Multi-objective optimization of p-xylene oxidation process using an improved self-adaptive differential evolution algorithm☆

Lili Tao *,Bin Xu ,Zhihua Hu Weimin Zhong *

1 College of Engineering,Shanghai Polytechnic University,Shanghai 201209,China

2 School of Mechanical Engineering,Shanghai University of Engineering Science,Shanghai 201620,China

3 Key Laboratory of Advanced Control and Optimization for Chemical Processes,Ministry of Education,East China University of Science and Technology,Shanghai 200237,China

1.Introduction

Terephthalic acid(TPA)can be produced to a wide variety of products for various applications,such as polyester fiber and polyester film[1].In recent years,the wide application of the polyester materials contributes to the great demand of the TPA.The production for the liquidphase catalytic oxidation ofp-xylene(PX)to TPA plays an important role in chemical industry.Obviously,how to improve both the product quality and yield is of great interest to the current market.

As a typical free radical chain reaction,the PX oxidation reaction generates several intermediates,e.g.4-methylbenzyl alcohol(TALC),ptolualdehyde (TALD),p-toluic acid (p-TA),and 4-carboxybenzaldehyde(4-CBA).The structure of 4-CBA is extremely similar to TPA,thus the concentration of 4-CBA is a key index of the product quality[2].It has been proved[3]that the quality and yield of TPA depend on the operating conditions,such as concentrations of feed compounds,solvent ratio,temperatures of the reactor and residence time.Even a slight change of the operating conditions leads to a more pronounced change of the final product TPA.Therefore,optimized operating variables should be obtained to target the economic benefits of the process.Besides,the PX oxidation reaction is accompanied by many side reactions,the amount of the by-products is up to 32[4].Fromthe economic pointofview,the acetic acid combustion and PXcombustion are the mostsignificantside reactions.Unfortunately,minimization of the overall combustion loss and maximization of the TPA yield with acceptable TA quality are in conflict with each other.Thus,the operation optimization problem of the PX oxidation process is a complex multi-objective optimization problem with constraints.

Multi-objective optimization(MOO)using evolutionary algorithms(EAs)have gained popularity in the recent years because of the ability of EAs to yield a set of Pareto optimal solutions for problems with more than one objective simultaneously in a single run.Popular EAs such as the non-dominated sorting genetic algorithm-II(NSGA-II),multi-objective particle swarm algorithm(MPSO),and multi-objective differential evolution(MODE)and its different strategies have been proposed by researchers[5–7].As a new branch of evolutionary algorithm(EA),MODE were widely concerned by researchers in many applications because it has many advantages over traditional multi-objective optimization algorithms.Original Differential Evolution(DE)was proposed to solve real-parameter optimization problems by Storn and Price[8]in 1995.The algorithm uses weighted difference between solutions to perturb the population and to create candidate solutions.The offspring,named trial solutions,are partly from the candidate solutions and partly from the old population according the select operator.Abbasset al.[9]proposed a Pareto-frontier differential evolution(PDE)algorithm to solve MOPs.The novel PDE algorithm randomly sets the step length parameterFfrom a Gaussian distribution and keeps only the non-dominated ones as the basis for the next generation.Gonget al.[10]presented an improved differential evolution algorithm.The major modifications are characterized by:(1)the initial population is generated by employing the orthogonal design method,(2)adopting an archive to store the non-dominated solutions and employing the new Pareto-adaptive ε-dominance method to update the archive at each generation,and(3)alternatively selecting parents by a randomscheme and an elitistselection scheme,to improve its performance.A MODE with a diversity enhancement mechanism is proposed[11]to prevent the algorithm becoming trapped in a locally optimal Pareto front.A set of randomly generated parameter vectors are inserted into the current population to increase the diversity of the newly generated offspring.Self-adaption strategy has attracted much attention recently due to the difficulty in the determination of the two control parameters in the MODE.Wanget al.[12]presented a selfadaptive differentialevolution algorithmincorporate Pareto dominance.A crowding entropy diversity measure tactic is proposed to maintain the diversity of Pareto optimality in an external elitist archive and the two control parameters are randomly picked up from predefined ranges.Qianet al.[13]put forward a new approach which integrated selfadaptive differential evolution algorithm with α-constraineddomination principle,named SADE-αCD,the trial vector generation strategies and the DE parameters are gradually self-adjusted adaptively.The advantageous performance of SADE-αCD is validated by comparisonswith NSGA-IIand constrained multi-objective differentialevolution through eighteen test problems.The performance indicators showed that SADE-αCD is an effective approach in solving constrained multiobjective problems.

