research article steady modeling for an ammonia synthesis ... · 2. ammonia synthesis production...

Research ArticleSteady Modeling for an Ammonia Synthesis Reactor Based ona Novel CDEAS-LS-SVM Model

Zhuoqian Liu,1 Lingbo Zhang,1 Wei Xu,2 and Xingsheng Gu1

1 Key Laboratory of Advanced Control and Optimization for Chemical Process, Ministry of Education, Shanghai 200237, China2 Shanghai Electric Group Co. Ltd., Central Academe, Shanghai 200070, China

Correspondence should be addressed to Xingsheng Gu; [email protected]

Received 6 December 2013; Accepted 5 February 2014; Published 18 March 2014

Academic Editor: Huaicheng Yan

Copyright © 2014 Zhuoqian Liu et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

A steady-state mathematical model is built in order to represent plant behavior under stationary operating conditions. A novelmodeling using LS-SVR based on Cultural Differential Evolution with Ant Search is proposed. LS-SVM is adopted to establish themodel of the net value of ammonia.Themodelingmethod has fast convergence speed and good global adaptability for identificationof the ammonia synthesis process.The LS-SVRmodel was established using the above-mentionedmethod. Simulation results verifythe validity of the method.

1. Introduction

Ammonia is one of the important chemicals that has innu-merable uses in a wide range of areas, that is, explosivematerials, pharmaceuticals, polymers, acids and coolers,particularly in synthetic fertilizers. It is produced worldwideon a large scale with capacities extending to about 159milliontons at 2010. Generally, the average energy consumption ofammonia production per ton is 1900KG of standard coal inChina, which is much higher than the advanced standard of1570KG around the world. At the same time, the haze andparticulate matter 2.5 has been serious exceeded in big citiesin China at recent years, and one of the important reasons isthe emission of coal chemical factories. Thus, an economicpotential exists in energy consumption of the ammoniasynthesis as prices of energy rise and reduce the ammoniasynthesis pollution to protect the environment. Ammoniasynthesis process has the characteristics of nonlinearity,strong coupling, large time-delay and great inertia load, andso forth. Steady-state operation-optimization can be a reliabletechnique for output improvement and energy reductionwithout changing any devices.

The optimization of ammonia synthesis process highlyrelies on the accurate system model. To establish an appro-priate mathematical model of ammonia synthesis process is a

principal problem of operation optimization. It has receivedconsiderable attention since last century. Heterogeneoussimulation models imitating different types of ammonia syn-thesis reactors have been developed for design, optimizationand control [1]. Elnashaie et al. [2] studied the optimizationof an ammonia synthesis reactor which has three adiabaticbeds.The optimal temperature profile was obtained using theorthogonal collocation method in the paper. Pedemera et al.[3] studied the steady state analysis and optimization of aradial-flow ammonia synthesis reactor.

The above study indicated that both the productive capac-ity and the stability of the ammonia reactor are influenced bythe cold quench and the feed temperature significantly. Babuand Angira [4] described the simulation and optimizationdesign of an auto-thermal ammonia synthesis reactor usingQuasi-Newton and NAG subroutine method. The optimaltemperature trajectory along the reactor and optimal flowsthroughput 3.3% additional ammonia production. Sadeghiand Kavianiboroujeni [1] evaluated the process behavior ofan industrial ammonia synthesis reactorby one-dimensionalmodel and two-dimensional model; genetic algorithm (GA)was applied to optimize the reactor performance in varyingits quench flows. FromThe above literatures we can find thatmost models are built based on thermodynamic, kinetic andmass equilibria calculations. It is very difficult to simulate the

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014, Article ID 168371, 18 pageshttp://dx.doi.org/10.1155/2014/168371

2 Mathematical Problems in Engineering

specific internal mechanism because a lot of parameters areunknown in real industrial process.

In order to achieve the required accuracy of the model,some researches focus on the novel modeling methodscombining some heuristic methods such as ANN (ArtificialNeural Network), LS-SVM (Least Squares Support VectorMachine) with Evolutionary Algorithm, for example, geneticalgorithm, ant colony optimization (ACO), particle swarmoptimization (PSO), differential evolution (DE), and so forth.DE is one of themost popular algorithms for this problemandhas been applied in many fields. Sacco and Hendersonb [5]introduced a variant of the differential evolution algorithmwith a new mutation operator based on a topographicalheuristic, and used it to solve the nuclear reactor core designoptimization problem. Rout et al. [6] proposed a simple butpromising hybrid prediction model by suitably combiningan adaptive autoregressive moving average architecture anddifferential evolution for forecasting of exchange rates. Ozcanet al. [7] carried out the cost optimization of an air coolingsystem by using Lagrange multipliers method, differentialevolution algorithm and particle swarm optimization forvarious temperatures andmass flow rates.The results showedthat the method gives high accuracy results within a shorttime interval. Zhang et al. [8] proposed a hybrid differentialevolution algorithm for the job shop scheduling problemwithrandom processing times under the objective of minimizingthe expected total tardiness. Arya and Choube [9] describeda methodology for allocating repair time and failure rates tosegments of a meshed distribution system using differentialevolution technique. Xu et al. [10] proposed a model ofammonia conversion rate by LS-SVM and a hybrid algorithmof PSO and DE is described to identify the hyper-parametersof LS-SVM.

To describe the relationship between net value of ammo-nia in ammonia synthesis reactor and the key operationalparameters, least squares support vectormachine is employedto build the structure of the relationship model, in which anovel algorithm called CDEAS is proposed to identify theparameters.The experiment results showed that the proposedCDEAS-LS-SVM optimizing model is very effective of beingused to obtain the optimal operational parameters of ammo-nia synthesis converter.

The remaining of the paper is organized as follows.Section 2 describes the ammonia synthesis production pro-cess. Section 3 proposes a novel Cultural Differential Evolu-tion with Ant Colony Search (CDEAS) algorithm. Section 4constructs a model using LS-SVM based on the proposedCDEAS algorithm. Section 5 presents the experiments andcomputational results and discussion. Finally, Section 6 sum-marizes the above results and presents several problemswhich remain to be solved.

2. Ammonia Synthesis Production Process

A normal ammonia production flow chart includes thesynthesis gas production, purification, gas compression, andammonia synthesis. Ammonia synthesis loop is one of themost critical units in the entire process. The system has

been realized by LuHua Inc., a medium fertilizers factory ofYanKuang Group, China.

