Cuiwen Cao ,Yakun Zhang ,Teng Yu ,Xingsheng Gu ,Zhong Xin ,Jie Li

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 State Key Laboratory of Chemical Engineering,East China University of Science and Technology,Shanghai 200237,China

3 State Key Laboratory of Multiphase Complex Systems,Institute of Processing Engineering,Chinese Academy of Sciences,Beijing 100190,China

Keywords:3-Layer mixed cultural evolutionary framework Optimal operation Syngas production Coal-water slurry gasifier

A B S T R A C T Optimizing operational parameters for syngas production of Texaco coal-water slurry gasifeir studied in this paper is a complicated nonlinear constrained problem concerning 3 BP(Error Back Propagation)neural networks.To solve this model,a new 3-layer cultural evolving algorithm framework which has a population space,a medium space and a belief space is firstly conceived.Standard differential evolution algorithm(DE),genetic algorithm(GA),and particle swarm optimization algorithm(PSO)are embedded in this framework to build 3-layer mixed cultural DE/GA/PSO(3LM-CDE,3LM-CGA,and 3LM-CPSO)algorithms.The accuracy and efficiency of the proposed hybrid algorithms are firstly tested in 20 benchmark nonlinear constrained functions.Then,the operational optimization model for syngas production in a Texaco coal-water slurry gasifier of a real-world chemical plant is solved effectively.The simulation results are encouraging that the 3-layer cultural algorithm evolving framework suggests ways in which the performance of DE,GA,PSO and other population-based evolutionary algorithms(EAs)can be improved,and the optimal operational parameters based on 3LM-CDE algorithm of the syngas production in the Texaco coalwater slurry gasifier shows outstanding computing results than actual industry use and other algorithms.

1.Introduction

Gasification is a vital component of“clean coal”technology[1].Gasifiers are the most important upstream units in gasification processes and they generate syngases which can be used to produce a wide range of chemical products.Optimal operation for gasifiers can greatly increase the production efficiency and economic benefits.

In this paper,the parameter optimization for syngas production in a Texaco coal-water slurry gasifier from a realistic chemical plant is addressed.In this industrial gasifier,the inlet“total oxygen”flowrate,the inlet“central oxygen”flowrate,and the inlet flowrate of quench water(in the mid of the gasifier)(see Section 4)are the three mostfrequently-used operating parameters.The operators in real practice adjust those three parameters based on their experiences to improve the production of“efficient gas”(CO+H2compositions in syngas),which often provides suboptimal operational conditions for the gasifier.Therefore,it is significant to develop models and algorithms to obtain optimal operational conditions for it.

Many researchers studied syngas production in different gasifiers[1–8].However,very few attempted to generate optimal process conditions for maximum benefits[9].Among these few studies on optimization[9–18],most of them[9–16]are limited to experimental-scale gasifiers.For instance,Larentis et al.[11]developed empirical and phenomenological models for syngas production and optimized the process parameters by experimental methods.Mohanty[9]extended the model proposed by Larentis et al.[11]to three-objective parameter optimization one and then used non-dominated sorting genetic algorithm(NSGA)to solve it.

In the work of Song&Cao et al.[17]and Sun&Gu[18],the former assessed three operational variables(the same as this paper)by using standard PSO algorithm and the latter optimized one operational variable(the gasifier inlet“total oxygen”flowrate).

Evolutionary algorithms(EAs)have attracted significant attention from both the researchers and the industry users for their most striking characteristic of population-based search.Among EAs,genetic algorithm(GA)[19],particle swarm optimization algorithm(PSO)[20]and differential evolution algorithm(DE)[21,22]were three of the most efficient and robust EAs for solving optimization problems over continuous space[23].However,how to avoid the local optima,how to manage the knowledge of elite individuals,and how to increase the convergence velocity are three of the major obstacles for these algorithms.Many strategies such as parameters adjustment[23–28]and hybrid with other heuristic algorithms[29–34]have been developed to improve the performance of these algorithms.In this paper,the“hybrid algorithms with cultural algorithm(CA)”by using GA,PSO,and DE algorithms are focused on.