For the multi-criteria nature of most real-world problems,there have been many attempts of the application of MODE in industrial process for modeling,optimal design and operation.Babuet al.[14]applied the MODE algorithmto optimize the industrialadiabatic styrene reactor considering productivity,selectivity and yield as the main objectives.Pareto optimal front was obtained to provide wide-ranging optimal operating conditions.Reddyet al.[15]studied the MODE algorithm with an application to a case study in reservoir system optimization.Itwas found that MODE provided many alternative Pareto optimal solutions with uniform coverage and convergence to true Pareto optimal fronts.Gujarathiet al.[16]carried out the multiobjective optimization of the purified terephthalic acid(PTA)oxidation process.Four industrially important cases were studied and the Pareto optimal fronts were calculated.The results showed that the MODE covered a better range than NSGA-II.

In this paper,an improved self-adaptive multi-objective differential evolution(ISADE)algorithm is proposed.In order to overcome the problems of premature convergence and falling into the local optimum,immune operation is introduced to the SADE algorithm to strengthen the local search ability and optimization accuracy.The proposed algorithm is successfully tested on several benchmark test problems,and the performance measures such as convergence and divergence metrics are calculated.Then,the multi-objective optimization of an industrial PX oxidation process is carried out by the proposed ISADE algorithm.

The organization of this paper is as below.Section 1 briefly presents some chemical mechanisms and technological process of the PX oxidation with modeling approach.Section 2 describes the ISADE algorithm realization steps in detail and the performance metrics.Then,Section 3 presents the implementation of multi-objective optimal design for the PX oxidation process and simulation results.Finally,some concluding remarks and suggestions will be given in Section 4.

2.Description and Modeling of the PX Oxidation Process

2.1.Description of the PX oxidation process

In general,there are four production technologies for PX oxidation process in terms of their reaction temperatures(i.e.BP-Amoco,Du pont-ICI,Mitsui and Lurgi-Eastman).In this paper,the reactor model of Mitsui PX oxidation process was used for the production of TPA.The PX oxidation process has been described in our previous work[17].The basic flowsheet is shown as Fig.1.

As a typical free radical chain reaction,the overall process of the oxidation reaction includes three steps[2]:chain initiation,chainpropagation and chain termination.PX oxidation reaction involves many side reactions,among which,acetic acid combustion and PXcombustion are the most crucial.As is known,the main products of the side reactions are COx(CO and CO2).In literature,the amountof the COxwas obtained by neural network or support vector machine technique and the effect of the side reactions was neglected[18].In our previous work,an industrial PX oxidation process model based on an extended kinetic model combining the main reactions and side reactions was established and proved to be more reliable.The process model is simulated in Aspen Plus using the user defined model because the PX oxidation modelbased on the free radicalmechanismcannotbe written by built-in models provided by Aspen Plus software.Detailed description of the modeling process can be referred to our previous work and is not repeated here[19].

2.2.Model equations for process parameters optimization

The objective of optimization is supposed to reflect some economic criterion[20].In industrialPX oxidation process,the overallcombustion loss should be minimized and the output of TA should be maximized to obtain greater economic pro fits.Although a single-objective optimization has been realized in our previous work.The objective was to minimize the acetic acid and PX combustion loss while constraining the quality specifications.Actually,we neglected to take into account the yield of the TPA which is conflict with the combustion loss in this optimization process.To solve this problem,multi-objective optimization strategies have more advantages.Multiple conflicting objectives can be optimized simultaneously by a multi-objective algorithm and suitable multi-criterion decision-making techniques can be employed to selectsome of the optimalsolutions according to personal preference[21].

The concentration of 4-CBA in the outlet of the reactor is referred to the key index of the product quality since the structure of 4-CBA is extremely similar to TPA.Thus,a multi-objective problem which involves three conflicting objectives(namely the yield of TPA,the combustion loss of PX and the combustion loss of HAC)is established.For simplicity,the yield of TPA can be measured in terms of the conversion ratio of flowrate of TPA to the flowrate of the feed PX.

wherexPX(%)is the percentage of COxgenerated byp-xylene combustion in total amount of COxof the off-gas,wHAC(g·mol-1)is the which are 61%and 75%,respectively.

