Research Article A Chaos-Enhanced Particle Swarm Optimization …downloads.hindawi.com/journals/mpe/2016/6519678.pdf · 2019-07-30 · algorithms can be found in the literature, such

Research ArticleA Chaos-Enhanced Particle Swarm Optimization withAdaptive Parameters and Its Application in MaximumPower Point Tracking

Ying-Yi Hong1 Angelo A Beltran Jr23 and Arnold C Paglinawan34

1Department of Electrical Engineering Chung Yuan Christian University Chung Li District Taoyuan City 320 Taiwan2Center for Research amp Development and Department of Electronics Engineering Adamson University 1000 Manila Philippines3School of Graduate Studies Mapua Institute of Technology 1002 Manila Philippines4School of Electrical Electronics Computer Engineering Mapua Institute of Technology 1002 Manila Philippines

Correspondence should be addressed to Ying-Yi Hong yyh10632yahoocomtw

Received 11 April 2016 Accepted 5 July 2016

Academic Editor Zhen-Lai Han

Copyright copy 2016 Ying-Yi Hong et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This work proposes an enhanced particle swarm optimization scheme that improves upon the performance of the standardparticle swarm optimization algorithm The proposed algorithm is based on chaos search to solve the problems of stagnationwhich is the problem of being trapped in a local optimum and with the risk of premature convergence Type 110158401015840 constriction isincorporated to help strengthen the stability and quality of convergence and adaptive learning coefficients are utilized to intensifythe exploitation and exploration search characteristics of the algorithm Several well known benchmark functions are operated toverify the effectiveness of the proposed method The test performance of the proposed method is compared with those of otherpopular population-based algorithms in the literature Simulation results clearly demonstrate that the proposed method exhibitsfaster convergence escapes local minima and avoids premature convergence and stagnation in a high-dimensional problem spaceThe validity of the proposed PSO algorithm is demonstrated using a fuzzy logic-based maximum power point tracking controlmodel for a standalone solar photovoltaic system

1 Introduction

Swarm intelligence is becoming one of the hottest areas ofresearch in the field of computational intelligence especiallywith regard to self-organizing and decentralized systemsSwarm intelligence simulates the behavior of human andanimal populations Several swarm intelligence optimizationalgorithms can be found in the literature such as antcolony optimization artificial bee colony optimization thefirefly algorithm differential evolution and others Theseare biologically inspired optimization and computationaltechniques that are based on the social behaviors of fishbirds and humans Particle swarm optimization (PSO) isa nature-inspired algorithm that draws on the behaviorof flocking birds social interactions among humans andthe schooling of fish In fish schooling bird flocking andhuman social interactions the population is called a swarm

and candidate solutions corresponding to the individuals ormembers in the swarm are called particles Birds and fishesgenerally travel in a group without collision Accordinglyusing the group information for finding the shelter andfood each particle adjusts its corresponding position andvelocity representing a candidate solution The position ofa particle is influenced by neighbors and the best foundsolution by any particle PSO is a population-based searchtechnique that involves stochastic evolutionary optimizationIt is originally developed in 1995 by Eberhart and Kennedy[1 2] to optimize constrained and unconstrained continuousnonlinear and nondifferentiable multimodal functions [1 3]PSO is a metaheuristic algorithm that was inspired by thecollaborative or swarming behavior of biological popula-tions [4] Since then it has been applied to solve a widerange of optimization problems such as constrained andunconstrained problems multiobjective problems problems

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 6519678 19 pageshttpdxdoiorg10115520166519678

2 Mathematical Problems in Engineering

with multiple solutions and optimization in dynamic envi-ronments [5ndash8] Some of the advantages of particle swarmoptimization are the following (a) computational efficiency[6] (b) effective convergence and parameter selection [7] (c)simplicity flexibility robustness and ease of implementations[9] (d) ability to hybridize with other algorithms [10] andmany others PSO has few parameters to adjust and a smallmemory requirement and uses few CPU resources making itcomputationally efficient Unlike simulated annealing whichcan work only with discrete variables PSO can work forboth discrete and analog variables without ADC or DACconversion Also genetic algorithm optimization requirescrossover selection and mutation operators whereas PSOutilizes only the exchange of information among individualssearching the problem space repeatedly [11] In recent yearsthe use of particle swarm optimization has been investigatedwith focus on its use to solve a wide range of scientific andengineering problems such as fault detection [12] parameteridentification [13 14] power systems [15ndash17] transportation[18] electronic circuit design [19] and plant control design[20] Most relevant research focuses on either constrained orunconstrained optimization problems