Figure 1 represents a flow sheet for the ammonia syn-thesis process. The ammonia synthesis reactor is a one-axialflow and two-radial flow three-bed quench-type unit [11].Hydrogen-nitrogen mixture is reacted in the catalyst bedunder high temperature and pressure. The temperature inthe reactor is sustained by the heat of reaction because thereaction is exothermic [1].The reaction of ammonia synthesisprocess contains

3

2H2+1

2N2 NH

3+Q. (1)

The reaction is limited by the unfavorable position ofthe chemical equilibrium and by the low activity of thepromoted iron catalysts with high pressure and temperature[12]. In general, no more than 20% of the synthesis gas isconverted into ammonia per pass even at high pressure of30MPa [12]. As the ammonia reaction is exothermic, it isnecessary for removing the heat generated in the catalyst bedby the progress of the reaction to obtain a reasonable overallconversion rate as same as to protect the life of the catalyst[13]. The mixture gas from the condenser is divided into twoparts Q1 and Q2 to go to the converter. The first cold shotQ1 is recirculated to the annular space between the outershell reactor and catalyst bed from the top to the bottomto refrigerate the shell and remove the heat released by thereaction. Then the gas Q1 from the bottom of reactor goesthrough the preheater and is heated by the counter-currentflowing reacted gas from waste heat boiler. Q1 gas is dividedinto 4 cold quench gas (q1, q2, q3, and q4) and Q2 gas formixing with the gas between consecutive catalyst beds toquench the hot spots before entry to the subsequent catalystbeds. The hot spot temperatures (TIRA705, TIRA712N, andTIRA714) represent the highest reaction temperatures at eachstage of the catalyst bed.

Figure 2 represents the ammonia synthesis unit. Thereacted gas including N

2, H2, NH3, and inert gas after reactor

passes through the waste heat boiler. Then it goes throughthe preheater and the water cooler to be further cooled. Partof the ammonia is condensed and separated by ammoniaseparator I. Inert gas from the ammonia synthesis loop areejected by purge gas from separator to prevent accumulationof inert gas in the system. The fresh feed gas is producedby the Texaco coal gasification air separation section, aprocess that converts the Coal Water Slurry into synthesisgas for ammonia. The fresh gas consists of hydrogen andnitrogen in stoichiometric proportions of 3 : 1 approximatelyand mixes with small amounts of argon and methane. Thefresh gas which passes compressor is compounded with therecycle gas which comes from the circulator, and then themixture goes through oil separator and condenser. Mixturegas is further cooled by liquid ammonia and goes throughammonia separator II to separate the partial liquid ammonia,and then it goes out with very few ammonia. The liquidammonia from ammonia separator I and separator II flowsto the liquid ammonia jar. Mixture is heated in ammoniacondenser to about 25∘C and flows to the reactor and thewhole cycle starts again.

Mathematical Problems in Engineering 3

Ammoniareactor

CirculatorWaste heat

boiler Preheater Water cooler Ammonia separator I

Oil separator

Condenser

Ammonia cooler

Synthesis gas

Evaporator

Ammonia separator II

Hydrogen recovery unit

Ammonia recovery unit

TIRA705

TIR712N

TIRA714

AR701

AR701-4

FIR705 703

PI

725TI

Liquid ammonia

Compressor

Figure 1: Ammonia synthesis system.

Preheater

Q2

Preheaterq1

q4

q3

q2

Waste heat boiler

Q1

Q 1

I radial bed

II radial bed

Inter-changer

Axial bedFIR704

703

702

FIR705

FIR

FIR

Figure 2: The ammonia synthesis unit.

3. Proposed Cultural Differential Evolutionwith Ant Search Algorithm

3.1. Differential Evolution Algorithm. Evolutionary Algo-rithms, which are inspired by the evolution of species, havebeen adopted to solve a wide range of optimization problemssuccessfully in different fields. The primary advantage ofEvolutionaryAlgorithms is that they just require the objectivefunction values, while properties such as differentiability andcontinuity are not necessary [14].

Differential evolution, proposed by Storn and Price, is afast and simple population based stochastic search technique[15]. DE employs mutation, crossover, and selection opera-tions. It focuses on differential vectors of individuals with thecharacteristics of simple structure and rapid convergence.Thedetailed procedure of DE is presented below.

(1) Initialization. In a 𝐷-dimension space, NP parametervectors so-called individuals cover the entire search spaceby uniformly randomizing the initial individuals within thesearch space constrained by the minimum and maximumparameter bounds𝑋min and𝑋max:

𝑥0

𝑖,𝑑= 𝑥0

min,𝑑 + rand (0, 1) (𝑥0

max,𝑑 − 𝑥0

min,𝑑) 𝑗 = 1, 2, . . . , 𝐷.

(2)


(2)Mutation.DE employs themutation operation to produceamutant vector 𝑢𝑡

𝑖𝑑called target vector corresponding to each

individual 𝑥𝑡𝑖𝑑after initialization. In iteration 𝑡, the mutant

vector 𝑢𝑡𝑖𝑑

of individual 𝑥𝑡𝑖𝑑

can be generated according tocertain mutation strategies. Equations (3)–(7) indicate themost frequent mutation strategies version, respectively:

DE/rand/1 𝑢𝑡𝑖𝑑= 𝑋𝑡

𝑟1𝑖,𝑑+ 𝐹 (𝑋

𝑡

𝑟2𝑖,𝑑− 𝑋𝑡

𝑟3𝑖,𝑑) , (3)

DE/rand/2 𝑢𝑡𝑖𝑑= 𝑋𝑡

𝑟1𝑖,𝑑+ 𝐹 (𝑋

𝑡


𝑟3𝑖,𝑑)

+ 𝐹 (𝑋𝑡


𝑟5𝑖,𝑑) ,

(4)

DE/best/1 𝑢𝑡𝑖𝑑= 𝑋𝑡

best,𝑑 + 𝐹 (𝑋𝑡


𝑟2𝑖,𝑑) , (5)

DE/best/2 𝑢𝑡𝑖𝑑= 𝑋𝑡

best,𝑑 + 𝐹 (𝑋𝑡


𝑟2𝑖,𝑑)

+ 𝐹 (𝑋𝑡


𝑟4𝑖,𝑑) ,

(6)

DE/rand-to-best/1 𝑢𝑡𝑖𝑑= 𝑋𝑡

𝑖,𝑑+ 𝐹 (𝑋

𝑡

best,𝑑 − 𝑋𝑡

𝑖,𝑑)

+ 𝐹 (𝑋𝑡


𝑟2𝑖,𝑑) ,

(7)

where 𝑟1𝑖, 𝑟2𝑖, 𝑟3𝑖, 𝑟4𝑖, and 𝑟5𝑖are mutually exclusive integersrandomly generated within the range [1,NP] which shouldnot be 𝑖. 𝐹 is the mutation factor for scaling the differencevector, usually bounded in [0, 2]. 𝑋𝑡best is the best individualwith the best fitness value at generation 𝑡 in the population.

(3) Crossover. The individual 𝑋𝑡𝑖and mutant vector 𝑢𝑡

𝑖are

hybridized to compose the trial vector 𝑦𝑡𝑖after mutation

operation. The binomial crossover is adopted by the DE inthe paper, which is defined as

𝑦𝑡

𝑖𝑑= {𝑢𝑡

𝑖𝑑if rand ≤ 𝐶

𝑅or 𝑖 = 𝑖rand

𝑋𝑡

𝑖,𝑑otherwise,

(8)

where rand is a random number between in 0 and 1 dis-tributed uniformly. The crossover factor 𝐶

𝑅is a probability

rate within the range 0 and 1, which influences the tradeoffbetween the ability of exploration and exploitation. 𝑖rand is aninteger chosen randomly in [1, 𝐷]. To ensure that the trialvector (𝑦𝑡

𝑖) differs from its corresponding individual (𝑋𝑡

𝑖) by

at least one dimension, 𝑖 = 𝑖rand is recommended.