Reynolds[35],who was enlightened by the far more rapid evolving velocity of human society,developed CA in 1994.CA is a dual inheritance algorithm frame system that models evolution in human society at both the macro-evolutionary level(population space)and the microevolutionary level(belief space)[36].Strategies such as improving the knowledge in belief space[37–43]and tackling with individuals in population space[42,44–49,18]have been developed recently.Coello&Becerra[42],Gao et al.[44],and Iacoban et al.[45]substituted the evolutionary programming algorithm of the original CA population space for DE,GA and PSO,respectively.Digalaskis and Margaritis[46]proposed a parallel cooperating cultural algorithm(PARCA)with multi-population.Each population's solution was generated by GA.The populations of Guo et al.[47]were evolved by evolutionary programming and they tried to do the knowledge migration tasks among elites.Sun et al.[18]presented a multi-population competitive co-evolutionary cultural differential evolution(MCCDE)algorithm.They adopted 5 independent populations which shared the same competitive space(belief space).Xu et al.[48]proposed a two-population cultural DE(MCDE)algorithm.The two populations can exchange their knowledge in every generation.Daneshyari[49]studied single-and multi-objective NLP problems via the multiple-swarm PSO algorithm.The particles in population space were grouped by using K-means clustering and the five types of knowledge in belief space from Reynolds and Ali[43]were used.

To the best of our knowledge,few efforts have been reported in the literature for optimization syngas production in a Texaco coal-water slurry gasifier from a realistic plant.In this paper,to efficiently solve the single-objective optimization model of Song&Cao et al.[17]and to provide better results,a novel three-layer cultural evolutionary algorithm framework is proposed,which includes a population space,a medium space and a belief space.The standard DE,GA and PSO are integrated into the framework to develop three-layer mixed cultural DE/GA/PSO algorithms(3LM-CDE,3LM-CGA,and 3LM-CPSO),accordingly.The effectiveness of the proposed hybrid algorithms is illustrated by twenty general single-objective nonlinear constrained benchmark examples.The computational results demonstrate that the 3-layer cultural algorithm evolving framework is effective in three types'standard algorithms.Finally,3LM-CDE,3LM-CGA and 3LM-CPSO algorithms successfully solve the real-world example and 3LM-CDE generates better results compared to the other algorithms.

The contents of this paper are organized as follows.Section 1 is the introduction.Section 2 is the overview of standard GA,DE,PSO and CA algorithms.Section 3 presents 3LM-CDE,3LM-CGA and 3LM-CPSO algorithms,and computational results of twenty general single-objective nonlinear constrained examples.In Section 4,the nonlinear constrained model for syngas production in a Texaco coal-water slurry gasifier is effectively solved by all the above algorithms.Section 5 is the conclusions.

2.Overview of Standard GA,DE,PSO and CA Algorithms

Before 3LM-CDE,3LM-CGA and 3LM-CPSO algorithms are proposed,the brief reviews on the standard GA,DE,PSO and CA are described as follows[23]:

GA[19]is possibly the first algorithmic model developed to simulate genetic systems.The characteristics of individuals are expressed using genotypes.The main driving operators of GA are selection operator,crossover operator and mutation operator.

PSO[20]algorithm is a population-based search algorithm based on the simulation of the social behavior of birds with a flock.The initial intent of the particle swarm concept is to graphically simulate the graceful and unpredictable choreography of a bird flock with the aim of discovering patterns that govern the ability of birds to fly synchronously,and to suddenly change direction with a regrouping in an optimal formation.From this initial objective,the concept evolves into a simple and efficient optimization algorithm.

DE[21,22]is a stochastic population-based search strategy developed by Storn and Price.While DE shares similarities with GA,it differs significantly in the sense that distance and direction information from the current population is used to guide the search process.Furthermore,the original DE strategies are developed to be applied to continuous-valued landscapes.

In standard CA[35],hierarchical human social structure and knowledge information are described as three components of two layers:the population space,the belief space and the communication protocol through which the two spaces interact with and evolve each other.The general framework for CA is depicted in Fig.1.

Fig.1.Framework of CA.

