research article steady modeling for an ammonia...
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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 ยฉ 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
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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 25โC and flows to the reactor and thewhole cycle starts again.
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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 ๐ท-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 bounds๐min and๐max:
๐ฅ0
๐,๐= ๐ฅ0
min,๐ + rand (0, 1) (๐ฅ0
max,๐ โ ๐ฅ0
min,๐) ๐ = 1, 2, . . . , ๐ท.
(2)
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4 Mathematical Problems in Engineering
(2)Mutation.DE employs themutation operation to produceamutant vector ๐ข๐ก
๐๐called target vector corresponding to each
individual ๐ฅ๐ก๐๐after initialization. In iteration ๐ก, the mutant
vector ๐ข๐ก๐๐
of individual ๐ฅ๐ก๐๐
can be generated according tocertain mutation strategies. Equations (3)โ(7) indicate themost frequent mutation strategies version, respectively:
DE/rand/1 ๐ข๐ก๐๐= ๐๐ก
๐1๐,๐+ ๐น (๐
๐ก
๐2๐,๐โ ๐๐ก
๐3๐,๐) , (3)
DE/rand/2 ๐ข๐ก๐๐= ๐๐ก
๐1๐,๐+ ๐น (๐
๐ก
๐2๐,๐โ ๐๐ก
๐3๐,๐)
+ ๐น (๐๐ก
๐4๐,๐โ ๐๐ก
๐5๐,๐) ,
(4)
DE/best/1 ๐ข๐ก๐๐= ๐๐ก
best,๐ + ๐น (๐๐ก
๐1๐,๐โ ๐๐ก
๐2๐,๐) , (5)
DE/best/2 ๐ข๐ก๐๐= ๐๐ก
best,๐ + ๐น (๐๐ก
๐1๐,๐โ ๐๐ก
๐2๐,๐)
+ ๐น (๐๐ก
๐3๐,๐โ ๐๐ก
๐4๐,๐) ,
(6)
DE/rand-to-best/1 ๐ข๐ก๐๐= ๐๐ก
๐,๐+ ๐น (๐
๐ก
best,๐ โ ๐๐ก
๐,๐)
+ ๐น (๐๐ก
๐1๐,๐โ ๐๐ก
๐2๐,๐) ,
(7)
where ๐1๐, ๐2๐, ๐3๐, ๐4๐, and ๐5๐are mutually exclusive integersrandomly generated within the range [1,NP] which shouldnot be ๐. ๐น is the mutation factor for scaling the differencevector, usually bounded in [0, 2]. ๐๐กbest is the best individualwith the best fitness value at generation ๐ก in the population.
(3) Crossover. The individual ๐๐ก๐and mutant vector ๐ข๐ก
๐are
hybridized to compose the trial vector ๐ฆ๐ก๐after mutation
operation. The binomial crossover is adopted by the DE inthe paper, which is defined as
๐ฆ๐ก
๐๐= {๐ข๐ก
๐๐if rand โค ๐ถ
๐ or ๐ = ๐rand
๐๐ก
๐,๐otherwise,
(8)
where rand is a random number between in 0 and 1 dis-tributed uniformly. The crossover factor ๐ถ
๐ is a probability
rate within the range 0 and 1, which influences the tradeoffbetween the ability of exploration and exploitation. ๐rand is aninteger chosen randomly in [1, ๐ท]. To ensure that the trialvector (๐ฆ๐ก
๐) differs from its corresponding individual (๐๐ก
๐) by
at least one dimension, ๐ = ๐rand 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 individual๐ฆ๐ก
๐is comparedwith the individual๐๐ก
๐, and
the one with better fitness is selected as the new individual inthe next iteration:
๐๐ก+1
๐๐= {๐ฆ๐ก
๐๐if ๐ (๐ฆ๐ก
๐๐) โค ๐ (๐
๐ก
๐,๐)
๐๐ก
๐,๐otherwise.