The particle swarm optimization was developed to opti-mally search for the local best and the global best thesesearches are frequently known as the exploitation and explo-ration of the problem space respectively Hong et al [21]stated that exploitation involves an intense search of particlesin a local region while exploration is a long term searchwhose main objective is to find the global optimum ofthe fitness function Although particle swarm optimizationrapidly searches the solution of many complex optimizationproblems it suffers from premature convergence trappingat a local minimum the slowing down of convergence nearthe global optimum and stagnation in a particular region ofthe problem space especially in a multimodal functions andhigh-dimensional problem space If a particle is located atthe position of the global best and the preceding velocity andweight inertia are non-zero then the particle is moving awayfrom that particular point [16 22] Premature convergencehappens if no particle moves and the previous velocities arenear to zero Stagnation thus occurs if themajority of particlesare concentrated at the best position that is disclosed bythe neighbors or the swarm This fact has in recent yearsmotivated various investigations by several researchers onvariants of the particle swarm optimization in an attemptto improve the performance of exploitation and explorationand to eliminate the aforementioned problems The variousmethods of particle swarm optimization have been used forseveral purposes including scheduling classification featureselection and optimization

Mendes et al [23] presented fully informed particleswarm optimization in which during the optimizationsearch particles are influenced by the best particles intheir neighborhood and information is evenly distributedduring the generations of the algorithm Liang et al [24]proposed a comprehensive learning PSO in which eachparticle learns from the other neighborhood personal bestat different dimensions Accordingly particles update theirvelocity based on the history of the their own personal bests

Wang et al [25 26] developed the opposition-based PSOwith Cauchy mutation Their technique uses an oppositionlearning scheme in which the Cauchy mutation operatorhelps the particles move to the best positions Pant etal [27] modified the inertia weight to exhibit a Gaussiandistribution Xiang et al [28] applied the time delay conceptPSO to enable the processing of information by particlesto find the global best Cui et al [29] presented the fitnessuniform selection strategy (FUSS) and the random walkstrategy (RWS) to enhance the exploitation and explorationcapabilities of PSO Montes de Oca et al [30] developedFrankensteinrsquos PSO which incorporates several variants ofPSO in the literature such as constriction [31] the time-varying inertia weight optimizer [32 33] the fully informedparticle swarm optimizer [23] and the adaptive hierarchicalPSO [34] The adaptive PSO that was proposed by Zhan etal [35] utilized the information that was obtained from thepopulation distribution and fitness of particles to determinethe status of the swarm and an elitist learning strategy tosearch for the global optimum Juang et al [36] presented theuse of fuzzy set theory to tune automatically the accelerationcoefficients in the standard PSO A quadratic interpolationand crossover operator is also incorporated to improve theglobal search ability The literature includes hybridizationof particle swarm optimization with other stochastic orevolutionary techniques [10 37ndash39] to realize all of theirstrengths

Every modification of the particle swarm optimizationuses a different method to solve optimization problems Theinvestigations cited above therefore elucidate some improve-ments of the standard particle swarm optimization Howevervariants of particle swarm optimization generally exhibit thefollowing limitations (a) The particles may be positionedin a region that has a lower quality index than previouslyleading to a risk of premature convergence trapping in localoptima and the impossibility of further improvement of thebest positions of the particles because the inertia weightcognitive factors and social learning factors in the algorithmare not adaptive or self-organizing (b) The inclusion of themutation operator may improve the speed of convergenceNevertheless global convergence is not guaranteed becausethe method is likely to become trapped in the local optimumduring local searches of several functions (c)The probabilityin the algorithm may improve the updated positions ofparticles However the changes in the new positions ofparticles consistent with the probabilistic calculations canmove the particles into the worst positions (d) Improv-ing information sharing and the particle learning processcapability of the algorithm can provide several benefits butdoing so often increases CPU times for computing the globaloptimum (e) Integrating particle swarm optimization withother evolutionary or stochastic algorithms may increasethe number of required generations the complexity of thealgorithm and the number of required calculations

This paper proposes a novel particle swarm optimizationframework The primary merits of the proposed variant ofparticle swarm optimization are as follows (a) A modifiedsine chaos inertia weight operator is introduced overcomingthe drawback of trapping in a local minimum which is