For this process,six process variables are chosen by sensitivity analysis,i.e.the concentration of cobalt catalyst(xCo,%),the concentration of manganese catalyst(xMn,%),the concentration of bromide(xBr,%),temperature of the reactor(T,°C),residence time of reactor(τ,s)and the HAC/PX ratio(Sratio).Therefore,the mathematical expression of the optimization problem can be described as the following:

X=[xCo,xMn,xBr,T,τ,Sratio]Tare the operating conditionsto be optimized.g1(X)andg2(X)are the two constraints.As described above,this optimization problem is a typical constrained MOPs.A self-adaptive multi-objective differential evolution willbe used to solve this problem.

3.An Improved Immune Self-adaptive Differential Evolution Algorithm

3.1.Differential evolutionary algorithm

Nowadays,DE algorithm is considered to be a very effective alternative to solve complex multi-objective optimization problems.It provides fast convergence rate,robustness and global optimization ability.The basic operations of DE algorithm include the mutation,crossover and selection operations for generating the trial parameter vectors.The main procedure of the basic DE can be referred to[8].

Among all the operations mentioned above,mutation operation is the core of DE algorithm.Many different mutation strategies are proposed in the literature and the frequently used 5 strategies are listed as followed:

DE/rand/1:

wheret=1,2,3,...is generation(time),indicesri(i=1,…,5)are mutually exclusive integers randomly generated within[1,NP],respectively,and are also different from indexi;xtbestdenotes the best vector which gives best fitness value in the population at generationt;Fis the scale factor.

The crossover operation includes the binomial crossover method and the index cross method.The mostfrequently used binominal crossover operation is adopted in this paper:

If the performance of the trial vector is better than the target vector,then the trialvector will replace the targetvector in the nextgeneration by a greedy selection operator in the selection operation.

3.2.Improved immune self-adaptive differential evolution algorithm

3.2.1.Adaptive adjustment strategies for scale factor and crossover rate

ChoosingFandCRdepends on the specific problem applied,and is often difficult.In general,smallFandCRvalues benefit to a fast convergence speed,while largeFandCRvalues help to increase the diversity of population and avoid premature convergence.To get a balance between the convergence speed and premature convergence,it's kind of a necessity to conduct an adaptive adjustment strategy forFandCR.

In order to improve the global searching capability by escaping the local optima and avoiding premature convergence,the immune concepts and methods are introduced to improve the overall performance of the SADE algorithm[13]in this paper.Immune algorithm is a heuristic search and optimization technique based on the principles of genetics and biological immune system.In the immune system,the lymphocytes recognize invading antigens and produce antibodies to exclude the foreign antigens.In general,in an optimization problem,the antigen and antibody are equivalent to the objective function and the candidate solution.As a population-based search,IA maintains populations of potential solutions during the search procedures.Therefore,the immune conceptsand methods are applied to seek the optimal scale factor parameters.

The parameterFupdating strategies are controlled as follows:

(1)Calculate affinity values:IA uses affinity as a discriminator of the quality ofsolutions,and the antibody with higheraffinity is more likely to be selected and survives to the nextgeneration athigher probability.In this paper,affinity is defined as

wherer∈[0,0.5],andiis the numerical order of each antibody in the population by arranging them according to their function values.

The affinity values ofthe scale factorparameters are calculated in the following way

The scale factor parameters are arranged from the best to the worst according to their affinity.The best 20%mutation parametersMFT+1(new)are stored into the memory baseMF.

(2)Update of the mutation parameter memory baseMF

(3)Generation of scale factor parameters for the next generation

The scale factorparameters in the nextgeneration are updated as

3.2.2.Constraint-handling technique

Since constraints are frequently associated with real-world optimization problems,alpha constrained method is employed to handle constraints.The α-constrained method treats some infeasible solutions as feasible ones when these infeasible solution violations are under a specified level.The level α is controlled according to the following formula:

wherexθis the top θth solution at the initialization stage andcpis a parameter to control the speed of reducing relaxation of constraints.Here,θ is recommended to be 0.2?NPandcp∈ [2,10].

A pseudo feasible concept is defined to effectively exploit the information carried by the infeasible solutions.The pseudo feasible solution denotes the solution whose overall constraint violation is less than α.Details of the constrained-dominate relationship between individuals including pseudo feasible and pseudo infeasible solutions can be referred to our previous work[13].After sorting operation according to the constrained-dominate relationship criteria,we choose exactlyNPsolutions from combined populationXt∪Utto form new populationXt+1.