(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 ๐น, mutation strate-gies, and crossover factor ๐ถ
๐ have great influence on the bal-
ance ofDEโs exploration and exploitation ability.๐นdecides theamplification of differential variation; ๐ถ
๐ is used to control
the possibility of the crossover operation; mutation strategieshave great influence on the results of mutation operation. Insome literatures ๐น, ๐ถ
๐ , and mutation strategies are defined in
advance or varied by some specific regulations. But the factors๐น,๐ถ๐ , and strategies are very difficult to choose since the prior
knowledge is absent. Therefore, Ant Colony Search is usedto search the suitable combination of ๐น, ๐ถ
๐ , and mutation
strategies adaptively to accelerate the global search. Someresearchers have found an inevitable relationship betweenthe parameters (๐น, ๐ถ
๐ , and mutation strategies) and the
optimization results of DE [16โ18]. However, the approachesabove are not applying the most suitable ๐น, ๐ถ
๐ , 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 ๐น and ๐ถ
๐ in 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. ๐น and ๐ถ
๐ are defined to be the values between [0, 1],
๐น โ {0.1ร๐}, ๐ = 1, 2, . . . , 10 and๐ถ๐ โ {0.1ร๐}, ๐ = 1, 2, . . . , 10.
Each of the ants chooses a combination of๐น and๐ถ๐ according
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 ๐. By exchanginginformation according to pheromone, the ants cooperatewitheach other to choose appropriate combination of ๐น and ๐ถ
๐ .
Then ant colony renews the pheromone trails of all ants.Then, the pheromone trail ๐
๐๐is updated in the following
equation:
๐๐๐(๐ก + 1) = (1 โ ๐
1) ๐๐๐(๐ก) +
subpopulation1
โ
๐=1
ฮ๐๐
๐๐(๐ก) ,
(10)
where 0 โค ๐1< 1 means the pheromone trail evaporation
rate, ๐ = 1, 2, . . . , 10,๐ = 1, 2; 1st parameter represents ๐น and
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Mathematical Problems in Engineering 5
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
๐1,1 ๐1,2 ๐1,3 ๐1,10
F
F
CR
CR
๐2,1 ๐2,2 ๐2,3 ๐2,10
0.1
1.0
0.2
0.3
...
0.1
1.0
0.2
0.3
...
ยท ยท ยท
ยท ยท ยท
ยท ยท ยท
ยท ยท ยท
Figure 4: Relationship between pheromone and ant paths of ๐น, ๐ถ๐ .
2nd parameter represents ๐ถ๐ ; ฮ๐๐๐๐(๐ก) is the quantity of the
pheromone trail of ant ๐,
ฮ๐๐
๐๐(๐ก)
=
{{{{
{{{{
{
1 if ๐ โ ๐๐๐
and fitness (๐ฆ๐ก๐) < fitness (๐ฅbest๐ก) ,
0.5 if ๐ โ ๐๐๐
and fitness (๐ฅbest๐ก) < fitness (๐ฆ๐ก๐)
and fitness (๐ฆ๐ก๐) < fitness (๐ฅ๐ก
๐) ,
0 otherwise,(11)
where ๐๐๐
is the ant group that chooses ๐th value as theselection of๐th parameter;๐ฅbest๐ก denotes the best individualof ant colony till ๐กth 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 ๐th value of๐th parameter (๐น and ๐ถ
๐ ) in Figure 4 is set by
๐๐๐(๐ก) =
{{
{{
{
๐๐
๐๐(๐ก)
โ๐๐๐๐(๐ก)
if rand1< ๐๐
rand2
otherwise.(12)
Figure 4 illustrates the relationship between pheromonematrix and ant path of ๐น and ๐ถ
๐ , where ๐
๐ is 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 ๐น and ๐ถ
๐ depends on the
pheromone of each path. According to the performance ofall the individuals, the individual is chosen by the mostappropriate combination of ๐น and ๐ถ
๐ in each generation.