Mathematical Problems in Engineering 3

commonly associated with an inertia weight operator Chaossearch improves the best positions of the particles favorsrapid finding of solutions in the problem space and avoidsthe risk of premature convergence (b) Type 110158401015840 constrictioncoefficient [40] is incorporated to increase the convergencerate and stability of the particle swarm optimization (c)Self-organizing adaptive cognitive and social learning coef-ficients [41] are integrated to improve the exploitation andexploration search of the particle swarm optimization algo-rithm (d) The proposed optimization algorithm has simplestructure reducing the required memory demands and thecomputational burden on the CPU It can therefore easily berealized using a few low-cost test modules

The remainder of this paper is organized as followsSection 2 presents the standard particle swarm optimiza-tion algorithm Section 3 describes the proposed variant ofparticle swarm optimization algorithm Section 4 discussesthe performance of the proposed variant of the particleswarm optimization and compares results obtained whenwell known optimization methods are applied to benchmarkfunctions The proposed variant of particle swarm optimiza-tion is further utilized in maximum power point trackingcontrol using fuzzy logic for a standalone photovoltaic sys-tem Finally a brief conclusion is drawn

2 Standard Particle Swarm Optimization

The particle swarm optimization is a simulating algorithmevolutionary and a population-based stochastic optimizationmethod that originates in animal behaviors such as theschooling of fish and the flocking of bird as well as humanbehaviors It has best position memory of all optimizationmethods and a few adjustable parameters and is easy toimplementThe standard PSO does not use the gradient of anobjective function and mutation [11] Each particle randomlymoves throughout the problem space updating its positionand velocity with the best values Each particle representsa candidate solution to the problem and searches for thelocal or global optimum Every particle retains a memory ofthe best position achieved so far and it travels through theproblem space adaptivelyThe personal best (119901119905best) is the bestsolution so far achieved by an individual in the swarm withinthe problem space 119901119905best = (119901

119905

best1 119901119905

best2 119901119905

best3 119901119905

bestpop)

while the global best (119892best) refers to globally obtainedbest solution by any particle within the swarm 119892best =

(1198921198941 1198921198942 1198921198943 119892

119894119889) in the problem space dimension 119889 The

position and velocity of particle 119894 in the problem spacedimension 119889 are thus given by 119909

119901(119905) = (119909

1198941 1199091198942 1199091198943 119909

119894119889)

and V119901(119905) = (V

1198941 V1198942 V1198943 V

119894119889) respectively The velocity

and position of a particle 119901 are adjusted as follows [1 2]

V119905+1119901

= 120596 times V119905119901+ 1198881times 1199031times (119901119905

best minus 119909119905

119901) + 1198882times 1199032

times (119892best minus 119909119905

119901)

119909119905+1

119901= 119909119905

119901+ V119905+1119901

(1)

where the superscript 119905 is the generation indexwhereas 1198881and

1198882are cognitive and social parameters which are frequently

known as acceleration constants and which are mainlyresponsible for attracting the particles toward 119901119905best and 119892bestThe terms 119903

1 1199032 and 120596 denote uniform random numbers

[1199031 1199032] isin [0 1] and inertia weight 120596 isin [0 1] respectively

These factors are mainly responsible for balancing the localand global optima search capabilities of the particles inproblem space Every generation the velocity of individualsin the swarm is computed and which adjusted velocity is usedto compute the next position of the particle To determinewhether the best solution is achieved and to evaluate theperformance of each particle the fitness function is includedThe best position of each particle is relayed to all particlesin the neighborhood The velocity and the position of eachparticle are repeatedly adjusted until the halting criteria aresatisfied or convergence is obtained

3 Chaos-Enhanced Particle SwarmOptimization with Adaptive Parameters

This section demonstrates that the proposed variant of par-ticle swarm optimization improves upon the performance ofthe standard particle swarm optimization consistent with (1)The novel scheme improves upon the performance of otherpopulation-based algorithms in solving high-dimensional ormultimodal problems Chaos operates in a nonlinear fashionand is associated with complex behavior unpredictabilitydeterminism and high sensitivity to initial conditions Inchaos a small perturbation in the initial conditions canproduce dramatically different results [42 43] In 1963Lorenz [44] presented an autonomous nonlinear differentialequation that generated the first chaotic system In recentyears the scientific community has paid increasing attentionto the chaotic systems and their applications in variousareas of science and engineering Such systems have beeninvestigated in such fields as parameter identifications [14]optimizations [45] electronic circuits [46] electric motordrives [47 48] power electronics [49] communications [50]robotics [51] and many others