The population space comprises a set of possible solutions to the problem studied and can be modeled using any population-based approach of evolutionary programming(EP).The population space generates the initial solutions and evolves using any possible EPs.The belief space accepts a part of elite individuals selected from the population space through Function Acceptance,extracts problem solving knowledge,and then updates and stores the knowledge,which in return influences the evolution in population by Function Influence.In this way,the population space and belief space are connected and evolve each other through the communication protocol.More details of standard GA,DE,PSO and CA algorithms can be found in the book of Andries[23].

3.Three-layer Mixed Cultural Evolving Algorithm Framework

The proposed three-layer mixed cultural evolving algorithm framework is illustrated in Fig.2 involving three main components including a population space,a medium space and a belief space.In the following,each component is described in detail.

3.1.Basic layer:population space

In population space,an initial populationNP;j=1,2,…,n}=′|j=1,2,…,n}is generated randomly and then evolves into sub-populationsvia some evolving algorithms such as standard DE,GA and PSO algorithms.The population size(i.e.NP)in this paper is selected between the integer interval[40,100].It should be noted that i represents the number of individuals in each population,j(=1,2,…,n)is the dimension of each decision vector and ngis the number of descendant populations from the initial population.Then,these sub-populations accept knowledge from their upper space to improve themselves and evolve into several updated groups of sub-populations.

Fig.2.Three-layer mixed cultural evolving algorithm framework.

3.2.Medium layer:medium space

The population of medium space is developed by selecting a definite number of the best individuals from nggroups of species in population space.In Fig.2,is a population evolved from nggroups of species in population space.This population can be designed to evolve further via either accepting the knowledge of belief space which is described below,or using its own evolving rules,or using both of them.In this paper,the population of medium space is designed to evolve further via accepting the knowledge of belief space.

3.3.Top layer:belief space

The belief space is the knowledge management repository in which the entire population stores and distributes its former successful experiences.In belief space,effective knowledge is extracted and updated from the population in medium space using Function Accept(),which is defined in Section 3.4.The knowledge in belief space can be divided into five categories:situational knowledge,normative knowledge,domain knowledge,history knowledge and topographical knowledge.Since the domain knowledge and history knowledge are not used in this paper,the other three types of knowledge are explained below.

3.3.1.Situational knowledge

The situational knowledge is used to keep the best individual(an ndimension decision vector S)during the entire evolution process in three-layer mixed cultural evolving algorithms.It suggests all individuals to evolve toward elite individuals.Let S(t)denotes the best individual in the current evolving circle t.For each evolving circle t,it evaluates all individuals and updates S with the best one as given in Eq.(1).

where ybest(t)is the best individual at the current iteration t in medium space,which is defined asand Fit[·]is the fitness value of ybest(t)or S(t).The fitness value is assigned to the objective function value.

3.3.2.Normative knowledge

The normative knowledge consists of a set of promising ranges.It guides the individuals in medium space to jump into the good ranges.The normative knowledge at iteration t[denoted as N(t)]is described in Eq.(2).

The intervals of yj(t)and the normative knowledge are simultaneously updated based on the following rules(for the problem in Eq.(9)):

①If the i-th individual affects the lower bound of yj(t)[i.e.,lj(t)]at the tth iteration,then lj(t)and the fitness value are updated using Eqs.(3)and(4),respectively.

②If the i-th individual produces effect on the upper bound of yj(t)[i.e.,uj(t)]at the tth iteration,then uj(t)and the fitness value are updated using Eqs.(5)and(6).

3.3.3.Topographical knowledge

3.4.Information exchange in three spaces

Several functions below are defined for information communication among the above three spaces.

3.4.1.Acceptance function

Acceptance function denoted as Accept()is designed as the fixed ratio given in Eq.(7).

where ngstands for the population number in population space.It is used to select(1/ng)%individuals from population space to medium space and also select individuals from medium space that affect belief space.In this paper,ng=5 and Accept()=20%.

3.4.2.Influence functions

Two Influence function influences,namely Influence1()and Influence2()are designed.Function Influence1()is designed to guide evolving processes of the individuals in medium space based on the elite experiences stored in Belief Space.If the i-th individual affects the lower bound or upper bound of[ y1（ t）,y2（t）,…,yn（t）]}at the tth iteration,then the rules of Function Influence1()are presented in Eq.(8).where rand(0,1)is the random number between 0 and 1.With Eq.(8)the component j of individual i[i.e.,yi,j(t)]in medium space is updated.

