Solving chemical dynamic optimization problems with ranking-based differential evolution algorithms☆

Xu Chen ,Wenli Du *,Feng Qian

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

2 School of Electrical and Information Engineering,Jiangsu University,Zhenjiang 212013,China

1.Introduction

Dynamic optimization problems(DOPs)are often encountered in chemical engineering,as most industrial process models are time dependent and described by differential equations.The solution of DOPs is usually very difficult because of their highly nonlinear and multidimensional nature,as well as the presence of constraints on state and control variables and implicit process discontinuities[1].Given the profound importance of DOPs in industrial and engineering practices,developing efficient methods for DOPs has attracted great interest.Dynamic optimization methods can be roughly divided into three categories:dynamic programming(DP),indirect methods,and direct methods.

Classic DP method relies on Bellman's optimality[2].DP is a successful method for solving DOPs,except for dimension curse.To overcome this drawback,Luus[3]proposed iterative dynamic programming method by use of coarse grid points and search region reduction strategies.However,its high computational cost for systems involving a large number of differential-algebraic equations has restricted its application to problems on a smaller scale[4].

With indirect methods,DOPs are solved by using Pontryagin's maximum principle[5].It converts the original problem into a two-point boundary value problem,which rarely has an analytical solution and requires numerical alternative such as shooting method[6].The two point boundary value problem is always extremely difficult to solve,especially in the presence of pointor path constraints on state variables.Therefore,indirect methods are extremely complicated to apply in practice.

Direct methods transform the original dynamic problem(which is in finite dimensional)into a finite dimensional non-linear programming(NLP)problem,either using complete parameterization(CP)[7]or control vector parameterization(CVP)[8,9].CP method,also named simultaneous strategy,discretizes both state and control variables.This full discretization results in a large-scale NLP and specialized NLP solvers have to be used to solve the NLP efficiently.The CP method has been recently reviewed by Kameswaran and Biegler[7].CVP method only discretizes control variables and transforms original DOP into an NLP where the system dynamics(differential equations)must be solved for each evaluation of the performance index.The dimension of the NLP problems in CVP is much smaller than that in CP.Therefore,CVP is relatively easier to implement,and a large number of optimization algorithms,including deterministic gradient-based algorithms[8]and stochastic optimization algorithms[10],have been combined with CVP to deal with DOPs.

The NLPs from the application of direct approaches(such as CVP)are frequently multimodal.Deterministic gradient-based algorithms may converge to local optima,especially if they are started far away from the global solution[9].In addition,explicit mathematical descriptions of industrial model sometimes do not exist,and methods based on gradient information may become incapable.To surmount these difficulties,stochastic optimization algorithms based on evolutionary computing can be used as robust alternatives.The use of evolutionary methods to optimize DOPs,including genetic algorithm[10–12],simulated annealing[13,14],particle swarm optimization[15–17],and scatter search[18],has received increasing interest.

Differential evolution(DE)is a population based stochastic optimization technique,invented by Storn and Price[19].Since its inception in 1995,it has emerged as a very competitive form of evolutionary computing.The use of DE algorithms to solve DOPs also has drawn the attention of many researchers[20].Chiou and Wang[20]developed a hybrid DE algorithm by embedding an acceleration phase and a migration phase into the original DE algorithm to solve DOPs in fed-batch fermentation process.Kapadi and Gudi[21]employed standard DE(SDE)to solve optimal control and parameter selection problems of fed-batch fermentation involving general constraints on state variables.To speed up DE and solve DOPs,Babu and Angira[22]introduced modified DE(MDE)that utilizes only one set of population compared with the two sets in the original DE at any given time point in a generation.Angira and Santosh[1]suggested the use of trigonometric DE(TDE)to solve DOPs encountered in chemical engineering.Fan et al.[23]proposed a hybrid DE algorithm called Alopex-DE by integrating Alopex to solve DOPs of chemical processes.

Das and Suganthan[24]pointed out that DE's weak selective pressure(due to unbiased selection of parents or target vectors)may result in inefficient exploitation.To overcome this weakness,Gong and Cai[25]presented a ranking-based mutation operator(RMO)for DE algorithms,in which better individuals have larger possibilities to be selected in mutation operator.This study deals with the utilization of RMO to enhance the performance of CVP-based DE algorithms for DOPs.The DOPs are first converted into NLP problems by CVP approach;then the RMO is incorporated into three DE algorithms,SDE[21],MDE[22],and TDE[1],to obtain three ranking-based differential evolution(DE-RMO)algorithms,i.e.,SDE-RMO,MDE-RMO,and TDE-RMO,to solve DOPs.Three DERMO algorithms and their non-ranking DE algorithms are applied to solve four constrained DOPs from previous studies.The simulation results indicate that the DE-RMO algorithms can provide better findings in terms of solution accuracy and convergence speed compared with previous non-ranking DE algorithms.