Feng et al [52] introduced two means of modifying theinertia weight of a PSO using chaos The first type is thechaotic decreasing inertia weight and the second type is thechaotic random inertia weight In this paper the latter isconsidered intensifying the inertia weight parameter of thePSO The dynamic chaos random inertia weight is used toensure a balance between exploitation and exploration Alow inertia weight favors exploitation while a high inertiaweight favors exploration A static inertia weight influencesthe convergence rate of the algorithm and often leads topremature convergence Chaotic search optimization in allinstances was used herein because of its highly dynamicproperty which ensures the diversity of the particles andescape from local optimum in the process of searching for theglobal optimum

The logistic map 119911119905+1

= 120583119911119905(1 minus 119911

119905) [53 54] where 120583 = 4

is a very common chaotic map which is found in much ofthe literature on chaotic inertia weight it does not guaranteechaos on initial values of 119911

0notin 0 025 05 075 1 that may

arise during the initial generation process In this paper


1

08

06

04

02

0

Popu

latio

n siz

e

Reproduction rate1080604020

Figure 1 Bifurcations of sine map (119911119905+1

) at interval [0 1]

the sine chaotic map [54] given by (2) was utilized toavoid this shortcoming Its simplicity eliminates complexcalculations reducing the CPU time

119911119905+1

= 120573 sin (120587119911119905) (2)

where 120573 gt 0 119911119905 119911119905+1

isin [0 1] and 119905 is the generation numberFigure 1 presents the bifurcation diagram of the sine chaoticmap In some instances of generations 119911

119905+1has relatively very

small values Hence to improve the effectiveness of the chaosrandom inertia weight of particle swarm optimization theoriginal sine chaotic map is lightly modified as follows

119911119905+1

=

10038161003816100381610038161003816100381610038161003816

sin(120587119911119905

rand (sdot))

10038161003816100381610038161003816100381610038161003816

(3)

where 120573 = 1 and 119911119905 119911119905+1

isin [0 1] the absolute sign ensuresthat the next-generation process in chaos space has 119911

119905+1isin

[0 1] Therefore the chaotic random inertia weight 120596119905chaos isgiven by

120596119905

chaos = 05 times rand (sdot) + 05 times 119911119905+1 (4)

Figure 2 plots the dynamics of the modified sine chaoticmap while Figure 3 displays the bifurcation diagram

Type 110158401015840 constriction coefficient is integrated to the

proposed variant of PSO to prevent the divergence of theparticles during the search for solutions in problem spaceThe coefficient is used to fine-tune the convergence of particleswarm optimization Consider

120594 =2

(120601 minus 2 + radic1206012minus 4120601)

(5)

where the parameter 120601 = 1198881+ 1198882depends on the cognitive

and social parameters and the criterion 120601 gt 4 guaranteesthe effectiveness of the constriction coefficient Incorporatingthe above coefficient ensures the quality of convergence andthe stability of the generation process for particle swarmoptimization

0 100 200 300 400 5000

02

04

06

08

1

Chao

tic v

alue

Generations

Figure 2 Chaotic value (119911119905+1

) with 1199110= 07 obtained usingmodified

sine map (3) after 500 generations

1

08

06

04

02

0

Popu

latio

n siz

e


Figure 3 Bifurcations of modified sine map (119911119905+1

) at interval [0 1]

Time-varying cognitive and social parameters are incor-porated into PSO to improve its local and the global searchby making the cognitive component large and the socialcomponent small at the initialization or in the early part of theevolutionary process A linearly decreasing cognitive compo-nent and a linearly increasing social component in the evo-lutionary process enhance the exploitation and explorationof the PSO helping the particle swarm to converge at theglobal optimum The mathematical equation is representedas follows

119888119905

1= 1198881119891

minus119905

MAXITR(1198881119891

minus 1198881119894)

119888119905

2= 1198882119894+

119905

MAXITR(1198882119891

minus 1198882119894)

(6)

where 1198881119894 1198882119894 1198881119891

and 1198882119891

are the initial and final values of thecognitive parameters and the social parameters respectively119905 is the current generation and the MAXITR is the value inthe final generation


Table 1 Performance of different methods in minimizing sphere function based on ten simulation results (number of generations = 500)

Proposed PSO FA ACO DE

Best Fitness 74310119864 minus 14 11579119864 minus 11 31532119864 minus 08 11929119864 minus 07 12010119864 minus 08

Time 12669 14928 79243 73712 12056

Mean Fitness 460054119864 minus 13 33756119864 minus 07 430642119864 minus 08 346940119864 minus 07 278219119864 minus 08