In subpopulation 2, the individual is set as ant at eachgeneration. Mutation strategies which are listed at (3)โ(7) are
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6 Mathematical Problems in Engineering
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
๐1 ๐2 ๐3 ๐4 ๐5
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 ๐.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 ๐ is updated in the followingequation:
๐๐(๐ก + 1) = (1 โ ๐
2) ๐๐(๐ก) +
subpopulation2
โ
๐=1
ฮ๐๐
๐(๐ก) , (13)
where 0 โค ๐2< 1 means the pheromone trail evaporation
rate and ฮ๐๐๐(๐ก) is the quantity of the pheromone trail of ant ๐,
๐๐
๐(๐ก)
=
{{{{
{{{{
{
1 if ๐ โ ๐๐and fitness (๐ฆ๐ก
๐) < fitness (๐ฅbest๐ก) ,
0.5 if ๐ โ ๐๐and fitness (๐ฅbest๐ก) < fitness (๐ฆ๐ก
๐)
and fitness (๐ฆ๐ก๐) < fitness (๐ฅ๐ก
๐) ,
0 otherwise,(14)
where ๐๐is the ant group that chooses ๐th value as the
selection of parameter; ๐ฅbest๐ก denotes the best individual ofant colony till ๐กth 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 ๐th valueof๐th parameter (mutation strategies) is set by
๐๐(๐ก) =
{{
{{
{
๐๐
๐(๐ก)
โ๐๐๐(๐ก)
if rand3< ๐
๐
rand4
otherwise,(15)
where๐๐ is 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 ๐ธ 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 ๐ธ.
The normative knowledge contains the intervals thatdecide the individuals of population space where to move. ๐
๐
and ๐ข๐are the lower and upper bounds of the search range
in population space. ๐ฟ๐and ๐
๐are the value of the fitness
function associated with that bound. If the ๐๐and ๐ข
๐are
updated, the ๐ฟ๐and ๐
๐must be updated too.
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Mathematical Problems in Engineering 7
๐๐and ๐ข
๐are set by
๐๐= {๐ฅ๐,min, if ๐ฅ๐,min < ๐๐ or ๐ (๐ฅ๐,min) < ๐ฟ ๐๐๐
otherwise,
๐ข๐= {๐ฅ๐,max, if ๐ฅ๐,max > ๐ข๐ or ๐ (๐ฅ๐,max) > ๐๐๐ข๐
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:
๐ฅ๐ก+1
๐,๐=
{{{{{{{{{{{{{{{{{{{
{{{{{{{{{{{{{{{{{{{
{
๐ฅ๐ก
๐,๐+๐ (0.5, 0.3) โ (๐
๐ก
๐2๐,๐โ ๐๐ก
๐3๐,๐)ร rand,
if ๐ฅ๐ก๐,๐โค ๐ธ๐, ๐ฅ๐ก
๐,๐โฅ ๐ข๐,
๐ฅ๐ก
๐,๐โ๐ (0.5, 0.3) โ (๐
๐ก
๐2๐,๐โ ๐๐ก
๐3๐,๐)ร rand,
if ๐ฅ๐ก๐,๐> ๐ธ๐, ๐ฅ๐ก
๐,๐< ๐ข๐,
๐๐ก
๐1๐,๐+๐ (0.5, 0.3) โ (๐ข
๐โ ๐๐ก
๐3๐,๐)ร rand,
if ๐ฅ๐ก๐,๐โค ๐ธ๐, ๐ฅ๐ก
๐,๐โฅ ๐๐,
๐๐ก
๐1๐,๐โ๐ (0.5, 0.3) โ (๐
๐โ ๐๐ก
๐3๐,๐)ร rand,
if ๐ฅ๐ก๐,๐> ๐ธ๐, ๐ฅ๐ก
๐,๐> ๐๐,
๐ฅ๐ก+1
๐,๐=
{{{
{{{
{
๐๐ก
๐1๐,๐+ ๐น โ (๐ข
๐โ ๐๐ก
๐1๐,๐) ร rand, if ๐ฅ
๐,๐> ๐๐
๐๐ก
๐1๐,๐โ ๐น โ (๐
๐ก
๐1๐,๐โ ๐๐) ร rand, if ๐ฅ
๐,๐< ๐ข๐
๐๐ก
๐1๐,๐+ ๐น โ (๐ข
๐โ ๐๐) ร rand, if ๐
๐< ๐ฅ๐,๐< ๐ข๐,
(17)where ๐น is a constant of 0.2.
3.2.5. Knowledge Exchange. After ๐ก steps, the ๐น and ๐ถ๐
of subpopulation 2 are replaced by the suitable ๐น and ๐ถ๐
calculated by subpopulation 1 and the mutation strategyof subpopulation 1 is displaced by the suitable mutationstrategy calculated by subpopulation 2 simultaneously. Sothe ๐น and ๐ถ
๐ and 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 individualโs fitness.