(4) Selection. When a newly generated trial vector exceedsits corresponding upper and lower bounds, it is reinitializedwithin the presetting range uniformly and randomly. Thenthe trial individual𝑦𝑡

𝑖is comparedwith the individual𝑋𝑡

𝑖, and

the one with better fitness is selected as the new individual inthe next iteration:

𝑋𝑡+1

𝑖𝑑= {𝑦𝑡

𝑖𝑑if 𝑓 (𝑦𝑡

𝑖𝑑) ≤ 𝑓 (𝑋

𝑡

𝑖,𝑑)

𝑋𝑡

𝑖,𝑑otherwise.

(9)

(5) Termination. All above three evolutionary operationscontinue until termination criterion is achieved, such asthe evolution reaching the maximum/minimum of functionevaluations.

As an effective and powerful random optimizationmethod, DE has been successfully used to solve real worldproblems in diverse fields both unconstrained and con-strained optimization problems.

3.2. Cultural Differential Evolution with Ant Search. As wementioned in Section 3.1, mutation factor 𝐹, mutation strate-gies, and crossover factor 𝐶

𝑅have great influence on the bal-

ance ofDE’s exploration and exploitation ability.𝐹decides theamplification of differential variation; 𝐶

𝑅is used to control

the possibility of the crossover operation; mutation strategieshave great influence on the results of mutation operation. Insome literatures 𝐹, 𝐶

𝑅, and mutation strategies are defined in

advance or varied by some specific regulations. But the factors𝐹,𝐶𝑅, and strategies are very difficult to choose since the prior

knowledge is absent. Therefore, Ant Colony Search is usedto search the suitable combination of 𝐹, 𝐶

𝑅, and mutation

strategies adaptively to accelerate the global search. Someresearchers have found an inevitable relationship betweenthe parameters (𝐹, 𝐶

𝑅, and mutation strategies) and the

optimization results of DE [16–18]. However, the approachesabove are not applying the most suitable 𝐹, 𝐶

𝑅, and mutation

strategies simultaneously.In this paper, based on the theory of Cultural Algorithm

and Ant Colony Optimization (ACO), an improved Cul-tural Differential Algorithm incorporation with Ant ColonySearch is presented. In order to accelerate searching out theglobal solution, the Ant Colony Search is used to searchthe optimal combination of 𝐹 and 𝐶

𝑅in subpopulation 1 as

well as mutation strategy in subpopulation 2. The frameworkof Cultural Differential Evolution with Ant Search is brieflydescribed in Figure 3.

3.2.1. Population Space. The population space is divided intotwo parts: subpopulation 1 and subpopulation 2. The twosubpopulations contain equal number of the individuals.

In subpopulation 1, the individual is set as ant at each gen-eration. 𝐹 and 𝐶

𝑅are defined to be the values between [0, 1],

𝐹 ∈ {0.1×𝑖}, 𝑖 = 1, 2, . . . , 10 and𝐶𝑅∈ {0.1×𝑖}, 𝑖 = 1, 2, . . . , 10.

Each of the ants chooses a combination of𝐹 and𝐶𝑅according

to the information which is calculated by the fitness functionof ants. During search process, the information gathered bythe ants is preserved in the pheromone trails 𝜏. By exchanginginformation according to pheromone, the ants cooperatewitheach other to choose appropriate combination of 𝐹 and 𝐶

𝑅.

Then ant colony renews the pheromone trails of all ants.Then, the pheromone trail 𝜏

𝑚𝑛is updated in the following

equation:

𝜏𝑚𝑛(𝑡 + 1) = (1 − 𝜌

1) 𝜏𝑚𝑛(𝑡) +

subpopulation1

∑

𝑖=1

Δ𝜏𝑖

𝑚𝑛(𝑡) ,

(10)

where 0 ≤ 𝜌1< 1 means the pheromone trail evaporation

rate, 𝑛 = 1, 2, . . . , 10,𝑚 = 1, 2; 1st parameter represents 𝐹 and


Subpopulation 1 Subpopulation 2

Belief space

Influencefunction

Acceptancefunction

Select Performancefunction

Ant Search of mutation strategy

Knowledge exchange

Population space

Ant Search ofF and CR

(situational knowledge and normative knowledge)

Figure 3: The framework of CDEAS algorithm.

0.1 0.2 0.3 1.0

0.1 0.2 0.3 1.0

𝜏1,1 𝜏1,2 𝜏1,3 𝜏1,10

F

F

CR

CR

𝜏2,1 𝜏2,2 𝜏2,3 𝜏2,10

0.1

1.0

0.2

0.3

...

0.1

1.0

0.2

0.3

...

· · ·

· · ·

· · ·

· · ·

Figure 4: Relationship between pheromone and ant paths of 𝐹, 𝐶𝑅.

2nd parameter represents 𝐶𝑅; Δ𝜏𝑖𝑚𝑛(𝑡) is the quantity of the

pheromone trail of ant 𝑖,

Δ𝜏𝑖

𝑚𝑛(𝑡)

=

{{{{

{{{{

{

1 if 𝑖 ∈ 𝑋𝑚𝑛

and fitness (𝑦𝑡𝑖) < fitness (𝑥best𝑡) ,

0.5 if 𝑖 ∈ 𝑋𝑚𝑛

and fitness (𝑥best𝑡) < fitness (𝑦𝑡𝑖)

and fitness (𝑦𝑡𝑖) < fitness (𝑥𝑡

𝑖) ,

0 otherwise,(11)

where 𝑋𝑚𝑛

is the ant group that chooses 𝑛th value as theselection of𝑚th parameter;𝑥best𝑡 denotes the best individualof ant colony till 𝑡th generation.

In order to prevent the ants from being limited to oneant path and improve the possibility of choosing other paths

considerably, the probability of each ant chooses 𝑛th value of𝑚th parameter (𝐹 and 𝐶

𝑅) in Figure 4 is set by

𝑝𝑚𝑛(𝑡) =

{{

{{

{

𝜏𝑖

𝑚𝑛(𝑡)

∑𝑛𝜏𝑚𝑛(𝑡)

if rand1< 𝑃𝑠

rand2

otherwise.(12)

Figure 4 illustrates the relationship between pheromonematrix and ant path of 𝐹 and 𝐶

𝑅, where 𝑃

𝑠is a constant

which is defined as selection parameter and rand1and rand

2

are two random values which are uniformly distributed in[0, 1]. Selection of the values of 𝐹 and 𝐶

𝑅depends on the

pheromone of each path. According to the performance ofall the individuals, the individual is chosen by the mostappropriate combination of 𝐹 and 𝐶

𝑅in each generation.

In subpopulation 2, the individual is set as ant at eachgeneration. Mutation strategies which are listed at (3)–(7) are


Mutation strategy

Mutation strategy

DE/rand/1 DE/rand/2 DE/best/1 DE/best/2 DE/rand-to-best/1

0.2 0.4 0.6 0.8 1

𝜀1 𝜀2 𝜀3 𝜀4 𝜀5

0.4

0.6

0.2

1.0

...