Time 12845 14131 80510 73559 12117

Worst Fitness 12253119864 minus 12 23454119864 minus 06 57355119864 minus 08 88534119864 minus 07 54930119864 minus 08

Time 12849 14669 80826 74529 11936

Table 2 Shortest CPU times of different methods in minimizing sphere function based on ten simulation results (number of generations =500)



Time 12669 13332 79243 72014 11381


Time 13084 14928 82801 74778 12438

The above components improve the performance ofthe standard PSO Therefore the proposed mathematicalequation for the velocity and position of the particle swarmoptimization are as follows

V119905+1119901

= 120594 times 120596119905

chaos times V119905119901+ 119888119905

1times (119901119905

best minus 119909119905

119901) + 119888119905

2

times (119892119905

best minus 119909119905

119901)

119909119905+1

119901= 119909119905

119901+ V119905+1119901

(7)

The uniform random numbers [1199031 1199032] isin [0 1] from the

velocity equation of the standard PSO are not included inthe proposed PSO Figure 4 displays the flowchart of theproposed chaos-enhanced PSO

The mathematical representations and algorithmic stepsrepresent a significant improvement on the performance ofthe standard PSO A numerical benchmark test was carriedout using the unimodal and multimodal functions Thefollowing section presents and discusses the results

4 Simulation Results

In this section four benchmark test functions are used totest the performance of the proposed algorithm Subsectionselucidate the names of the benchmark test functions theirsearch spaces their mathematical representations and vari-able domains

Five programming codes were developed in Matlab 712to minimize the above benchmark functions These codescorrespond to the proposed PSO the standard PSO the fireflyalgorithm (FA) [55 56] ant colony optimization (ACO)[57 58] and differential evolution (DE) [59 60] which areevaluated using the benchmark test functions for comparativepurposes

The parameter settings for the above algorithms are asfollows For the PSO the inertia weight 120596 cognitive learning1198881 and social learning 119888

2factors are given as 099 15

and 20 respectively For the FA the light absorption 120574

attraction 120573 and mutation 120572 coefficients are 10 20 and02 respectively For ACO the selection pressure 119902 anddeviation-to-distance ratio 120589 are 05 and 10 respectivelyThe roulette wheel selection method is used for ACO Themutation coefficient 120573 and the crossover rate for DE are 08and 02 respectively The population size and the number ofunknowns (dimensions) for all population-based algorithmsthat were used in the benchmark test are 20 Ten simulationstests are performed using each of the algorithms in order toevaluate their performance in minimization

To verify the optimality and robustness of the algorithmstwo convergence criteria are adopted the convergence tol-erance and the fixed maximum number of generations Thedesktop computer that was used in the benchmark testfunction experiments ran the Microsoft Windows 7 64-BitOperating System and had an Intel (R) Core 1198945 (330GHz)processor with 80GB RAM installed

41 Benchmark Testing Sphere Function The sphere functionis given by 119891

1(119909) = sum

119889

119894=11199092

119894 minus100 lt 119909

119894lt 100 where

119889 = 20 and 119909 = (1199091 1199092 119909

20) The sphere function 119891

1(119909)

is a unimodal test function whose global optimum value is1198911(119909) = 0 and 119909

1= 1199092= 1199093= sdot sdot sdot = 119909

20= 0 Table 1 presents

the best mean and worst values obtained by running allthe algorithms through 500 generations Figure 5 showsthe performance for the maximum number of generationsof each population-based algorithms in minimizing 119891

1(119909)

Table 2 presents the shortest CPU times that were requiredto minimize 119891

1(119909) under 500 generations and Figure 6 plots

this information for all algorithmsIn the next experimental test the convergence tolerance

was set to 0001 Table 3 presents the best mean andworst values that were used to minimize 119891

1(119909) using all

techniques based on ten simulation results Almost alltechniques provide similar solutions As presented in Table 3the proposed method gives smaller values of 119891

1(119909) in the

fewest generations and in the shortest CPU times Figure 7shows the convergence performance in minimizing 119891

1(119909)


Initialize velocities and population of particles with random positions

Update the modified sine chaotic map and chaos random

inertia weight according to (3) and (4)

Update the velocity and position of particles using

(7)

End of criterion

Initialization control parameters and settings (3) (4)

and (5)

Initialization

No

Evaluate the fitness value of the particles

Return

Start

Update the cognitive and social learning coefficients according to

(6)