Function Influence2()is designed to guide the evolving processes of the individuals in population space based on the topographical knowledge.The aim of this function is to update the best individualsof population space with

3.5.3LM-CDE/CGA/CPSO algorithms

The general single-objective nonlinear constrained problem addressed in this paper is given in Eq.(9).

where X=(x1,x2,…,xn)is the decision vector in the n-dimensional real continuous space Rnand f is the objective function.

Standard DE,GA and PSO algorithms are embedded into the proposed framework to develop 3LM-CDE,3LM-CGA and 3LM-CPSO algorithms accordingly.The entire procedure for 3LM-CDE,3LM-CGA and 3LMCPSO algorithms is in Fig.3.The maximum of iris ng,which is the number of sub-populations in population space.The maximum iteration number of t(=max-iter)is designed to finish the online computing task.

3.6.Evaluation

Twenty single-objective nonlinear constrained benchmark examples are used[27,49](see Appendix A)to evaluate the proposed 3LM-CDE/3LM-CGA/3LM-CPSO algorithms.The proposed 3LM-CDE/3LM-CGA/3LM-CPSO algorithms with standard DE/GA/PSO algorithms are also compared.Table 1 presents the related parameters used in 3LM-CDE/3LM-CGA/3LM-CPSO algorithms and DE/GA/PSO algorithms.It should be noted that each function in 3LM-CDE/3LM-CGA/3LM-CPSO algorithms independently runs for 200 times.The population size in all algorithms of g01–g12 is 40 and that in g12–g20 is 100.The maximum iterations for 3LM-CDE/3LM-CGA/3LM-CPSO algorithms for twenty examples are given in Table 2.To evaluate pros and cons of those algorithms,six performance indicators[27]are employed,which are best solutions,mean solutions,worst solutions,medium solutions,standard deviation and average iterations for global optima or best known solutions.Tables 3–8 present those indicators for the proposed algorithms.Best solutions for twenty examples from 3LM-CDE/3LM-CGA/3LM-CPSO and DE/GA/PSO algorithms are illustrated in Figs.4–8,respectively.

From Table 3,it can be seen that the best solutions found for examples g01–g20 from 3LM-CDE are global optima,whereas the global optima are obtained only for Examples g01–g03,g11–g12 and g18 from 3LM–CGA,Examples g01–g03,g06,g09–g13,g15–g16 and g18 from 3LM-CPSO,Examples g01–g04,g06 and g09–g20 from DE,Examples g02–g03,g11–g12 and g18 from GA,and Examples g01–g03,g06,g11–13,g15–g16 and g18 from PSO.From Table 4,nineteen mean solutions obtained for Examples g01–g21 from 3LM-CDE are global optima compared to 1 from 3LM-CGA,9 from 3LM-CPSO,16 from DE,1 from GA and 8 from PSO.In Table 5,although the worst solutions for each example from 3LM-CDE/CGA/CPSO and DE/GA/PSO are obtained,19 worst solutions from 3LM-CDE are global optima compared to 1 global optimum from 3LM-CGA,10 from 3LM-CPSO,16 from DE,1 from GA and 9 from PSO.Similar results can be observed from Table 6 for medium solutions and Table 7 for standard deviations.

From Table 8,the average 11.72 iterations are needed to converge for Example g01 from 3LM-CDE,whereas it is 56.385 iterations from DE,85.36 iterations from 3LM-CPSO and 205.935 from PSO.The 3LM-CGA and GA algorithms cannot converge after their maximum iterations(i.e.,500).Similar results are applied to other examples except Examples g13 and g15.It should be noted that for Examples g07 and g08,all algorithms cannot converge to the global optima after their maximum iterations.

Fig.3.Entire procedure for 3LM-CDE,3LM-CGA,and 3LM-CPSO algorithms.