Step 3. To find the proper ๐น, ๐ถ๐ , 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 ๐น, ๐ถ
๐ , 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 ๐น = 0.3 and ๐ถ
๐ = 0.5; for CDEAS, the size
of both two subpopulations is 25; the initial ๐น and ๐ถ๐ are
randomly selected in (0, 1) and the initial mutation strategyis DE/rand/1; the interval information exchanges between thetwo subpopulations ๐ก is 50 generations; the thresholds ๐
๐ =
๐
๐ = 0.5 and ๐
1= ๐2= 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 for๐2, ๐4, ๐7, ๐9, ๐11, ๐13, and ๐
15is shifted. Functions ๐
1โผ๐7are
unimodal and functions๐8โผ๐18are 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 ๐
2and ๐7in all trials,
and the success rate reached 100% of functions ๐1, ๐2, ๐3, ๐4,
๐6, ๐7, and ๐
18. For most of the test functions, the success
-
8 Mathematical Problems in Engineering
Table 1: The comparison results of the CDEAS algorithm and original DE algorithm.
Original DE CDEASSphere function ๐
1
Best 1.1746 ร 10โ65 5.0147 ร 10โ79
Worst 1.0815 ร 10โ23 9.3244 ร 10โ75
Mean 3.6052 ร 10โ25 1.6390 ร 10โ75
Std. 1.9746 ร 10โ24 2.2315 ร 10โ75
Success rate (%) 100 100Times (s) 1.8803 14.6017
Shifted sphere function ๐2
Best 0 0Worst 8.0779 ร 10โ28 0Mean 3.3658 ร 10โ29 0Std. 1.5078 ร 10โ28 0Success rate (%) 100 100Times (s) 2.1788 18.1117
Schwefelโs Problem 1.2 ๐3
Best 2.4386 ร 10โ65 3.0368 ร 10โ78
Worst 2.4820 ร 10โ22 9.2902 ร 10โ73
Mean 8.2736 ร 10โ24 7.2341 ร 10โ74
Std. 4.5316 ร 10โ23 2.0187 ร 10โ73
Success rate (%) 100 100Times (s) 3.1647 24.1178
Shifted Schwefelโs Problem 1.2 ๐4
Best 0 0Worst 5.6545 ร 10โ27 3.4331 ร 10โ27
Mean 2.0868 ร 10โ28 1.8848 ร 10โ28
Std. 1.0323 ร 10โ27 7.9813 ร 10โ28
Success rate (%) 100 100Times (s) 3.3956 27.7058
Rosenbrockโs function ๐5
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
Schwefelโs Problem 1.2 with noise in fitness ๐6
Best 3.1344 ร 10โ39 3.98838 ร 10โ49
Worst 3.61389 ร 10โ36 1.6124 ร 10โ43
Mean 5.7744 ร 10โ37 7.4656 ร 10โ45
Std. 9.5348 ร 10โ37 2.9722 ร 10โ44
Success rate (%) 100 100Times (s) 3.2141 24.2426
Shifted Schwefelโs Problem 1.2 with noise in fitness ๐7
Best 0 0Worst 0 0Mean 0 0
-
Mathematical Problems in Engineering 9
Table 1: Continued.
Original DE CDEASStd. 0 0Success rate (%) 100 100Times (s) 3.3374 28.5638
Ackleyโs function ๐8
Best 7.1054 ร 10โ15 3.5527 ร 10โ15
Worst 4.8999 ร 10โ7 1.3404Mean 1.6332 ร 10โ8 0.1763Std. 8.9457 ร 10โ8 0.4068Success rate (%) 100 83.33Times (s) 2.4820 20.9353
Shifted Ackleyโs function ๐9
Best 7.1054 ร 10โ15 3.5527 ร 10โ15
Worst 0.9313 0.9313Mean 0.0310 0.0620Std. 0.1700 0.2362Success rate (%) 96.67 93.33Times (s) 2.7337 21.6841
Griewankโs function ๐10
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 Griewankโs function ๐11
Best 0 0Worst 0.0319 0.0343Mean 0.0056 0.0060Std 0.0089 0.0088Success rate (%) 80 76.67Times (s) 2.7768 22.8541
Rastriginโs function ๐12
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 Rastriginโs function ๐13
Best 5.9725 0.9949Worst 36.9923 6.7657Mean 19.4719 8.2581Std. 8.9164 3.8680Success rate (%) 16.67 76.67Times (s) 2.9313 23.8838
Noncontiguous Rastriginโs function ๐14
Best 20.7617 3.9949Worst 29.9112 11.9899Mean 25.4556 8.1947
-
10 Mathematical Problems in Engineering
Table 1: Continued.