Calculate the fitness value for each particles

No

YesEnd

Yes

No

Reproduction and updating

loop

Yes

t = 0

i = 1

Set Pbest and Gbest

t = 1

i = 1

with best fitnessUpdate Pbest and Gbest

Gbest

t = t + 1

i = i + 1

i gt pop

i = i + 1

i gt pop

Figure 4 Flowchart of chaos-enhanced PSO with adaptive parameters


Table 3 Performance of methods in minimizing sphere function based on ten simulation results (convergence tolerance = 0001)


BestFitness 68672119864 minus 04 43526119864 minus 04 70053119864 minus 04 56728119864 minus 04 65308119864 minus 04

Generations 162 191 252 354 306Time 04266 04942 41338 51278 07515

MeanFitness 86914119864 minus 04 70088119864 minus 04 849667119864 minus 04 779853119864 minus 04 906594119864 minus 04


WorstFitness 995120119864 minus 04 92794119864 minus 04 94996119864 minus 04 99648119864 minus 04 99164119864 minus 04


Table 4 Shortest CPU times for minimizing sphere function based on ten simulation results (convergence tolerance = 0001)






10 20 30 40 50 60 70 800

05

1

15

2

25

3

35

4

Generations

Fitn

ess v

alue

times104

ACODEFA

PSOProposed

Figure 5 Maximum number of generations in which differentalgorithms minimize sphere function

Table 4 illustrates the shortest and the longest CPUtimes based on ten simulation tests Table 4 reveals thatthe proposed method yields a small fitness of 119891

1(119909)

in the shortest CPU times and in the fewest genera-tions Figure 8 displays the number of generations ofthe different algorithmic methods and their shortest CPUtimes

42 Benchmark Testing Powell Function The Powell 1198912(119909) =

sum1198894

119894=1[(1199094119894minus3

+ 101199094119894minus2

)2+ 5(119909

4119894minus1minus 1199094119894)2+ (1199094119894minus2

minus 21199094119894minus1

)4

10 20 30 40 50 60 70 80Generations

ACODEFA

PSOProposed

0

1

2

3

4

5

6

Fitn

ess v

alue

times104

Figure 6 Shortest CPU times in terms of maximum numberof generations for minimizing sphere function using differentalgorithms

+ 10(1199094119894minus3

minus 1199094119894)4] minus4 lt 119909

119894lt 5 where 119889 = 20 and

119909 = (1199091 1199092 119909

20) is a multimodal function whose global

optimum value is 1198912(119909) = 0 and 119909

1= 1199092

= 1199093

=

sdot sdot sdot = 11990920

= 0 Table 5 presents the best mean andworst values of minimizing Powell function obtained in 500generations using all population-based algorithms Figure 9plots the performance of the algorithms inminimizing 119891

2(119909)

Table 6 and Figure 10 display the shortest CPU times of thealgorithm


Table 5 Performance of methods used to minimize Powell function based on ten simulation results (number of generations = 500)



Time 17946 17312 117101 83304 16328


Time 17969 18037 116532 82524 16128


Time 18766 18609 117172 81815 15889

Table 6 Shortest CPU times forminimizing Powell function using variousmethods based on ten simulation results (number of generations =500)



Time 17684 17275 115836 81192 15722


Time 18766 18609 117172 83445 16441

10 20 30 40 50 60 70 800

1

2

3

4

5

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed

times104

Figure 7 Convergence performance of different algorithms tominimize sphere function

Table 7 represents the best mean and worst values of theminimum 119891

2(119909) that is obtained using all techniques with

the convergence tolerance set to 0001The proposed methodyields the best value 119891

2(119909) in fewest generations based on ten

simulation results Figure 11 displays the minimum 1198912(119909) at

convergence Table 8 provides the shortest and longest CPUtimes of the algorithms based on the ten simulation resultsTable 8 indicates that the proposed method has shortestCPU times Figure 12 displays the generation of the differentpopulation-based algorithms

43 Benchmark Testing Griewank Function The Griewankfunction is given by 119891

3(119909) = sum

119889

119894=1(1199092

1198944000) minus prod

119889

119894=1cos(119909119894

radic119894) + 1 minus600 lt 119909119894lt 600 where 119889 = 20 and 119909 = (119909

1

10 20 30 40 50 60 70 800

1

2

3

4

5

Generations

Fitn

ess v

alue

times104

ACODEFA

PSOProposed

Figure 8 Shortest CPU times for convergence inminimizing spherefunction using various algorithms