Table 1 Related parameters for Examples g01–g20 used in 3LM-CDE/CGA/CPSO and DE/GA/PSO algorithms

4.Single Objective Parameter Optimization for Syngas Production of the Texaco Coal-water Slurry Gasifier

For the reason of reference[17]being written in Chinese,the singleobjective parameter optimization model of it is explained in detail as follows.

The pressurization gasification process of the Texaco water-coal slurry gasifier is depicted in Fig.9.During the steady-state operation,there are three most-frequently-used operating variables which are“total oxygen”feed flowrate(denoted as xFo)from the three-stream burner port on the top of the gasifier(the oxygen stream in the outer annular tube),“central oxygen”feed flowrate(xFoc)(the oxygen flow in the central tube)and“quench water”feed flowrate(xFw)(in the mid of the gasifier).15 types of inlet variables greatly affect the yield of the syngas and its effective components(CO+H2).Since therelationships among them are too complicated to be formulated,the BP neural network soft sensor models are developed.Each of them has 18 variables:15 inlet variables and 3 outlet variables listed in Table 9.The three-stream burner port on the top of the Texaco coal-water slurry gasifier is shown in Fig.10.

Table 2 Maximum iterations for 3LM-CDE/CGA/CPSO,and DE/GA/PSO algorithms for Examples g01–g20

Table 3 Best solutions for Examples g01–g20 from 3LM-CDE/CGA/CPSO and DE/GA/PSO algorithms

4.1.Training and testing the three BP neural networks

After date preprocessing,310 groups of historical steady-state data for eighteen variables in one month are collected and normalized.Then,15 inlet variables are reduced to 8 principal variables using principal component analysis.The three BP neural networks(Fig.11)are established,each having 8 input nodes,8 hidden nodes and 1 output node.

The steps of training each of the three BP neural networks are given below:

Step 1 Initialize wij,vj,θjand r(i=1,2,…,8;j=1,2,…,8)randomly between the interval[?1,1].wijand vjare the weights of the hidden and output layer,θjand r are thresholds of the hidden and output layer.

Step 2 Select randomly a set of samples for the BP neural networks.The input of the network are the 8 principal components denoted as Q=(q1,q2,…,q8)and the objective output is oout.

Step 3 Compute the input(sj)and output(bj)of the nodes of the hidden layer using Eq.(10).

Step 4 Calculate the input l and output out of the nodes of the output layer using Eq.(11).

Table 4 Mean solutions for Examples g01–g20 from 3LM-CDE/CGA/CPSO and DE/GA/PSO algorithms

Table 5 Worst solutions for Examples g01–g20 from 3LM-CDE/CGA/CPSO and DE/GA/PSO algorithms

Step 5 Compute the error for the output node(denoted as d)using Eq.(12).

Step 6 Compute the error for nodes of the hidden layer(denoted as ej)using Eq.(13).

Step 7 Update vjand r using d and bjaccording to Eq.(14).

Step 8 Update wijand θjwith ejand qiaccording to Eq.(15).

Step 9 Calculate global relative error(Err)of the network based on Ns(=310)groups of samples using Eq.(16).ooutkis the objective output of the kth set of training samples and outkis the actual output of the kth set of training samples from the BP neural network.

Table 6 Medium solutions for Examples g01–g20 from 3LM-CDE/CGA/CPSO and DE/GA/PSO algorithms

Finally,the 3 predicted actual values for the outlet flowrate of sgngas(fFg),[CO]in the outlet syngas(fCO),and[H2]in the outlet syngas(fH2)are gained via the denormalized process in Eq.(17).

The trained values of the weights and thresholds for each BP neural network are listed in Table 10.Figs.12–14 show the testing fitting curve of the outlet flowrate of syngas,[CO]in the outlet syngas and[H2]in the outlet syngas from each BP neural network.

4.2.Single-objective operation optimization model

The single-objective optimization model for syngas production of the Texaco coal-water slurry gasifier is presented in Eq.(18).

where Yeg,Fc,Cc,and ρcare the effective gas yield,the current inlet flowrate of the coal slurry,current inlet concentration of the coal slurry and the current density of the coal slurry,respectively.fFg,fCO,and fH2denotes the outlet flowrate of syngas,the[CO]in the outlet syngas and the[H2]in the outlet syngas which are determined by the above trained three BP neural networks.xFo,xFoc,and xFware the three operating variables,representing the inlet“total oxygen”flowrate,inlet“central oxygen”flowrate and inlet flowrate of quench water.