2.Formulation of Dynamic Optimization Problems

Dynamic optimization allows the computation of optimal operating policies to minimize(or maximize)a predefined performance index[18].The objective function is formulated as

where J is the objective function,is a Mayer item,andis a Lagrangian term.

The dynamic model of a chemical process is often described by differential equations as follows:

where x(t)∈Rndenotes the vector of state variables,u(t)∈Rmdenotes the vector of control variables(or operational variables),and x(t0)=x0is the initial conditions.

Four kinds of constraints may exist in the DOPs.They are path inequality constraints,path equality constraints,terminal inequality constraints,and terminal equality constraints.Path constraints should be satisfied in the entire time interval[t0,tf],i.e.,

3.Differential Evolution Algorithms

DE is a population-based stochastic optimizer in the continuous search domain,proposed by Storn and Price[19].It is capable of handling non-differentiable,non-linear and multi-modal optimization problems.DE initializes a population of NP individuals and employs mutation,crossover,and selection operators at each generation to evolve its population toward the optimal direction.DE population initializes NP individuals(each individualis called a targetvector)from the search space

where g denotes the generation counter and D defines the number of variables.

After initialization,the mutation operator is applied to generate mutant vectorfor each target vectorMany mutation strategies are described in previous studies.A classical strategy is“DE/rand/1”:

where F is a scale factor,r1,r2,and r3are three mutually different integers randomly generated from[1,NP]and different from the target vector index i.

After mutation,DE employs a crossover operator to produce the trial vectorbetween.The crossover operator performed on each component is

where CR is the crossover rate and jrandis a randomly generated integer in{1,D}.

A selection operator then adopts a one-to-one competition between

DE repeats these three operators until a termination criterion is satisfied.Due to its simple structure,ease of use and good performance,several DE algorithms have been proposed to deal with DOPs,such as SDE[21],MDE[22],and TDE[1].

MDE is proposed to reduce the computational time of original DE and optimize DOPs[22].It utilizes only one set of population compared with the two sets in the original DE.If a generated trial vectoris better than its corresponding target vectorthe former is immediately copied into the current population and participates in the mutation.Such an improvement enhances the convergence speed,with less function evaluations as compared to DE.The detailed implementation of MDE can be found in[22].

TDE was proposed by Fan and Lampinen[26],in which a new local search operation,i.e.,trigonometric mutation operation(TMO),is embedded into the original DE.Angira and Santosh[1]dealt with the application and evaluation of TDE to solve DOPs.In TDE,the TMOis performed according to the following equation

TDE performs the TMO with a probability Mt and performs the“DE/rand/1”using Eq.(10),with a probability 1-Mt.The detailed procedure of TDE can be found in literature[1].

4.Ranking-based Differential Evolution Algorithms for DOPs

In this section,we present the ranking-based DE algorithms for constrained DOPs.First,we state the CVP approach.Next,the RMO technology is described.Finally,we integrate RMO into previous DE algorithms to form three ranking-based differential evolution(DE-RMO)algorithms for constrained DOPs.

4.1.CVP approach

The original DOP is an in finite-dimensional optimization problem,as the control vector u(t)to be optimized is a continuous function of time t.Therefore,CVP[8]is required to transform original problem into a finite-dimensional NLP problem.In the present work,time interval[t0,tf]is divided into N stages and the ith time stage is[ti?1,ti](i=1,2,...,N).Control variables are approximated by constant functions in the ith stage[ti?1,ti].Hence,the coding of optimization variables are expressed as

where uji(j=1,2,...,m;i=1,2,...,N)is the approximation of the j th control variable ujin the i th time stage.After discretization,an optimization algorithm can be applied to select the optimal values of X.During optimization,the differential equations[i.e.,Eqs.(2)and(3)]must be solved using a differential equations solver for each evaluation of the objective function value(OFV).

There are four kinds of constraints in the DOPs:path inequality constraints,path equality constraints,terminal inequality constraints,and terminal equality constraints.To deal with the path inequality constraints(i.e.Eq.(4))and path equality constraints(i.e.Eq.(5)),p additional variablescalled state constraint variables[27],are introduced using the following relationship

The final value xn+i(tf)gives the total violation of ith path constraint integrated over the entire time interval.