Original DE CDEASStd. 2.9078 2.2473Success rate (%) 0 86.67Times (s) 3.1663 25.5374
Shifted noncontiguous Rastriginโs function ๐15
Best 0 0Worst 16 6Mean 6.7666 1.5333Std. 3.4509 1.8519Success rate (%) 40 96.67Times (s) 3.3374 25.9430
Schwefelโs function ๐16
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
Schwefelโs Problem 2.21 ๐17
Best 0.1640 0.3254Worst 4.5102 4.7086Mean 1.1077 1.9849Std. 0.8652 1.16418Success rate (%) 53.33 23.33Times (s) 2.3806 19.2505
Schwefelโs Problem 2.22 ๐18
Best 1.2706 ร 10โ35 8.5946 ร 10โ45
Worst 1.6842 ร 10โ34 1.8362 ร 10โ42
Mean 6.1883 ร 10โ35 2.6992 ร 10โ43
Std. 3.4937 ร 10โ35 4.6257 ร 10โ43
Success rate (%) 100 100Times (s) 2.6297 20.8573
rate of CDEAS is higher in comparison with original DE.Moreover, CDEAS gets very close to the global optimum insome other functions ๐
1, ๐3, ๐4, ๐6, and ๐
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 ๐
1, ๐3, ๐4, ๐5, ๐6, ๐12, ๐13, ๐14, ๐15, and ๐
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 ๐
1, ๐2, ๐3, ๐4,
๐6, ๐7, ๐11, ๐12, ๐13, ๐14, ๐15, and ๐
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]
ฮ (NH3) = ๐ดNH3OUT โ ๐ดNH3IN. (18)
-
Mathematical Problems in Engineering 11
0 400 800 1200 1600 2000Evolution generation
โ80
โ70
โ60
โ50
โ40
โ30
โ20
โ10
0
10
log(fi
tnes
s val
ue)
Convergence figure of originalDE and CDEAS for f1
0 400 800 1200 1600 2000Evolution generation
log(fi
tnes
s val
ue)
โ30
โ25
โ20
โ15
โ10
โ5
0
5
Convergence figure of originalDE and CDEAS for f2
0 400 800 1200 1600 2000Evolution generation
0
โ80
โ70
โ60
โ50
โ40
โ30
โ20
โ10
10
log(fi
tnes
s val
ue)
Convergence figure of originalDE and CDEAS for f3
0 400 800 1200 1600 2000Evolution generation
10
log(fi
tnes
s val
ue)
โ30
โ25
โ20
โ15
โ10
โ5
0
5
Convergence figure of originalDE and CDEAS for f4
123456789
1011
0 400 800 1200 1600 2000Evolution generation
log(fi
tnes
s val
ue)
Convergence figure of originalDE and CDEAS for f5
0 400 800 1200 1600 2000Evolution generation
log(fi
tnes
s val
ue)
โ30
โ35
โ40
โ45
โ25
โ20
โ15
โ10
โ5
0
5
Convergence figure of originalDE and CDEAS for f6
10
Original DECDEAS
0 400 800 1200 1600 2000Evolution generation
log(fi
tnes
s val
ue)
โ30
โ35
โ25
โ20
โ15
โ10
โ5
0
5
Convergence fgure of originalDE and CDEAS for f7
Original DECDEAS
0 400 800 1200 1600 2000Evolution generation
โ16
โ14
โ12
โ10
โ8
โ6
โ4
โ2
0
2
log(fi
tnes
s val
ue)
Convergence fgure of originalDE and CDEAS for f8
Figure 6: Continued.