4.3.Single-objective optimization process

At first,the actual effective gas fields(,i=1,2,…,310)from the Ns(=310)historical steady-state samples are calculated using Eq.(19)and then classified into 4 clustering centers with 4-grade working conditions:excellent(Yeg≥3657.5),good(3562.5 ≤Yeg＜3657.5),medium(3477.8 ≤Yeg＜3562.5)and poor(Yeg＜3477.8).

All parameters on the right hand side of Eq.(19)are actual values and have the same meanings with those in Table 9.If the grade of the new working condition is medium or poor,the optimization process is triggered,and else,the current status is held.

Now a group of new data is obtained and the Texaco gasifier is working in a steady-state condition.The values for those 18 variables are:M*=1.16 wt.%,A*=7.68 wt.%,Va*=41.39 wt.%,C*=49.77 wt.%,ST*=1210°C,Cc*=65.05%,Fc*=68.69 m3·h?1,Pc*=5.74 MPa,Tc*=42.91°C,xFo*=28,550.82 m3·h?1,xFoc*=4340.00 m3·h?1,Po*=7.40 MPa,To*=33.55°C,xFw*=337.55 m3·h?1,Tw*=229.45°C,Fg*=210528.09 m3·h?1,[CO]*=39.96 vol%,and[H2]*=40.28 vol%.Yeg*is 3481.37 m3·h?1from Eq.(19)and the grade of its working condition is medium.Then,the optimization process begins.

Step 1 Randomly initialize the populations of xFo,xFocand xFwwithin[15000,35000],[3000,6000]and[250,400],respectively.

Step 2 Repeat

Step 3 Obtain a new group of the 15 inlet variables(M*,A*,Va*,C*,ST*,Cc*,Fc*,Pc*,Tc*,xFo,xFoc,Po*,To*,xFw,Tw*)by replacing xFo*,xFoc*and xFw*with xFo,xFocand xFw.

Step 4 Each inlet variable in Step 3 is normalized using Eq.(20).is denoted as the kth inlet variable,is the normalized value andis the variation range of(See Table 9).

Table 7 Standard deviations for Examples g01–g20 from 3LM-CDE/CGA/CPSO and DE/GA/PSO algorithms

Table 8 Average iterations to converge for Examples g01–g20 from 3LM-CDE/CGA/CPSO and DE/GA/PSO algorithms

Step 5 Reduce the normalized inlet variables to 8 principal variables(q1*,q2*,…,q8*)using the method of Song et al.[17].

Step 6 Compute the 3 predicted actual values of the 3 populations for fFg,fCOand fH2using each BP neural network trained in Section 4.1.

Step 7 Evaluate the fitness values[i.e.,the objective function in Eq.(18)]for the 3 populations(xFo,xFocand xFw).

Step 8 Update the 3 populations using PSO(in this paper 3LMCDE/3LM-CGA/3LM-CPSO and DE/GA/PSO algorithms are used).

Step 9 Check the termination criteria.If not satisfied,go to Step 2,and else continue.

Step 10 Output the final normalizedand,and the normalized outFg,outCOand outH2.

Step 11 Output the final denormalized fFg,fCOand fH2using Eq.(17),xFo,xFoc,and xFwby Eq.(21),and Yeg.

Fig.4.Best solutions for Examples g01–g04 from 3LM-CDE,3LM-CGA,3LM-CPSO,DE,GA and PSO algorithms.

Fig.5.Best solutions for Examples g05–g08 from 3LM-CDE,3LM-CGA,3LM-CPSO,DE,GA and PSO algorithms.

Fig.6.Best solutions for Examples g09–g12 from 3LM-CDE,3LM-CGA,3LM-CPSO,DE,GA and PSO algorithms.

Fig.7.Best solutions for Examples g13–g16 from 3LM-CDE,3LM-CGA,3LM-CPSO,DE,GA and PSO algorithms.

Fig.8.Best solutions for Examples g17–g20 from 3LM-CDE,3LM-CGA,3LM-CPSO,DE,GA and PSO algorithms.