For terminal inequality constraints(i.e.Eq.(6))and terminal equality constraints(i.e.Eq.(7)),the constraint violations can be calculated as

After obtaining the objective values and overall constraint violations of all individuals in the DE algorithms,Deb's feasibility-based rule[28]is used to compare two solutions.

Given two individuals X1and X2,their corresponding objective function values and overall constraint violations are(OFV1,OCV1)and(OFV2,OCV2).X1is said to be better than X2,if any of the following three conditions is met:

(1)both X1and X2are infeasible,and OCV1＜OCV2;

(2)X1is feasible,but X2is infeasible;

(3)both X1and X2are feasible,and OFV1＜OFV2.

Compared with the penalty function constraint-handling method,Deb's feasibility-based rule does not introduce any sensitive parameter.Interested readers can read the survey paper of constraint-handling methods by Mezura-Montes and Coello[29].

4.2.Ranking-based mutation operator

Previous studies indicate that the DE algorithms may suffer from low convergence speed,because of the unbiased selection of parents in mutation operators[24].Recently,Gong and Cai[25]introduced the ranking-based mutation operator into DE.In the RMO,better individuals have more chance to be utilized to produce offspring,which is helpful for the performance enhancement of DE algorithms.

This paper uses RMO to enhance the performance of DE to solve constrained DOPs.Therefore,the RMO technology is integrated into CVP-based DE algorithms.A feasibility-based ranking technique is used to sort the DE population from best to worst.The feasibility based ranking technique is based on Deb's feasibility-based rule[28]described in subsection 4.1.It is described as follows:

(1)feasible individuals are sorted before infeasible individuals;

(2)feasible individuals are sorted on the basis of their OFVs;

(3)infeasible individuals are sorted on the basis of their OCVs.

Gong et al.[30,31]have proposed some adaptive ranking mutation technique for constrained DE,but our feasibility-based ranking technique is much simpler compared with the adaptive ranking mutation technique,which may be more suitable for chemical DOPs.

Subsequently,the ranking Riof i th vector is assigned as

where NP is the population size.According to Eq.(21),the best vector in the current population will give the highest ranking.

The selection probability of i th individual is calculated as

The individuals in the RMO are finally selected according to the selection probabilities.The ranking-based individual selection in RMO is illustrated in Fig.1.The individuals with higher rankings(or selection probabilities)are more likely to be chosen in the mutation operator.This is beneficial for the performance enhancement of DE algorithms.

Fig.1.Ranking-based individual selection in RMO.

4.3.Implementations of DE-RMO algorithms

In this work,the RMO is incorporated into SDE[21],MDE[22],and TDE[1]to form three DE-RMO algorithms,namely SDE-RMO,MDE-RMO,and TDE-RMO,to solve constrained DOPs.We choose these three DE algorithms because all of them can be realized easily,which is beneficial for chemical DOPs.Here we only describe TDERMO in detail.The main procedure is given as follows.

Step 1:Initial a population with NP individuals.Set scale factor F,crossover rate CR,and trigonometric mutation probability Mt.

Step 2:Calculate the OFV and OCV for each individual using the CVP approach.The explicit Runge–Kutta method is used to solve the differential equations.

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Step 3.Sortthe population according to the feasibility-based ranking technique.

Step 4:For each individual,do Steps 5–8.

Step 5:If rand＜Mt,perform the TMO according to Eq.(13);else,perform the RMO according to Fig.1 and Eq.(10).

Step 6:Perform the crossover operator according to Eq.(15).

Step 7:Calculate the OFV and OCV for each individual using the CVP approach.

Step 8:Perform the selection operator Eq.(11)based on Deb's feasibility-based rule.

Step 9:Return to Step 3 until the termination condition is met.

Step 10:Output the best solutions.

Fig.2.illustrates the flowchart of TDE and TDE–RMO for DOPs.From Fig.2,we can see that the differences between TDE and TDE–RMO are:(1)TDE–RMO needs to sort the DE population before the mutation operators;(2)TDE–RMO employs RMO,while TDE employs the classic"DE/rand/1".However,both modifications can be easily realized.Therefore,TDE-RMO does not significantly increase the complexity of TDE algorithm.

5.Case Studies

In this section,the DE-RMO algorithms are applied to four constrained DOPs by CVP.First,we state the simulation setup for this study.Subsequently,simulations are conducted to compare the DE-RMO algorithms with their non-ranking DE algorithms.Finally,TDE-RMO is compared with some other stochastic optimization algorithms.

5.1.Simulation setup

Four constrained DOPs from the literature are used to evaluate the performance of the DE-RMO algorithms.The parameter settings of these DE algorithms are in Table 1.