By combining the SADE algorithm and immune operation,the main step and the pseudocode of the new ISADE is as below:

?

3.3.Experimental analysis

3.3.1.Experimental setup

Some applications of multi-objective methods in chemical engineering have been collected by Rangaiah[23].On the base of these typical chemical process,14 benchmark problems are selected to test the proposed ISADE algorithm which are reported in[13].These are TNK,SRN,CONSTR,OSY,CTP1-CTP7,BINH4,TAMAKI and VIENNET4.Among these 14 problems,the first 11 are two-objective problems and the next 3 are three-objective problems.The results of other algorithms for comparison purpose are taken from literature[24].These two tables show comparisons of the MODE,MODE III,and hybrid-MODE algorithms with other well-known algorithms.The results for the other algorithms,namely,NSGA-II(both binary-and real-coded versions),SPEA,and PAES,were taken from the literature.Note that the SADE was shown to be superior to the three powerful MOEAs and NSGA-II.The population size is fixed at 100.The initial crossover probability is CRmin=0.50 and CRmax=1.0.The maximal number of function evaluations FES is 25,000.Each algorithm independently runs 25 times.The immune scale factor and crossover probability are obtained adaptively during each call of the mutation and crossover operations,respectively.The experimental development environments are as below:Matlab 2009,running on the Intel(R)Core(TM)2 Quad CPU 2.50GHz,3.0GB of RAM.

3.3.2.Performance measures

In general,there are two goals in a multi-objective optimization as below:(1)convergence to the Pareto optimal front,and(2)diversity of solutions in the Pareto optimal set.To compare the performances of the improved SADE algorithm with those of other algorithms such as SADE and NSGA-II,two widely used metrics are used in this paper,i.e.,convergence metric γ,and divergence metric Δ.The first metric,γ,measures the extent of convergence to a known Pareto set of solutions,while the second metric,Δ,measures the extent of the spread achieved among the Pareto optimal solutions.In our experiments we use 500 uniformly spaced Pareto optimal solutions as the approximation of the true Pareto front corresponding to each problem for comparison.

The metric γ is defined as[25]:

whereNis the number of non-dominated solutions found by the algorithm being analyzed anddiis the minimum Euclidean distancebetween theith solution of obtained non-dominated front and the solutions in the optimal Pareto front.The smaller the value of the convergence metric,the better the convergence to the true Pareto optimal front.

Table 1Comparison of the convergence metric γ between ISADE and other MOEAs

The metric Δ is calculated using the equation[25]:

wheredfe+d1erepresents the sum of the Euclidean distances between the currently obtained extreme solutions and the extreme solutions of the Pareto set anddiis the Euclidean distance between the consecutive non-dominated solutions.The parameterdis the average of all Euclidean distancesdi,i=1,2,…,(N-1).A smaller value ofthe metric Δ shows a better distribution of solutions within the extreme Pareto optimal solutions.

3.3.3.Experimental results and analysis

Each testproblem is run for 25 times forallalgorithms.The statistical values of the convergence and divergence metrics,i.e.,mean values(Mean)and standard deviations(SD),are analyzed in details.The convergence metrics of the SADE algorithm and the proposed ISADE algorithm are presented in Table 1,while Table 2 presents the divergence metrics.The better results are shown in bold.To provide a visualization of these results,graphical results for problems OSY,CTP2,CTP5,CTP6,and TAMAKI are shown in Figs.2–5,respectively.Since the results showed that both ISADE and SADE are superior to MOEA/D,the results for MOEA/D are not depicted in these figures.In each figure,the result for each algorithm corresponds to the run equal to the median value with respect to γ metric.It should be noted that both algorithms take a maximum size of 100 for the external archives in the experiments.