-
12 Mathematical Problems in Engineering
0 400 800 1200 1600 2000Evolution generation
0 400 800 1200 1600 2000Evolution generation
0 400 800 1200 1600 2000Evolution generation
0.81
1.21.41.61.8
22.22.42.62.8
0 400 800 1200 1600 2000Evolution generation
0.81
1.21.41.61.8
22.22.42.62.8
0 400 800 1200 1600 2000Evolution generation
0.81
1.21.41.61.8
22.22.42.62.8
0 400 800 1200 1600 2000Evolution generation
0
0.5
1
1.5
2
2.5
3
0 400 800 1200 1600 2000Evolution generation
Original DECDEAS
2.22.42.62.8
33.23.43.63.8
44.2
0 400 800 1200 1600 2000Evolution generation
Original DECDEAS
02
โ16
โ14
โ12
โ10
โ8
โ6
โ4
โ2 024
โ16
โ14
โ12
โ10
โ8
โ6
โ4
โ2
0
1
2
3
โ3
โ2
โ1
Convergence figure of originalDE and CDEAS for f9
Convergence figure of originalDE and CDEAS for f10
Convergence figure of originalDE and CDEAS for f11
Convergence figure of originalDE and CDEAS for f12
Convergence figure of originalDE and CDEAS for f13
Convergence figure of originalDE and CDEAS for f14
Convergence figure of originalDE and CDEAS for f15
Convergence figure of originalDE and CDEAS for f16
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.
-
Mathematical Problems in Engineering 13
0
0.5
1
1.5
2
Original DECDEAS
0 400 800 1200 1600 2000Evolution generation
Original DECDEAS
0 400 800 1200 1600 2000Evolution generation
โ50 โ0.5
โ40
โ30
โ20
โ10
0
10
log(fi
tnes
s val
ue)
log(fi
tnes
s val
ue)
Convergence figure of originalDE and CDEAS for f17
Convergence figure of originalDE and CDEAS for f18
Figure 6: Convergence figure of CDEAS comparing with original DE for ๐1โผ๐18.
Table 2: Auxiliary variables of model of net value of Ammonia.
List Symbols Name Unit1 ๐ H/N H/N ratio %2 ๐ดCH4 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 ๐
๐System pressure Mpa
5 ๐น๐
Recycle gas flow rate Nm3/h6 ๐น
๐1Quench gas flows of axial layer Nm3/h
7 ๐น๐2
Cold quench gas flows of 1st radial layers Nm3/h8 ๐น
๐3Quench gas flows of 2nd radial layers Nm3/h
9 ๐น๐4
Hot quench gas flows of 1st radial layers Nm3/h10 ๐
๐ดHot-spot temperatures of axial bed โC
11 ๐๐ 1 Hot-spot temperatures of radial bed I
โC12 ๐
๐ 2 Hot-spot temperatures of radial bed IIโC
13 ๐EO Outlet gas temperature of evaporatorโC
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 ๐ผ
๐
๐
๐ผ๐๐พ + ๐พ
โ1๐ผ] [๐0
๐] = [
0
๐ฆ] , (19)
where ๐ผ๐is a [๐ ร 1] vector of ones, ๐ is the transpose of a
matrix or vector, ๐พ is a weight vector, ๐0means the model
offset, and ๐ is regression vector.๐พ is Mercer kernel matrix, which is defined as
๐พ = (
๐1,1
โ โ โ ๐1,๐
... d...
๐๐,1
โ โ โ ๐๐,๐
), (20)
where ๐๐,๐is 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:
๐๐,๐= ๐โ|๐ฅ๐โ๐ฅ๐|
2/2๐2
, (21)
where ๐ฅ๐and ๐ฅ
๐indicated different training samples, ๐ is the
kernel width parameter.
-
14 Mathematical Problems in Engineering
Table 3: The comparisons of training error and testing error of LS-SVM.
Method Type of error REโ MAEโ MSEโ
BP-NN Training error 9.4422 ร 10โ04 1.0544 ร 10โ04 1.3970 ร 10โ04
Testing error 0.008085 8.9666 ร 10โ04 0.001188
LS-SVM Training error 0.002231 2.4785 ร 10โ04
4.1672 ร 10โ04
Testing error 0.005328 5.9038 ร 10โ04 7.8169 ร 10โ04
DE-LS-SVM Training error 0.002739 3.04286 ร 10โ04
4.08512 ร 10โ04
Testing error 0.005252 5.8241 ร 10โ04 7.7032 ร 10โ04
CDEAS-LS-SVM Training error 0.002830 3.1415 ร 10โ04
3.3131 ร 10โ04
Testing error 0.004661 5.1752 ร 10โ04 6.8952 ร 10โ04โRE: relative error; MAE: mean absolute error; MSE: mean square error.