Three evaluation criteria are adopted to measure the performance of the algorithms.

BOFV(best objective function value):it records the best objective function values when the maximal number of function evaluation maxNFES is reached.The best,mean,and worst BOFV,as well as the standard deviations(std),are presented.BOFV measures the solution accuracy of an algorithm.

SR(success rate):It is equal to the number of successful runs over total runs.A success run means that within maxNFES,the algorithm finds a solution with satisfactory precision Js.

ANFES:It is used to record the average number of function evaluations for an algorithm to find a solution with satisfactory precision Js.ANFES measures the convergence speed of an algorithm.

5.2.Simulation results

The DE-RMO algorithms and their non-ranking DE algorithms are used to solve four constrained DOPs.All algorithms are coded in matlab.The built-in routine “ode45”is chosen as differential equation solver.

5.2.1.Problem 1—constrained van der Pol oscillator

This problem is extracted from a previous study[32]and described as follows.

For problem 1,the time interval is divided into N=20 stages in CVP approach.BOFV is recorded when maxNFES reaches 30000,and the satisfactory solution precision is setas Js=2.97500.The values of maxNFES and Jsare set based on the numeric experiments,because we cannot know the best objective function values in advance for a real-world DOP.

Fig.2.Flowcharts(a)TDE for DOPs(b)TDE-RMO for DOPs.

Table 2 shows the results of these DE algorithms.The boldface means that DE-RMO algorithms are better than their corresponding non-ranking DE algorithms.Thus our proposed DE-RMO algorithms provide consistently more accurate solutions than their corresponding non-ranking DE algorithms with respect to BOFV.Considering the std.of BOFV,we find that all of the DE-RMO algorithms give smaller std.values than their corresponding non-ranking DE algorithms.Thus the RMO is able to enhance the robustness of the previous DE algorithms.All three DE-RMO algorithms succeed in finding satisfactory solutions for each of 20 runs.In terms of ANFES,DE-RMO algorithms get less ANFES compared with their corresponding non-ranking DE algorithms,so that DE-RMO algorithms converge faster.Considering the overall performance,TDE-RMO ranks the first,followed by MDE-RMO,TDE,SDERMO,MDE,and SDE.

The reported best result 2.95436 is obtained by Vassiliadis et al.[32],with a gradient-based algorithm.Ourbestresult2.97228 is within 0.61%of the reported best value.Fig.3 illustrates the optimal control profile and the path constraint trajectory obtained by the TDE-RMO.The optimal control pro file is in agreement with that in[32].The path constraint x1(t)≥?0.4 is satisfied,and it is active in the interval[0.494,1.650].

Table 1 Parameter settings for the DE algorithms

Table 2 Numerical results for problem 1

5.2.2.Problem 2—mathematical system with nonlinear inequality constraint

The mathematical model of this problem is presented as[27]

For problem 2,the time interval is divided into N=20 stages,and other parameters are:D=20,maxNFES=60000,and Js=0.17500.Table 3 shows the results of these DE algorithms.The DERMO algorithms consistently give better results than their corresponding non-ranking DE algorithms in terms of BOFV,SR and ANFES.Hence,the RMO is able to enhance the solution accuracy,success rate,and convergence speed of DE algorithms.TDE-RMO gives the best performance in this problem,followed by MDE-RMO,TDE,SDE-RMO,MDE,and SDE.Mekarapiruk and Luus[27]divided the interval into N=20 stages and achieved a value of0.17266 with the iterative dynamic programming algorithm.Ourbestresult0.17272 is very close to theirresult.The optimal control pro file and the path constraint trajectory obtained by the TDERMO are plotted in Fig.4.The optimal control pro file is in agreement with that in[27],and the path constraint h(x,t)≤0 is active in the interval[0.290,0.706].

Fig.3.The optimal control pro file and the path constraint trajectory for problem 1.

Table 3 Numerical results for problem 2

5.2.3.Problem 3—optimal operation for a batch reactor with a cooling jacket

This problem formulates a first-order consecutive exothermic reaction,A→P→S,occurring in a batch reactor with a cooling jacket[33].The objective is to determine the optimal coolant flow rate u so that product P is maximized.The problem is described as

where x1,x2,and x3denote the concentrations of A,P and S,respectively;x4,x5,and x6are the temperatures(in Kelvin)of contents,C,and jacket,respectively;u is the coolant flow rate(m3·h?1);k1and k2are the reaction rate constants.The first three differential equations describe the material balances,and other three differential equations describe the energy balances.

Two constraints are path and terminal constraints.