1199092 119909

20) The Griewank function 119891

3(119909) is a highly multi-

modal function whose global optimum value is 1198913(119909) = 0

and 1199091= 1199092= 1199093= sdot sdot sdot = 119909


the best mean and worst values obtained using all of thetested algorithms in 500 generations Figure 13 presents theperformance of the different algorithms inminimizing119891

3(119909)

Table 10 and Figure 14 provide the shortest CPU timesrequired by the various algorithms

Table 11 presents the best mean and worst values thatare used to minimize 119891

3(119909) using all techniques based on

ten simulation results The convergence tolerance was setto 0001 As presented the proposed method provides thebest mean value of 119891

3(119909) in the fewest generations and

the shortest CPU time Figure 15 shows the convergence


Table 7 Performance of methods used to minimize Powell function based on ten simulation results (convergence tolerance = 0001)








Table 8 Shortest CPU times of methods used to minimize Powell function based on ten simulation results (convergence tolerance = 0001)






Table 9 Performance of various methods used to minimize Griewank function based on ten simulation results (number of generations =500)



Time 15032 15415 87746 76647 13218


Time 14964 15460 89015 76075 13114


Time 15037 15000 89016 75366 13135

Table 10 Shortest CPU times in which variousmethodsminimizeGriewank function based on ten simulation results (number of generations= 500)



Time 14636 15000 87636 73618 12926


Time 15142 15975 90928 78357 13246

Table 11 Performance of different methods used to minimize Griewank function based on ten simulation results (convergence tolerance =0001)









Table 12 Shortest CPU times of different methods used to minimize Griewank function based on ten simulation results (convergencetolerance = 0001)






20 40 60 80 100 120 1400

2000

4000

6000

8000

10000

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed

Figure 9 Maximum number of generations in which algorithmsminimize Powell function

50 100 150 2000

2000

4000

6000

8000

10000

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed

Figure 10 Shortest CPU times in terms of maximum number ofgenerations in which algorithms minimize Powell function

performance of each method in minimizing 1198913(119909) Table 12

presents the shortest and the longest CPU times based on theten simulation tests of the different algorithms Table 12 showsthat the proposed method yields the smallest value of 119891

3(119909)

in the shortest CPU times Figure 16 plots the generation ofthe different algorithms

44 Benchmark Testing Ackley Function The Ackley func-

tion1198914(119909) = minus20119890

minus02timesradic(1119889)sum119889

119894=11199092

119894 minus119890(1119889)sum

119889

119894=1cos(2120587times119909119894)+20+1198901

minus32 lt 119909119894lt 32 where 119889 = 20 and 119909 = (119909

1 1199092 119909

20) is a

multimodal function whose global optimum value is 1198914(119909) =

0 and 1199091= 1199092= 1199093= sdot sdot sdot = 119909

20= 0 Table 13 presents the best

mean and worst values obtained using all of the algorithmsin 500 generations Figure 17 presents the performance ofthe algorithms in minimizing 119891

4(119909) Table 14 and Figure 18

highlight the shortest CPU times of the different population-based algorithms

Table 15 presents the best mean and worst values of theminimum 119891

4(119909) that are obtained using all techniques when

the convergence tolerance was set to 0001 Based on theten simulation results the proposed method provides bestoptimal value of 119891

4(119909) in the fewest generations Figure 19

plots the convergence value in the minimizing of 1198914(119909)

Table 16 presents the shortest and longest CPU times requiredby the different algorithms based on the ten simulationresults Table 16 reveals that the proposed method yields asmallest fitness value of 119891

4(119909) Figure 20 plots the generation

of the different population-based algorithms

45 FLC Optimized by Chaos-Enhanced PSO with AdaptiveParameters for Maximum Power Point Tracking in Stan-dalone Photovoltaic System Developing fuzzy logic controlfor theMPPT [61ndash67] involves determining the scaling factorparameters and the shape of the fuzzymembership functionsThe two inputs and one output for this purpose are 119864(119899)which is tracking error Δ119864(119899) which is change of trackingerror and Δ119863(119899) which is the change of the duty cycleThey are selected to tune optimally the fuzzy logic controllerThe mathematical description are given as 119864(119899) = (119901(119899) minus

119901(119899 minus 1))(V(119899) minus V(119899 minus 1)) and Δ119864(119899) = 119864(119899) minus 119864(119899 minus 1)In this case 119901(119899) and V(119899) are the instantaneous power andvoltage of the PV respectively 119864(119899) represents the operatingpower point of the load whether it is currently located onthe left hand side or right hand side while Δ119864(119899) denotes


20 40 60 80 100 120 1400

1000

2000

3000

4000

5000

6000

7000

8000

Generations

Fitn

ess v

alue

ACODE

(a)