Fig.9.Schematic of the pressurization gasification process of the Texaco coal-water slurry gasifier.

Fig.10.Three-stream burner port on the top of the gasifier.

It should be pointed out that the three NLP BP neural networks are data-driven approaches in which the input and output data decide the type of the networks'parameters.Furthermore,if the trained BP neural networks are fed with a new set of data and predictions(using the trained weights)cannot provide accurate objectives compared to the target outputs(actual outputs),the three BP neural networks should be updated and training process is restarted.

The 3LM-CDE/3LM-CGA/3LM-CPSO and DE/GA/PSO algorithms use the population size of 50 and the maximum iteration number of 500.The Function Accept()for all 3LM-CDE/3LM-CGA/3LM-CPSO algorithms is 20%.In addition,the mutation factors(F)and crossover probability(C)are both 0.618 for DE and 3LM-CDE algorithms.While the probability of mutation(P.M.)is 0.318 for GA and 3LM-CGA algorithms,the probability of crossover(P.C.)is 0.852 for them.The inertia weight(denoted as w_pso)is allowed to vary from 0.2 to 0.9 for PSO and 3LM-CPSO algorithms,and the learning operators(i.e.,c1 and c2)are 2.0.The simulation results from 3LM-CDE/3LM-CGA/3LM-CPSO and DE/GA/PSO algorithms are listed in Table 11.The convergence curves from 3LMCDE/3LM-CGA/3LM-CPSO and DE/GA/PSO algorithms are depicted in Fig.15.

Fig.11.Structure of the three BP neural networks.

Table 10 Trained weights and thresholds for the three BP neural networks of Fig.11

From Table 11,the effective gas yield for syngas production from the real-world gasifier is 3481.37 with medium grade.While the effective gas yield increases to 3707.93 belonging to excellent grade from 3LM-CDE algorithm,it is only 3649.52 with good grade using DE algorithm.Using 3LM-CGA algorithm,the effective gas yield of 3696.40 belonging to excellent grade is obtained compared to 3642.69 for the effective gas yield with good grade from GA algorithm.Similarly,with 3LM-CPSO algorithm,the effective gas yield of 3702.42 belonging to excellent grade is found,but it is 3652.76 belonging to good grade with PSO algorithm.Therefore,it can be concluded that 3LM-CDE/CGA/CPSO algorithms are superior to DE/GA/PSO algorithms.

Fig.12.Fitting curve of the outlet flowrate of syngas.

Although 3LM-CDE/CGA/CPSO algorithms obtain the same grade of effective gas yield,3LM-CDE generates better effective gas yield compared to 3LM-CGA and 3LM-CPSO algorithms.In other words,3LM-CDE performs the best among 3LM-CDE/CGA/CPSO algorithms.

Meanwhile,under the condition of the other 12 inlet variables(See Table 9)being constant,the effective gas yield increases with the sum of the inlet“total oxygen”and the inlet“central oxygen”,and there exist more margins for the inlet“central oxygen”which can be further explored and tested in our future work.

Fig.13.Fitting curve of the[CO]in the outlet syngas.

Fig.14.Fitting curve of the[H2]in the outlet syngas.

Table 11 The effective gas yield for the large-scale single-objective model from the real-world and 3LM-CDE/CGA/CPSO

Fig.15.Convergence curves for the large-scale industrial model from 3LM-CDE,3LM-CGA,3LM-CPSO,DE,GA,and PSO algorithms.

5.Conclusions

In this paper,the parameter optimization problem for syngas production in a realistic Texaco coal-water slurry gasifier is studied.To solve this nonlinear constrained problem,a novel three-layer cultural algorithm evolving framework is proposed,which includes a population space,a medium space and a belief space.Standard DE,GA and PSO are embedded into this framework to develop 3LMCDE,3LM-CGA and 3LM-CPSO algorithms.The 3LM-CDE,3LM-CGA and 3LM-CPSO algorithms are evaluated using twenty singleobjective nonlinear constrained benchmark examples.The computational results demonstrate that the 3-layer cultural algorithm evolving framework effectively provides better evolving directions and succeeds in three types'standard EA algorithms.The proposed 3LM-CDE,3LM-CGA and 3LM-CPSO algorithms are then applied to a large-scale realistic case for syngas production of the Texaco coal-water slurry gasifier from a real-world chemical plant.The computing results demonstrated that this large-scale real example is successfully solved by the proposed algorithms and generated better optimal solutions compared to that using standard GA,DE and PSO.