Figure 5: Relationship between and ant paths of mutation strategy.

defined to be of the values {0.2, 0.4, 0.6, 0.8, 1.0}, respectively.For example, 0.2 means the first mutation strategy equation(3) is selected. Each of the ants chooses a mutation strategyaccording to the informationwhich is calculated by the fitnessfunction of ants. During search process, the informationgathered by the ants is preserved in the pheromone trails 𝜀.By exchanging information according to pheromone, the antscooperate with each other to choose appropriate mutationstrategy. Then ant colony renews the pheromone trails of allants.

Then, the pheromone trail 𝜀 is updated in the followingequation:

𝜀𝑘(𝑡 + 1) = (1 − 𝜌

2) 𝜀𝑘(𝑡) +

subpopulation2

∑

𝑖=1

Δ𝜀𝑖

𝑘(𝑡) , (13)

where 0 ≤ 𝜌2< 1 means the pheromone trail evaporation

rate and Δ𝜀𝑖𝑘(𝑡) is the quantity of the pheromone trail of ant 𝑖,

𝜀𝑖

𝑘(𝑡)

=

{{{{

{{{{

{

1 if 𝑖 ∈ 𝑋𝑘and fitness (𝑦𝑡

𝑖) < fitness (𝑥best𝑡) ,

0.5 if 𝑖 ∈ 𝑋𝑘and fitness (𝑥best𝑡) < fitness (𝑦𝑡

𝑖)

and fitness (𝑦𝑡𝑖) < fitness (𝑥𝑡

𝑖) ,

0 otherwise,(14)

where 𝑋𝑘is the ant group that chooses 𝑘th value as the

selection of parameter; 𝑥best𝑡 denotes the best individual ofant colony till 𝑡th generation.

In order to prevent the ants from being limited to oneant path and improve the possibility of choosing other paths

considerably, the probability of each ant choosing 𝑛th valueof𝑚th parameter (mutation strategies) is set by

𝑝𝑘(𝑡) =

{{

{{

{

𝜀𝑖

𝑘(𝑡)

∑𝑛𝜀𝑘(𝑡)

if rand3< 𝑃

𝑠

rand4

otherwise,(15)

where𝑃𝑠is a constant which is defined as selection parameter

and rand3and rand

4are two random values which are

uniformly distributed in [0, 1]. Selection of the values ofmutation strategies depends on the pheromone of each path.According to the performance of all the individuals, theindividual is chosen by the most appropriate combination ofmutation strategies in each generation.

Figure 5 illustrates the relationship between pheromonematrix and ant path of mutation strategies.

3.2.2. Belief Space. In our approach, the belief space isdivided into two knowledge sources, situational knowledgeand normative knowledge.

Situational knowledge consists of the global best exem-plar 𝐸 which is found along the searching process andprovides guidance for individuals of population space. Theupdate of the situational knowledge is done if the bestindividual found in the current populations space is betterthan 𝐸.

The normative knowledge contains the intervals thatdecide the individuals of population space where to move. 𝑙

𝑖

and 𝑢𝑖are the lower and upper bounds of the search range

in population space. 𝐿𝑖and 𝑈

𝑖are the value of the fitness

function associated with that bound. If the 𝑙𝑖and 𝑢

𝑖are

updated, the 𝐿𝑖and 𝑈

𝑖must be updated too.


𝑙𝑖and 𝑢

𝑖are set by

𝑙𝑖= {𝑥𝑖,min, if 𝑥𝑖,min < 𝑙𝑖 or 𝑓 (𝑥𝑖,min) < 𝐿 𝑖𝑙𝑖

otherwise,

𝑢𝑖= {𝑥𝑖,max, if 𝑥𝑖,max > 𝑢𝑖 or 𝑓 (𝑥𝑖,max) > 𝑈𝑖𝑢𝑖

otherwise.

(16)

3.2.3. Acceptance Function. Acceptance function controls theamount of good individuals which impact on the update ofbelief space [19]. In this paper, 30% of the individuals inthe belief space are replaced by the good ones in populationspace.

3.2.4. Influence Function. In the CDEAS, situational knowl-edge and normative knowledge are involved to influence eachindividual in the population space, and then population spaceis updated.

The individuals in population space are updated in thefollowing equation:

𝑥𝑡+1

𝑖,𝑑=

{{{{{{{{{{{{{{{{{{{

{{{{{{{{{{{{{{{{{{{

{

𝑥𝑡

𝑖,𝑑+𝑁 (0.5, 0.3) ∗ (𝑋

𝑡


𝑟3𝑖,𝑑)× rand,

if 𝑥𝑡𝑖,𝑑≤ 𝐸𝑗, 𝑥𝑡

𝑖,𝑑≥ 𝑢𝑖,

𝑥𝑡

𝑖,𝑑−𝑁 (0.5, 0.3) ∗ (𝑋

𝑡



if 𝑥𝑡𝑖,𝑑> 𝐸𝑗, 𝑥𝑡

𝑖,𝑑< 𝑢𝑖,

𝑋𝑡

𝑟1𝑖,𝑑+𝑁 (0.5, 0.3) ∗ (𝑢

𝑖− 𝑋𝑡


if 𝑥𝑡𝑖,𝑑≤ 𝐸𝑗, 𝑥𝑡

𝑖,𝑑≥ 𝑙𝑖,

𝑋𝑡

𝑟1𝑖,𝑑−𝑁 (0.5, 0.3) ∗ (𝑙

𝑖− 𝑋𝑡


if 𝑥𝑡𝑖,𝑑> 𝐸𝑗, 𝑥𝑡

𝑖,𝑑> 𝑙𝑖,

𝑥𝑡+1

𝑖,𝑑=

{{{

{{{

{

𝑋𝑡

𝑟1𝑖,𝑑+ 𝐹 ∗ (𝑢

𝑖− 𝑋𝑡

𝑟1𝑖,𝑑) × rand, if 𝑥

𝑖,𝑑> 𝑙𝑖

𝑋𝑡

𝑟1𝑖,𝑑− 𝐹 ∗ (𝑋

𝑡

𝑟1𝑖,𝑑− 𝑙𝑖) × rand, if 𝑥

𝑖,𝑑< 𝑢𝑖

𝑋𝑡

𝑟1𝑖,𝑑+ 𝐹 ∗ (𝑢

𝑖− 𝑙𝑖) × rand, if 𝑙

𝑖< 𝑥𝑖,𝑑< 𝑢𝑖,

(17)where 𝐹 is a constant of 0.2.

3.2.5. Knowledge Exchange. After 𝑡 steps, the 𝐹 and 𝐶𝑅

of subpopulation 2 are replaced by the suitable 𝐹 and 𝐶𝑅

calculated by subpopulation 1 and the mutation strategyof subpopulation 1 is displaced by the suitable mutationstrategy calculated by subpopulation 2 simultaneously. Sothe 𝐹 and 𝐶

𝑅and mutation strategy are varying in the two

subpopulations to enable the individuals to converge globallyand fast.