As we can see from (19)โผ(21), only two parameters(๐พ, ๐) 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 (๐พ, ๐) of LS-SVM.
5. Results and Discussion
Operational parameters such as ๐ดH2 , ๐ดCH4 , and ๐๐ werecollected and acquired from plant DCS from the year 2011-2012. In addition, data on the inlet ammonia concentrationof recycle gas ๐ดNH3 were simulated by mechanism and softsensor model [20].
The extreme values are eliminated from the data using the3๐ 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 (๐พ, ๐) 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
๐1(๐ฅ) =
๐ท
โ
๐=1
๐ฅ2
๐,
๐ = [0, 0, . . . , 0] : the global optimum.
(A.1)
(2) Shifted sphere function
๐2(๐ฅ) =
๐ท
โ
๐=1
๐ง2
๐,
๐ง = ๐ฅ โ ๐,
๐ = [๐1, ๐2, . . . , ๐
๐ท] : the shifted global optimum.
(A.2)
-
Mathematical Problems in Engineering 15
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.
-
16 Mathematical Problems in Engineering
Table 4: Global optimum, search ranges, and initialization ranges of the test functions.
๐ Dimension Global optimum โ๐ฅ ๐(โ๐ฅ) Search range Target๐1
30
0 0 [โ100, 100]๐ท 10โ5
๐2
๐ 0 [โ100, 100]๐ท 10โ5
๐3
0 0 [โ100, 100]๐ท 10โ5
๐4
๐ 0 [โ100, 100]๐ท 10โ5
๐5
1 1 [โ100, 100]๐ท 100๐6
0 0 [โ32, 32]๐ท 10โ5
๐7
๐ 0 [โ32, 32]๐ท 10โ5
๐8
0 0 [โ32, 32]๐ท 10โ5
๐9
๐ 0 [โ32, 32]๐ท 0.1๐10
0 0 [0, 600]๐ท 0.001๐11
๐ 0 [โ600, 600]๐ท 0.01๐12
0 0 [โ5, 5]๐ท 10๐13
๐ 0 [โ5, 5]๐ท 10๐14
0 0 [โ5, 5]๐ท 10๐15
๐ 0 [โ5, 5]๐ท 5๐16
418.9829 0 [โ500, 500]๐ท 500๐17
0 0 [โ100, 100]๐ท 1๐18
0 0 [โ10, 10]๐ท 10โ5
๐ is the shifted vector.
(3) Schwefelโs Problem 1.2
๐3(๐ฅ) =
๐ท
โ
๐=1
(
๐
โ
๐=1
๐ฅ๐)
2
,
๐ = [0, 0, . . . , 0] : the global optimum.
(A.3)
(4) Shifted Schwefelโs Problem 1.2
๐4(๐ฅ) =
๐ท
โ
๐=1
(
๐
โ
๐=1
๐ง๐)
2
,
๐ง = ๐ฅ โ ๐,
๐ = [๐1, ๐2, . . . , ๐
๐ท] : the shifted global optimum.
(A.4)
(5) Rosenbrockโs function
๐5(๐ฅ) =
๐ทโ1
โ
๐=1
(100(๐ฅ2
๐โ ๐ฅ2
๐+1)2
+ (๐ฅ๐โ 1)2
) ,
๐ = [1, 1, . . . , 1] : the global optimum.
(A.5)
(6) Schwefelโs Problem 1.2 with noise in fitness
๐6(๐ฅ) = (
๐ท
โ
๐=1
(
๐
โ
๐=1
๐ฅ๐)
2
) โ (1 + 0.4 |๐ (0, 1)|) ,
๐ = [0, 0, . . . , 0] : the global optimum.
(A.6)
(7) Shifted Schwefelโs Problem 1.2 with noise in fitness
๐7(๐ฅ) = (
๐ท
โ
๐=1
(
๐
โ
๐=1
๐ง๐)
2
) โ (1 + 0.4 |๐ (0, 1)|) ,
๐ง = ๐ฅ โ ๐,
๐ = [๐1, ๐2, . . . , ๐
๐ท] : the shifted global optimum.