It can be seen from Tables 1 and 2 that our ISADE method obtains best performances for all the test problems in terms of the two metrics.The ISADE algorithm outperforms NSGA-II especially on SRN,CTP1,CTP2,CTP5,CTP7 BINH4,TAMAKI and VIENNET4.Although the SADE attains better results on CTP6,ISADE is still superior to SADE for most ofthe testproblemsin terms ofconvergence metrics.Forthe divergence metric,Table 2 indicates that the SADE algorithm resulted in better average divergence metric value on CTP5 and TAMAKI.Although both convergence and diversity issues are equally important for MOO studies,it is also a fact that obtaining a better diversity of solutions is insignificant if the algorithm does not converge to the true Pareto front[18].Take the CTP5 for example,for Fig.3,it can be observed that the convergence of the SADE algorithm to the true Pareto sets is worse than that of ISADE algorithm.It is therefore important for MOEAs to aim firstfor good convergence and then forsatisfactory divergence rather than searching for good divergence and then satisfactory convergence.It is noteworthy that the ISADE algorithm shows good search capability for the boundary points of the Pareto front.Superior performance ofISADE is due to its betterlocalresearch ability as a result of immune operation.In addition,variance values resulting from the ISADE algorithm are very small in every case,demonstrating the robustness of the ISADE algorithm.

Fig.2.Comparison of Pareto fronts between different algorithms for CTP2.

Table 2Comparison of the divergence metric Δ between ISADE and other MOEAs

With the successful results obtained from application of the ISADE algorithm to the test problems and comparative numerical studies,it can be concluded that the proposed ISADE algorithm is an advantageous approach to solving CMOPs.We apply this algorithm to an industrial application,namely,multi-objective optimization of the PX oxidation,the results of which are discussed in the next section.

Fig.3.Comparison of Pareto fronts between different algorithms for CTP5.

Fig.4.Comparison of Pareto fronts between different algorithms for CTP6.

4.Multi-objective Optimization of the PX Oxidation

Now the ISADE algorithm is adopted to optimize the PX oxidation operation.A population size of 100 and different operations are performed for 30 generations to obtain the non-dominated Pareto optimal solutions.

For this MOP,six process variables are chosen by sensitivity analysis,i.e.the concentration of cobalt catalyst,the concentration of manganese catalyst,the concentration of bromide promoter temperature of the reactor,residence time of reactor and the ratio of HAC/PX.

The Pareto optimal front obtained by a single run of ISADE is shown in Fig.6.The points in the Pareto sets indicate the maximum possible of TPA yield,minimum possible of minimization of the PX combustion loss,or minimization ofthe HAC combustion loss with the given operating constraints.Fig.6 illustrates the solutions achieved by ISADE spread along the Pareto optimal front with diversity.It can be seen from Fig.7 thatthe increase ofconversion ofPXresultsin highercombustion lossof both HAC and PX.This also proves that the yield of TPA conflicts with the combustion loss of PX and HAC.Therefore,it is unrealistic to optimize all the objectives simultaneously.Table 3 lists the operation parameters and optimal objectives corresponding to 3 selected solutions from the Pareto set.The first two are the boundary points and the other is an intermediate point.It can be seen from Table 3 that the lower concentration of 4-CBA results in the higher combustion loss.Operating under the conditions predicted will enhance productivity and reduce the consumption and thereby increase pro fit.Different optimal operation conditions can also be obtained by ISADE according to the market price or person preference.Since the ISADE method is a general algorithm,the described procedure is suitable for maximizing the benefits of any operating industrial plant.

Fig.5.Comparison of Pareto fronts between different algorithms for TAMAKI.

Fig.6.The Pareto optimal front of the PX oxidation process.

Fig.7.Pair interactions of the three object functions.

5.Conclusions

This work aims at maximizing the efficient TPA yield that is produced from the industrial PX oxidation process and minimizing the HAC and PX combustion simultaneously.A constrained multi-objective self-adaptive differential evolution algorithm is proposed.Immune concept is used to control the scale factor in the evolution process.This operation not only enhances the algorithm's local searching ability,but helps to maintain diversity of the Pareto optimal solution set.The performance of the proposed ISADE algorithm is evaluated on 14 benchmark problems and the numerical experimental results are compared with the SADE and MOEA/D algorithms.The performance metrics obtained in this study are found to be better in terms of convergence and comparable in terms of divergence,i.e.,the ISADE algorithm outperforms the other two algorithms.Multi-objective optimization problem of an industrial PX oxidation process is then solved using the proposed ISADE algorithm.Operating under the conditions predicted will enhance productivity and reduce the consumption and thereby increase pro fit.Since the ISADE method is a general algorithm,the described procedure is suitable for maximizing the benefits ofany operating industrial plant.

Table 3The operation parameters and optimal objectives corresponding to 3 selected solutions from the Pareto set

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