3.2.6. Procedure of CDEAS. The procedure of CDEAS isproposed as follows.

Step 1. Initialize the population spaces and the belief spaces;the population space is divided into subpopulation 1 andsubpopulation 2.

Step 2. Evaluate each individual’s fitness.

Step 3. To find the proper 𝐹, 𝐶𝑅, and mutation strategy, the

Ant Colony Search strategy is used in subpopulation 1 andsubpopulation 2, respectively.

Step 4. According to acceptance function, choose good indi-viduals from subpopulation 1 and subpopulation 2, and thenupdate the normative knowledge and situational knowledge.

Step 5. Adopt the normative knowledge and situationalknowledge to influence each individual in population spacethrough the influence functions, and generate two corre-sponding subpopulations.

Step 6. Select individuals from subpopulation 1 and subpop-ulation 2, and update the belief spaces including the twoknowledge sources for the next generation.

Step 7. If the algorithm reaches the given times, exchangethe knowledge of 𝐹, 𝐶

𝑅, and mutation strategy between

subpopulation 1 and subpopulation 2; otherwise, go to Step 8.

Step 8. If the stop criteria are achieved, terminate the itera-tion; otherwise, go back to Step 2.

3.3. Simulation Results of CDEAS. The proposed CDEASalgorithm is compared with original DE algorithm. To getthe average performance of the CDEAS algorithm 30 runson each problem instance were performed and the solutionquality was averaged.The parameters of CDEAS and originalDE algorithm are set as follows: the maximum evolutiongeneration is 2000; the size of the population is 50; for originalDE algorithm 𝐹 = 0.3 and 𝐶

𝑅= 0.5; for CDEAS, the size

of both two subpopulations is 25; the initial 𝐹 and 𝐶𝑅are

randomly selected in (0, 1) and the initial mutation strategyis DE/rand/1; the interval information exchanges between thetwo subpopulations 𝑡 is 50 generations; the thresholds 𝑃

𝑠=

𝑃

𝑠= 0.5 and 𝜌

1= 𝜌2= 0.1.

To illustrate the effectiveness and performance of CDEASalgorithm for optimization problems, a set of 18 representa-tive benchmark functions which were listed in the appendixwere employed to evaluate them in comparison with originalDE. The test problems are heterogeneous, nonlinear, andnumerical benchmark functions and the global optimum for𝑓2, 𝑓4, 𝑓7, 𝑓9, 𝑓11, 𝑓13, and 𝑓

15is shifted. Functions 𝑓

1∼𝑓7are

unimodal and functions𝑓8∼𝑓18are multimodal.The detailed

principle of functions is presented in [11]. The comparisonsresults of CDEAS and original DE algorithm are shown inTable 4 of the appendix. The experimental results of originalDE and CDEAS algorithm on each function are listed inTable 1. Mean, best, worst, std., success rate, time representthe mean minimum, best minimum, worst minimum, thestandard deviation of minimum, the success rate, and theaverage computing time in 30 trials, respectively.

From simulation results of Table 1 we can obtain thatCDEAS reached the global optimum of 𝑓

2and 𝑓7in all trials,

and the success rate reached 100% of functions 𝑓1, 𝑓2, 𝑓3, 𝑓4,

𝑓6, 𝑓7, and 𝑓

18. For most of the test functions, the success


Table 1: The comparison results of the CDEAS algorithm and original DE algorithm.

Original DE CDEASSphere function 𝑓

1

Best 1.1746 × 10−65 5.0147 × 10−79

Worst 1.0815 × 10−23 9.3244 × 10−75

Mean 3.6052 × 10−25 1.6390 × 10−75

Std. 1.9746 × 10−24 2.2315 × 10−75

Success rate (%) 100 100Times (s) 1.8803 14.6017

Shifted sphere function 𝑓2

Best 0 0Worst 8.0779 × 10−28 0Mean 3.3658 × 10−29 0Std. 1.5078 × 10−28 0Success rate (%) 100 100Times (s) 2.1788 18.1117

Schwefel’s Problem 1.2 𝑓3

Best 2.4386 × 10−65 3.0368 × 10−78

Worst 2.4820 × 10−22 9.2902 × 10−73

Mean 8.2736 × 10−24 7.2341 × 10−74

Std. 4.5316 × 10−23 2.0187 × 10−73


Shifted Schwefel’s Problem 1.2 𝑓4

Best 0 0Worst 5.6545 × 10−27 3.4331 × 10−27

Mean 2.0868 × 10−28 1.8848 × 10−28

Std. 1.0323 × 10−27 7.9813 × 10−28


Rosenbrock’s function 𝑓5

Best 13.0060 5.2659Worst 166.1159 139.1358Mean 70.9399 39.4936Std. 40.0052 31.2897Success rate (%) 86.67 96.67Times (s) 1.9594 16.7233

Schwefel’s Problem 1.2 with noise in fitness 𝑓6

Best 3.1344 × 10−39 3.98838 × 10−49

Worst 3.61389 × 10−36 1.6124 × 10−43

Mean 5.7744 × 10−37 7.4656 × 10−45

Std. 9.5348 × 10−37 2.9722 × 10−44


Shifted Schwefel’s Problem 1.2 with noise in fitness 𝑓7

Best 0 0Worst 0 0Mean 0 0


Table 1: Continued.

Original DE CDEASStd. 0 0Success rate (%) 100 100Times (s) 3.3374 28.5638

Ackley’s function 𝑓8

Best 7.1054 × 10−15 3.5527 × 10−15

Worst 4.8999 × 10−7 1.3404Mean 1.6332 × 10−8 0.1763Std. 8.9457 × 10−8 0.4068Success rate (%) 100 83.33Times (s) 2.4820 20.9353

Shifted Ackley’s function 𝑓9

Best 7.1054 × 10−15 3.5527 × 10−15

Worst 0.9313 0.9313Mean 0.0310 0.0620Std. 0.1700 0.2362Success rate (%) 96.67 93.33Times (s) 2.7337 21.6841

Griewank’s function 𝑓10

Best 0 0Worst 0.0367 0.0270Mean 0.0020 0.0054Std. 0.0074 0.0076Success rate (%) 90 56.67Times (s) 2.535 20.7793

Shifted Griewank’s function 𝑓11

Best 0 0Worst 0.0319 0.0343Mean 0.0056 0.0060Std 0.0089 0.0088Success rate (%) 80 76.67Times (s) 2.7768 22.8541

Rastrigin’s function 𝑓12

Best 8.1540 1.9899Worst 35.5878 12.9344Mean 20.3594 6.5003Std. 6.3072 2.6612Success rate (%) 3.33 90Times (s) 2.7264 22.3237

Shifted Rastrigin’s function 𝑓13


Noncontiguous Rastrigin’s function 𝑓14

Best 20.7617 3.9949Worst 29.9112 11.9899Mean 25.4556 8.1947


Table 1: Continued.