Nomenclature

bjoutput of the nodes of the hidden layer

Cc current inlet concentration of the coal slurry

d error for the output node

Errglobal relative error

ejerror for nodes of the hidden layer

Fc current inlet flowrate of the coal slurry

Fit[·] the fitness value calculated by objective function

fFgthe predicted outlet flowrate of syngas,m3·h?1

fCOthe CO volume fraction in the outlet syngas,vol%

f H2the H2volume fraction in the outlet syngas,vol%

ir1,2,…,ng,represents the inner iteration number in algorithm

l input of the nodes of the output layer

n the dimension of each decision vector

ngthe number of sub-populations in population space

ooutkobjective output of the kth set of training samples

Out output of the nodes of the output layer

outCOthe CO volume fraction in the outlet syngas before denormalized

outFgthe predicted outlet flowrate of syngas before denormalized

outH2the H2volume fraction in the outlet syngas before denormalized

outkactual output of the kth set of training samples from the BP neural network

r threshold of the output layer

sjinput of the nodes of the hidden layer

t represents the outer iteration number in algorithm

vjweight of the output layer

wijweight of the hidden layer

xFoinlet total oxygen flowrate,m3·h?1

xFocinlet central oxygen flowrate,m3·h?1

xFwinlet flowrate of quench water,m3·h?1

Yegeffective gas yield

θjthreshold of the hidden layer

ρccurrent density of the coal slurry

Appendix A.Twenty single-objective NLP benchmark examples[27,49]

The optimal solution is x*=(1,1),where f(x*)=0.

The optimal solution is x*=(0,0,0),where f(x*)=0.

The optimal solution is x*=(0,?1),where f(x*)=3.

The optimal solution is x*=(0,0…,0),where f(x*)=0.

The optimal solution is x*=(0,0,…,0),where f(x*)=0.

The optimal solution is x*=(420.8796,420.8796,…,420.8796),where f(x*)=0.

The optimal solution is x*=(0,0,…,0),where f(x*)=0.

The optimal solution is x*=(0,0,…,0),where f(x*)=0.

The optimal solution is x*=(0,0,…,0),where f(x*)=0.

The optimal solution is x*=(0,0,…,0),where f(x*)=0.

The optimal solution is x*=(0.08983,0.7126)or(0.08983,?0.7126),where f(x*)=?1.0316285.

The optimal solution is x*=(3.142,2.275),(3.142,?2.275)or(9.425,2.475),where f(x*)=0.398.

The optimal solution is x*=(1,1,1,1,1,1,1,1,1,3,3,3),where f(x*)=?15.Constraints g1,g2,g3,g4,g5,and g6are active.

The global maximum is unknown;the best report solution is f(x*)=0.803619.Constraints g1is closed to being active(g1=10?8).

The optimal solution is x*=(78,33,29.995256025682,45,36.775812905788),where f(x*)=?30665.539.Constraints g1and g6are active.

The global solution is x*=(14.095,0.84296),where f(x*)=?6961.81388.Both constraints are active.

The global solution is x*=(2.171996,2.363683,8.773926,5.095984,0.9906548,0.9906548,1.430574,1.321644,9.828726,8.280092,8.375927),where f(x*)=24.3062091.Constraints g1,g2,g3,g4,g5,and g6are active.

The global solution is x*=(1.2279713,4.2453733),where f(x*)=0.095825.Constraints g1,g2,g3,g4,g5,and g6are active.

The global solution is x*=(2.330449,1.951372,?0.477541,4.365642,?0.6244870,1.038131,1.594227),where f(x*)=680.6300573.

The global solution is x*=(579.19,1360.13,5109.92,182.0174,295.5985,217.9799,286.40,395,5979),where f(x*)=7049.25.Constraints g1,g2,and g3are active.