(A.7)
(8) Ackleyโs function
๐8(๐ฅ) = โ 20 exp(โ0.2โ 1
๐ท
๐ท
โ
๐=1
๐ฅ2
๐)
โ exp( 1๐ท
๐ท
โ
๐=1
cos (2๐๐ฅ๐)) + 20 + ๐,
๐ = [0, 0, . . . , 0] : the global optimum.
(A.8)
(9) Shifted Ackleyโs function
๐9(๐ฅ) = โ 20 exp(โ0.2โ 1
๐ท
๐ท
โ
๐=1
๐ง2
๐)
โ exp( 1๐ท
๐ท
โ
๐=1
cos (2๐๐ง๐)) + 20 + ๐,
๐ง = ๐ฅ โ ๐,
๐ = [๐1, ๐2, . . . , ๐
๐ท] : the shifted global optimum.
(A.9)
-
Mathematical Problems in Engineering 17
(10) Griewankโs function
๐10(๐ฅ) =
๐ท
โ
๐=1
๐ฅ2
๐
4000โ
๐ท
โ
๐=1
cos๐ฅ๐
โ๐
+ 1,
๐ = [0, 0, . . . , 0] : the global optimum.
(A.10)
(11) Shifted Griewankโs function
๐11(๐ฅ) =
๐ท
โ
๐=1
๐ง2
๐
4000โ
๐ท
โ
๐=1
cos๐ง๐
โ๐
+ 1,
๐ง = ๐ฅ โ ๐,
๐ = [๐1, ๐2, . . . , ๐
๐ท] : the shifted global optimum.
(A.11)
(12) Rastriginโs function
๐12(๐ฅ) =
๐ท
โ
๐=1
(๐ฅ2
๐โ 10 cos (2๐๐ฅ
๐) + 10) ,
๐ = [0, 0, . . . , 0] : the global optimum.
(A.12)
(13) Shifted Rastriginโs function
๐13(๐ฅ) =
๐ท
โ
๐=1
(๐ง2
๐โ 10 cos (2๐๐ง
๐) + 10) ,
๐ง = ๐ฅ โ ๐,
๐ = [๐1, ๐2, . . . , ๐
๐ท] : the shifted global optimum.
(A.13)
(14) Noncontiguous Rastriginโs function
๐14(๐ฅ) =
๐ท
โ
๐=1
(๐ฆ2
๐โ 10 cos (2๐๐ฆ
๐) + 10) ,
๐ฆ๐=
{{
{{
{
๐ฅ๐
๐ฅ๐ <1
2round (2๐ฅ
๐)
2
๐ฅ๐ โฅ1
2,
for ๐ = 1, 2, . . . , ๐ท,
๐ = [0, 0, . . . , 0] : the global optimum.(A.14)
(15) Shifted noncontiguous Rastriginโs function
๐15(๐ฅ) =
๐ท
โ
๐=1
(๐ฆ2
๐โ 10 cos (2๐๐ฆ
๐) + 10) ,
๐ฆ๐=
{{
{{
{
๐ง๐
๐ง๐ <1
2round (2๐ง
๐)
2
๐ง๐ โฅ1
2,
for ๐ = 1, 2, . . . , ๐ท,
๐ง = ๐ฅ โ ๐,
๐ = [๐1, ๐2, . . . , ๐
๐ท] : the shifted global optimum.
(A.15)
(16) Schwefelโs function
๐16(๐ฅ) = 418.9829 ร ๐ท โ
๐ท
โ
๐=1
๐ฅ๐sin (๐ฅ๐
1/2
) ,
๐=[418.9829, 418.9829, . . . , 418.9829]: the global optimum.(A.16)
(17) Schwefelโs Problem 2.21
๐18(๐ฅ) = max {๐ฅ๐
, 1 โค ๐ โค ๐ท} ,
๐ = [0, 0, . . . , 0] : the global optimum.(A.17)
(18) Schwefelโs Problem 2.22
๐17(๐ฅ) =
๐ท
โ
๐=1
๐ฅ๐ +
๐ท
โ
๐=1
๐ฅ๐,
๐ = [0, 0, . . . , 0] : the global optimum.
(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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