Original DE CDEASStd. 2.9078 2.2473Success rate (%) 0 86.67Times (s) 3.1663 25.5374

Shifted noncontiguous Rastrigin’s function 𝑓15

Best 0 0Worst 16 6Mean 6.7666 1.5333Std. 3.4509 1.8519Success rate (%) 40 96.67Times (s) 3.3374 25.9430

Schwefel’s function 𝑓16

Best 118.4387 236.8770Worst 710.6303 1362.0521Mean 357.61725 676.4166Std. 144.41244 324.2317Success rate (%) 90 40Times (s) 2.5028 19.0009




Best 1.2706 × 10−35 8.5946 × 10−45

Worst 1.6842 × 10−34 1.8362 × 10−42

Mean 6.1883 × 10−35 2.6992 × 10−43

Std. 3.4937 × 10−35 4.6257 × 10−43


rate of CDEAS is higher in comparison with original DE.Moreover, CDEAS gets very close to the global optimum insome other functions 𝑓

1, 𝑓3, 𝑓4, 𝑓6, and 𝑓

18. It also presents

that the mean minimum, best minimum, worst minimum,the standard deviation of minimum, and the success rate ofCDEAS algorithm are clearly better than the original DE forfunctions 𝑓

1, 𝑓3, 𝑓4, 𝑓5, 𝑓6, 𝑓12, 𝑓13, 𝑓14, 𝑓15, and 𝑓

18although

the computing time of CDEAS is longer than that of originalDE because of its complexity.

The convergence figures of CDEAS comparing withoriginal DE for 18 instances are listed as Figure 6.

From Figure 6 one can observe that the convergencespeed of CDEAS is faster than original DE for 𝑓

1, 𝑓2, 𝑓3, 𝑓4,

𝑓6, 𝑓7, 𝑓11, 𝑓12, 𝑓13, 𝑓14, 𝑓15, and 𝑓

18.

All these comparisons of CDEAS with original DE algo-rithm have shown that CDEAS is a competitive algorithmto solve all the unimodal function problems and most ofthe multimodal function optimization problems listed above.As shown in the descriptions and all the illustrations before,CDEAS is efficacious on those typical function optimizations.

4. Model of Net Value of Ammonia UsingCDEAS-LS-SVM

4.1. Auxiliary Variables Selection of theModel. There are someprocess variables which have the greatest influence on the netvalue of ammina, such as system pressure, recycle gas flowrate, feed composition (H/N ratio), ammonia and inert gascencetration in the gas of reactor inlet, hot spot temperatures,and so forth. The relations between the process variablesare coupling and the operational variables interact with eachother.

The inlet ammonia concentration is an important processvariable which is beneficial to operation-optimization but thedevice of online catharometer is very expensive. Accordingto the mechanism and soft sensor model, a IIO-BP modelwas built to get the more accurate value of the inlet ammoniaconcentration [20]

Δ (NH3) = 𝐴NH3OUT − 𝐴NH3IN. (18)


0 400 800 1200 1600 2000Evolution generation

−80

−70

−60

−50

−40

−30

−20

−10

0

10

log(fi

tnes

s val

ue)

Convergence figure of originalDE and CDEAS for f1


log(fi

tnes

s val

ue)

−30

−25

−20

−15

−10

−5

0

5



0

−80

−70

−60

−50

−40

−30

−20

−10

10

log(fi

tnes

s val

ue)



10

log(fi

tnes

s val

ue)

−30

−25

−20

−15

−10

−5

0

5


123456789

1011


log(fi

tnes

s val

ue)



log(fi

tnes

s val

ue)

−30

−35

−40

−45

−25

−20

−15

−10

−5

0

5


10

Original DECDEAS


log(fi

tnes

s val

ue)

−30

−35

−25

−20

−15

−10

−5

0

5

Convergence fgure of originalDE and CDEAS for f7

Original DECDEAS


−16

−14

−12

−10

−8

−6

−4

−2

0

2

log(fi

tnes

s val

ue)

Convergence fgure of originalDE and CDEAS for f8

Figure 6: Continued.





0.81

1.21.41.61.8

22.22.42.62.8


0.81

1.21.41.61.8

22.22.42.62.8


0.81

1.21.41.61.8

22.22.42.62.8


0

0.5

1

1.5

2

2.5

3


Original DECDEAS

2.22.42.62.8

33.23.43.63.8

44.2


Original DECDEAS

02

−16

−14

−12

−10

−8

−6

−4

−2 024

−16

−14

−12

−10

−8

−6

−4

−2

0

1

2

3

−3

−2

−1









log(fi

tnes

s val

ue)

log(fi

tnes

s val

ue)

log(fi

tnes

s val

ue)

log(ft

tnes

s val

ue)

log(ft

tnes

s val

ue)

log(ft

tnes

s val

ue)

log(ft

tnes

s val

ue)

log(ft

tnes

s val

ue)

Figure 6: Continued.


0

0.5

1

1.5

2

Original DECDEAS


Original DECDEAS


−50 −0.5

−40

−30

−20

−10

0

10

log(fi

tnes

s val

ue)

log(fi

tnes

s val

ue)



Figure 6: Convergence figure of CDEAS comparing with original DE for 𝑓1∼𝑓18.

Table 2: Auxiliary variables of model of net value of Ammonia.

List Symbols Name Unit1 𝑅H/N H/N ratio %2 𝐴CH4 Methane concentration in recycled synthesis gas at the reactor inlet Mole ratio3 ANH3 Ammonia concentration in recycled synthesis gas at the reactor inlet Mole ratio4 𝑃

𝑆System pressure Mpa

5 𝐹𝑄

Recycle gas flow rate Nm3/h6 𝐹

𝑄1Quench gas flows of axial layer Nm3/h

7 𝐹𝑞2

Cold quench gas flows of 1st radial layers Nm3/h8 𝐹

𝑞3Quench gas flows of 2nd radial layers Nm3/h

9 𝐹𝑞4

Hot quench gas flows of 1st radial layers Nm3/h10 𝑇

𝐴Hot-spot temperatures of axial bed ∘C

11 𝑇𝑅1 Hot-spot temperatures of radial bed I

∘C12 𝑇

𝑅2 Hot-spot temperatures of radial bed II∘C

13 𝑇EO Outlet gas temperature of evaporator∘C

From the analysis discussed above, some important vari-ables have significant effects on the net value of ammonia.By discussion with experienced engineers and taking intoconsideration a priori knowledge about the process, thesystem pressure, recycle gas flow rate, the H/N ratio, hot-spottemperatures in the catalyst bed, and ammonia and methaneconcentration in the recycle gas are identified as the keyauxiliary variables to model net value of ammonia which islisted in Table 2.

4.2. Modeling the Net Value of Ammonia Using CDEAS-LS-SVM. LS-SVM is an alternate formulation of SVM,which is proposed by Suykens. The e-insensitive loss func-tion is replaced by a squared loss function, which con-structs the Lagrange function by solving the problem linearKarush Kuhn Tucker (KKT)

[0 𝐼

𝑇

𝑛

𝐼𝑛𝐾 + 𝛾

−1𝐼] [𝑏0

𝑏] = [

0

𝑦] , (19)

where 𝐼𝑛is a [𝑛 × 1] vector of ones, 𝑇 is the transpose of a

matrix or vector, 𝛾 is a weight vector, 𝑏0means the model

offset, and 𝑏 is regression vector.𝐾 is Mercer kernel matrix, which is defined as

𝐾 = (

𝑘1,1

⋅ ⋅ ⋅ 𝑘1,𝑛

... d...

