 1.00.10-4 17876 2.00u1-2 1.00.10-4 0.6434rbfSolve 375.31 7.00 10-4 1.00.10-4 21018 2.71 10-2 1.46 10-2 0.5333

ego 444.59 7.00 10-4 1.00.10-4 6129 1.00.10-2 7.00 10-2 1.4722

Table 2. Values ofWWTP performance criteria for the referenceand the optimized decision vectorCriterion vref v* UnitsEQ Index 9031.7 8980.4 kg poll units/dayViol. N-level 11.16 9.97 % of timeViol. NH4+-level 25.92 25.15 % of timeAeration Energy 7172.6 7164.4 kWh/day

- Pumping Energy 1919.3 1984.6 kWh/day

VI. CONCLUSIONS

Optimization problems which arise during an integrateddesign and control approach are frequently non-convex.Thus, it is not surprising to that global optimization (GO)has received much attention during the last decade frommany researchers in computer aided process engineering.However, the current state of the art is far fromsatisfactory, especially when we consider the globaloptimization of complex process models which aretypically expensive to evaluate. In this contribution, wehave considered a number of recent global optimizationmethods, including several surrogate-based approaches,and we evaluate their performance based on their resultsfor a benchmark problem in the management ofwastewater treatment systems. Numerical experimentswith the different GO solvers indicate that the proposedoptimization problem is indeed non-convex and that, asexpected, standard (local) solvers converge to localoptima. In contrast, the results obtained indicate thatsurrogate-based methods, and in particular the ones basedon RBFs, can indeed reduce computation timessignificantly while ensuring convergence to the bestknown solutions.

REFERENCES

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[16] M. Schuitze, D. Butler, and M.B. Beck. "Optimisation of ControlStrategies for the Urban Wastewater System - an IntegratedApproach". Water Science and Technology, vol. 39(9), pp. 209-216, 1999.

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