For problem 3,the time inter val is divided into N=20 stages,and the other parameters are:D=20,maxNFES=60000,and Js=0.64500.The results of these DE algorithms are presented in Table 4.This problem is more difficult than problems 1 and 2,as DE and MDE cannot find satisfactory solutions within max NFES.By contrast,all three DE-RMO algorithms give SR of 20/20.Therefore,DE-RMO algorithms are more reliable than their non-ranking DE algorithms.Overall,the results indicate that our DE-RMO algorithms perform better than their corresponding nonranking DE algorithms in terms of BOFV,SR and ANFES.For this problem,TDE-RMO still ranks the first,followed by MDE-RMO,SDE-RMO,TDE,MDE,and SDE.

Table 4 Numerical results for problem 3

Fig.4.The optimal control pro file and the path constraint trajectory for problem 2.

Sun et al.[33]acquired a value of 0.6446,with the line-up competition algorithm.Our best result is 0.64586,better than that in[33].The optimal control pro file and the constraint trajectory obtained by the TDE-RMO are plotted in Fig.5.The path constraint x4(t)≤370K is active in the interval[1.151,2.848],and terminal constraint x4(tf)≤320K is active at the final time.

5.2.4.Problem 4—optimal monoclonal antibody production

This problem considers the optimal monoclonal antibody production in a hybridoma fed-batch reactor[34].The problem is described as

where Xv,Glc,Gln,Lac,Amm,and MAb are the concentrations in viable cells,glucose,glutamine,lactate,ammonia,and monoclonal antibodies,respectively;V is the fermenter volume;Glcmand Glnmare the concentrations of glucose and glutamine in the feed stream,respectively;the control variables F1and F2are the volumetric feed rate of glucose and glutamine,respectively.The mathematical expressions of the specificratesμ,kd,qGln,qGlc,qLac,qAmm,qMAb,and the system parameters are provided in[34].

In this problem,the value of V(t)is constrained by

For this problem,the time interval is divided into N=20 stages,and the other parameters are:D=20,maxNFES=150000,and Js=333.50000.Table 5 presents the results of the six DE algorithms.Our DE-RMO algorithms perform better than their non-ranking DE algorithms.The optimal control pro file and the constraint trajectory are plotted in Fig.6.The path constraint is active at the final time.

According to the simulation results,it can be concluded that:

?DE-RMO algorithms perform better than the non-ranking DE algorithms in terms of solution accuracy,success rate,and convergence speed for DOPs.The better per for mance of DE-RMO algorithms should be attributed to the introduction of RMO.

?The path constraints and terminal constraints are active at the optimal solutions in the four DOPs.

?TDE-RMO provides the best performance among all the used DE algorithms.It is suggested to be used as an efficient optimizer to solve DOPs in the future.

5.3.Further comparison with other evolutionary algorithms

In the previous subsection,the performance of DE-RMO algorithms verified by solving four DOPs.TDE-RMO provides the best results among three DE-RMO algorithms.To provide additional comparison for reference,TDE-RMO is compared with four other evolutionary algorithms,adaptive particle swarm optimization(APSO)[35],real-code genetic algorithm(RCGA)[36],artificial bee colony(ABC)[37],and teaching–learning-based optimization(TLBO)[38].The parameters of these four algorithms are set as recommended in their original literature.All the algorithms use the same constraint-handling method as that in TDE-RMO.Table 6 shows TDE-RMO is better than these four algorithms on all problems.BOFV of TDE-RMO are more accurate and better than those of other algorithms.TDE-RMO also provides the smallest std.of BOFV over 20 run.Considering the SR,TDE-RMO is the most reliable.In terms of ANFES,TDE-RMO gets less ANFES compared with their corresponding non-ranking DE algorithms,which means that DE-RMO algorithms converge faster.The ranks of five algorithms for the four DOPs are TDE-RMO,TLBO,APSO,RCGA,and ABC.

Fig.5.The optimal coolant flow rate and the constraint trajectory for problem 3.

Table 5 Numerical results for problem 4

Table 6 Comparison between TDE-RMO and other evolutionary algorithms

6.Conclusions

In this study,the RMO technology has been proposed to enhance the DE algorithms to solve DOPs.Three DE-RMO algorithms have been designed by incorporating the RMO.The DE-RMO algorithms and their non-ranking DE algorithms are evaluated by solving four constrained DOPs from the literature.The simulation results demonstrate that the DE-RMO algorithms perform better than previous non-ranking DE algorithms in terms of solution accuracy,success rate and convergence speed.Hence,the DE-RMO algorithms can be used as promising alternatives to solve DOPs in the future.

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