𝑘𝑛,1

⋅ ⋅ ⋅ 𝑘𝑛,𝑛

), (20)

where 𝑘𝑖,𝑗is defined by kernel function.

There are several kinds of kernel functions, such ashyperbolic tangent, polynomial, and Gaussian radial basisfunction (RBF) which are commonly used. Literatures haveproved that RBF kernel function has strong generalization,so in this study RBF kernel was used:

𝑘𝑖,𝑗= 𝑒−|𝑥𝑖−𝑥𝑗|

2/2𝜎2

, (21)

where 𝑥𝑖and 𝑥

𝑗indicated different training samples, 𝜎 is the

kernel width parameter.


Table 3: The comparisons of training error and testing error of LS-SVM.

Method Type of error RE∗ MAE∗ MSE∗

BP-NN Training error 9.4422 × 10−04 1.0544 × 10−04 1.3970 × 10−04

Testing error 0.008085 8.9666 × 10−04 0.001188

LS-SVM Training error 0.002231 2.4785 × 10−04

4.1672 × 10−04

Testing error 0.005328 5.9038 × 10−04 7.8169 × 10−04

DE-LS-SVM Training error 0.002739 3.04286 × 10−04

4.08512 × 10−04

Testing error 0.005252 5.8241 × 10−04 7.7032 × 10−04

CDEAS-LS-SVM Training error 0.002830 3.1415 × 10−04

3.3131 × 10−04

Testing error 0.004661 5.1752 × 10−04 6.8952 × 10−04∗RE: relative error; MAE: mean absolute error; MSE: mean square error.

As we can see from (19)∼(21), only two parameters(𝛾, 𝜎) are needed for LS-SVM. It makes LS-SVM problemcomputationally easier than SVR problem.

Grid search is a commonly used method to select theparameters of LS-SVM, but it is time-consuming and inef-ficient. CDEAS algorithm has strong search capabilities, andthe algorithm is simple and easy to implement.Therefore, thispaper proposes the CDEAS algorithm to calculate the bestparameters (𝛾, 𝜎) of LS-SVM.

5. Results and Discussion

Operational parameters such as 𝐴H2 , 𝐴CH4 , and 𝑃𝑆 werecollected and acquired from plant DCS from the year 2011-2012. In addition, data on the inlet ammonia concentrationof recycle gas 𝐴NH3 were simulated by mechanism and softsensor model [20].

The extreme values are eliminated from the data using the3𝜎 criterion. After the smoothing and normalization, eachdata group is divided into 2 parts: 223 groups of trainingsamples which are used to train model while 90 groups oftesting samples which are valuing the generalization of themodel for identifying the parameters of the LS-SVM, thekernel width parameter, and the weight vector.

BP-NN, LS-SVM, and DE-LS-SVM are also used tomodel the net value of ammonia, respectively. BP-NN isa 13-15-1 three-layer network with back-propagation algo-rithm. LS-SVM gains the (𝛾, 𝜎) with grid-search and cross-validation. The parameter settings of CDEAS-LS-SVM arethe same as those in the benchmark tests. Each model is run30 times and the best value is shown in Table 3. Descriptivestatistics of training results and testing results of modelinclude the relative error, absolute error, and mean squareerror. The performance of the four models is compared asshown in Table 3. The training and testing results of fourmodels are illustrated in Figure 7.

Despite the fact that the training error using BP-NN issmaller than that using CDEAS-LS-SVM, which is becauseBP-NN is overfitting to the training data, the mean squareerror (MSE) on training data using CDEAS-LS-SVM isreduced by 25.6% and 23.2% compared with LS-SVM andDE-LS-SVM, respectively. In comparison with the othermodels (BP-NN, LS-SVM, and DE-LS-SVM), testing errorusing CDEAS-LS-SVMmodel is reduced by 14.1% and 11.2%,

respectively. The results indicate that the proposed CDEAS-LS-SVM model has a good tracking precision performanceand guides production better.

6. Conclusion

In this paper, an optimizing model which describes therelationship between net value of ammonia and key opera-tional parameters in ammonia synthesis has been proposed.Some representative benchmark functions were employed toevaluate the performance of a novel algorithm CDEAS. Theobtained results show that CDEAS algorithm is efficaciousfor solving most of the optimization problems comparisonswith original DE. Least squares support vector machineis used to build the model while CDEAS algorithm isemployed to identify the parameters of LS-SVM. The sim-ulation results indicated that CDEAS-LS-SVM is superiorto other models (BP-NN, LS-SVM, and DE-LS-SVM) andmeets the requirements of ammonia synthesis process. TheCDEAS-LS-SVM optimizing model makes it a promisingcandidate for obtaining the optimal operational parametersof ammonia synthesis process and meets the maximumbenefit of ammonia synthesis production.

Appendix

(1) Sphere function

𝑓1(𝑥) =

𝐷

∑

𝑖=1

𝑥2

𝑖,

𝑜 = [0, 0, . . . , 0] : the global optimum.

(A.1)

(2) Shifted sphere function

𝑓2(𝑥) =

𝐷

∑

𝑖=1

𝑧2

𝑖,

𝑧 = 𝑥 − 𝜃,

𝜃 = [𝜃1, 𝜃2, . . . , 𝜃

𝐷] : the shifted global optimum.

(A.2)


0 50 100 150 200 2500.1060.1070.1080.109

0.110.1110.1120.1130.1140.1150.116

Sample number

Net

val

ue o

f am

mon

ia (%

)

0 10 20 30 40 50 60 70 80 900.1050.1060.1070.1080.109

0.110.1110.1120.1130.1140.115

Sample number

Net

val

ue o

f am

mon

ia (%

)

0 50 100 150 200 250Sample number

Net

val

ue o

f am

mon

ia (%

)

0.1060.1070.1080.109

0.110.1110.1120.1130.1140.1150.116

Net

val

ue o

f am

mon

ia (%

)

Sample number0 10 20 30 40 50 60 70 80 90

0.107

0.108

0.109

0.11

0.111

0.112

0.113

0.114

0.115

0 50 100 150 200 250Sample number

Net

val

ue o

f am

mon

ia (%

)

0.1060.1070.1080.109

0.110.1110.1120.1130.1140.1150.116

Net

val

ue o

f am

mon

ia (%

)

Sample number0 10 20 30 40 50 60 70 80 90

0.107

0.108

0.109

0.11

0.111

0.112

0.113

0.114

0.115

Net

val

ue o

f am

mon

ia (%

)

Sample number0 50 100 150 200 250

0.1060.1070.1080.109

0.110.1110.1120.1130.1140.1150.116

Actual valuesTraining results of CDEAS-LS-SVM

Net

val

ue o

f am

mon

ia (%

)

Sample number0 10 20 30 40 50 60 70 80 90

0.107

0.108

0.109

0.11

0.111

0.112

0.113

0.114

0.115

Actual valuesTraining results of CDEAS-LS-SVM

Figure 7: The analyzed results, training results, and testing results of BP-NN, LS-SVM, DE-LS-SVM, and CDEAS-LS-SVM.


Table 4: Global optimum, search ranges, and initialization ranges of the test functions.

𝑓 Dimension Global optimum ⇀𝑥 𝑓(⇀𝑥) Search range Target𝑓1

30

0 0 [−100, 100]𝐷 10−5

𝑓2

𝜃 0 [−100, 100]𝐷 10−5

𝑓3

0 0 [−100, 100]𝐷 10−5

𝑓4

𝜃 0 [−100, 100]𝐷 10−5

𝑓5

1 1 [−100, 100]𝐷 100𝑓6

0 0 [−32, 32]𝐷 10−5

𝑓7

𝜃 0 [−32, 32]𝐷 10−5

𝑓8

0 0 [−32, 32]𝐷 10−5

𝑓9

𝜃 0 [−32, 32]𝐷 0.1𝑓10

0 0 [0, 600]𝐷 0.001𝑓11

𝜃 0 [−600, 600]𝐷 0.01𝑓12

0 0 [−5, 5]𝐷 10𝑓13

𝜃 0 [−5, 5]𝐷 10𝑓14

0 0 [−5, 5]𝐷 10𝑓15

𝜃 0 [−5, 5]𝐷 5𝑓16

418.9829 0 [−500, 500]𝐷 500𝑓17

0 0 [−100, 100]𝐷 1𝑓18

0 0 [−10, 10]𝐷 10−5

𝜃 is the shifted vector.

(3) Schwefel’s Problem 1.2

𝑓3(𝑥) =

𝐷

∑

𝑖=1

(

𝑖

∑

𝑗=1

𝑥𝑖)

2

,


(A.3)

(4) Shifted Schwefel’s Problem 1.2

𝑓4(𝑥) =

𝐷

∑

𝑖=1

(

𝑖

∑

𝑗=1

𝑧𝑖)

2

,

𝑧 = 𝑥 − 𝜃,

𝜃 = [𝜃1, 𝜃2, . . . , 𝜃


(A.4)

(5) Rosenbrock’s function

𝑓5(𝑥) =

𝐷−1

∑

𝑖=1

(100(𝑥2

𝑖− 𝑥2

𝑖+1)2

+ (𝑥𝑖− 1)2

) ,


(A.5)

(6) Schwefel’s Problem 1.2 with noise in fitness

𝑓6(𝑥) = (

𝐷

∑

𝑖=1

(

𝑖

∑

𝑗=1

𝑥𝑖)

2

) ∗ (1 + 0.4 |𝑁 (0, 1)|) ,


(A.6)

(7) Shifted Schwefel’s Problem 1.2 with noise in fitness

𝑓7(𝑥) = (

𝐷

∑

𝑖=1

(

𝑖

∑

𝑗=1

𝑧𝑖)

2

) ∗ (1 + 0.4 |𝑁 (0, 1)|) ,

𝑧 = 𝑥 − 𝜃,

𝜃 = [𝜃1, 𝜃2, . . . , 𝜃


(A.7)

(8) Ackley’s function

𝑓8(𝑥) = − 20 exp(−0.2√ 1

𝐷

𝐷

∑

𝑖=1

𝑥2

𝑖)

− exp( 1𝐷

𝐷

∑

𝑖=1

cos (2𝜋𝑥𝑖)) + 20 + 𝑒,


(A.8)

(9) Shifted Ackley’s function

𝑓9(𝑥) = − 20 exp(−0.2√ 1

𝐷

𝐷

∑

𝑖=1

𝑧2

𝑖)

− exp( 1𝐷

𝐷

∑

𝑖=1

cos (2𝜋𝑧𝑖)) + 20 + 𝑒,

𝑧 = 𝑥 − 𝜃,

𝜃 = [𝜃1, 𝜃2, . . . , 𝜃


(A.9)


(10) Griewank’s function

𝑓10(𝑥) =

𝐷

∑

𝑖=1

𝑥2

𝑖

4000−

𝐷

∏

𝑖=1

cos𝑥𝑖

√𝑖

+ 1,


(A.10)

(11) Shifted Griewank’s function

𝑓11(𝑥) =

𝐷

∑

𝑖=1

𝑧2

𝑖

4000−

𝐷

∏

𝑖=1

cos𝑧𝑖

√𝑖

+ 1,

𝑧 = 𝑥 − 𝜃,

𝜃 = [𝜃1, 𝜃2, . . . , 𝜃


(A.11)

(12) Rastrigin’s function

𝑓12(𝑥) =

𝐷

∑

𝑖=1

(𝑥2

𝑖− 10 cos (2𝜋𝑥

𝑖) + 10) ,


(A.12)

(13) Shifted Rastrigin’s function

𝑓13(𝑥) =

𝐷

∑

𝑖=1

(𝑧2

𝑖− 10 cos (2𝜋𝑧

𝑖) + 10) ,

𝑧 = 𝑥 − 𝜃,

𝜃 = [𝜃1, 𝜃2, . . . , 𝜃


(A.13)

(14) Noncontiguous Rastrigin’s function

𝑓14(𝑥) =

𝐷

∑

𝑖=1

(𝑦2

𝑖− 10 cos (2𝜋𝑦

𝑖) + 10) ,

𝑦𝑖=

{{

{{

{

𝑥𝑖

𝑥𝑖 <1

2round (2𝑥

𝑖)

2

𝑥𝑖 ≥1

2,

for 𝑖 = 1, 2, . . . , 𝐷,

𝑜 = [0, 0, . . . , 0] : the global optimum.(A.14)

(15) Shifted noncontiguous Rastrigin’s function

𝑓15(𝑥) =

𝐷

∑

𝑖=1

(𝑦2

𝑖− 10 cos (2𝜋𝑦

𝑖) + 10) ,

𝑦𝑖=

{{

{{

{

𝑧𝑖

𝑧𝑖 <1

2round (2𝑧

𝑖)

2

𝑧𝑖 ≥1

2,

for 𝑖 = 1, 2, . . . , 𝐷,

𝑧 = 𝑥 − 𝜃,

𝜃 = [𝜃1, 𝜃2, . . . , 𝜃


(A.15)

(16) Schwefel’s function

𝑓16(𝑥) = 418.9829 × 𝐷 −

𝐷

∑

𝑖=1

𝑥𝑖sin (𝑥𝑖

1/2

) ,

𝑜=[418.9829, 418.9829, . . . , 418.9829]: the global optimum.(A.16)


𝑓18(𝑥) = max {𝑥𝑖

, 1 ≤ 𝑖 ≤ 𝐷} ,

𝑜 = [0, 0, . . . , 0] : the global optimum.(A.17)


𝑓17(𝑥) =

𝐷

∑

𝑖=1

𝑥𝑖 +

𝐷

∏

𝑖=1

𝑥𝑖,


(A.18)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgments

The authors are grateful to the anonymous reviewers forgiving us helpful suggestions. This work is supported byNational Natural Science Foundation of China (Grant nos.61174040 and 61104178) and Fundamental Research Funds forthe Central Universities, Shanghai Commission of Scienceand Technology (Grant no. 12JC1403400).

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