2 4 6 8 100

2000

4000

6000

8000

10000

Generations

Fitn

ess v

alue

FAPSOProposed

(b)

Figure 11 Convergence performance of algorithms used to minimize Powell function (a) ACO and DE (b) FA PSO and proposed

20 40 60 80 100 120 1400

5000

10000

15000

Generations

Fitn

ess v

alue

ACODE

(a)

2 4 6 8 100

2000

4000

6000

8000

Generations

Fitn

ess v

alue

FAPSOProposed

(b)

Figure 12 Shortest CPU times performance for convergence of algorithms used to minimize Powell function (a) ACO and DE (b) FA PSOand proposed

Table 13 Performance of different methods used in minimizing Ackley function based on ten simulation results (number of generations =500)



Time 14763 16495 95650 76734 14923


Time 14848 16477 95221 76975 14897


Time 15081 16460 95146 77872 14923


Table 14 Shortest CPU times of different methods in minimizing Ackley function based on ten simulation results (number of generations =500)



Time 14625 15487 94085 75984 14536


Time 14996 17014 97459 79017 16048

Table 15 Performance of different methods used to minimize Ackley function based on ten simulation results (convergence tolerance =0001)








20 40 60 80 1000

50

100

150

200

250

300

350

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed

Figure 13 Maximum number of generations in which algorithmsminimize Griewank function

the direction of motion of the operating point The fuzzyinference system (FIS) approach used herein for maximumpower point tracking was theMamdani system with the min-max fuzzy combination operation for maximum power pointtracking The defuzzification method that was used to obtainthe actual value of the duty cycle signal as a crisp outputwas the center of gravity-based method The equation is119863 =

sum119899

119894=1(120583(119863119894) minus119863119894)sum119899

119894=1120583(119863119894) The variable output is the pulse

width modulation signal which is transmitted to the DCDCboost converter to drive the necessary load (Table 18) Thechaos-enhanced particle swarm optimization with adaptive

20 40 60 80 1000

50

100

150

200

250

300

350

400

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed

Figure 14 Shortest CPU times in terms of maximum numberof generations in which various algorithms minimize Griewankfunction

parameters was utilized to determine the parameter of thescaling factors and to optimize the width of each inputsand output membership function Each of these inputs andoutputs includes five membership functions of the fuzzylogic

The operation of the chaos-enhanced PSO with adap-tive parameters begins by generating a solution from therandomly generated population with the best positions Thevelocity equation yields particles in better positions throughthe application of chaos search and the self-organizing


Table 16 Shortest CPU times of different methods used to minimize Ackley function based on ten simulation results (convergence tolerance= 0001)






20 40 60 80 1000

50

100

150

200

250

300

350

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed

Figure 15 Convergence performance of different algorithms usedto minimize Griewank function

20 40 60 80 1000

50

100

150

200

250

300

350

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed

Figure 16 Shortest CPU times for convergence of different algo-rithms to minimize Griewank function

50 100 150 2000

5

10

15

20

25

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed

Figure 17 Maximum number of generations in which differentalgorithms minimize Ackley function

50 100 150 2000

5

10

15

20

25

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed

Figure 18 Shortest CPU times in terms of maximum numberof generations in which different algorithms minimize Ackleyfunction


50 100 150 200 2500

5

10

15

20

25

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed

Figure 19 Convergence performance of different algorithms usedto minimize Ackley function

50 100 150 200 2500

5

10

15

20

25

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed

Figure 20 Shortest CPU times in terms of convergence for differentalgorithms used to minimize Ackley function

parameters In this paper the cost function that is usedas the performance index is based on the minimization ofthe integral absolute error (IAE) ObjFunc = int

infin

0|119901(119905)ref minus

119901(119905)out|119889119905 The fitness value is calculated using Fitness =

2000 minus ObjFunc The measured cost function yields thedynamic maximum output power of the boost converterThe maximum power point tracking control was carried outusing the fuzzy logic and scaling factor controllers Figure 21presents the model that was used to tune the parameters ofthe fuzzy logic controller during the process of optimizationThe benchmark test is conducted with a variable irradianceand temperature of PV module as operating conditionswhich is shown in Figure 22 and the DCDC boost converter

Table 17 SunPower SPR-305-WHT PV array electrical characteris-tics

Parameters Variable ValueTotal number of cells 119873cell 96

Number of series connectedmodules per string 119873ser 1

Number of parallel strings 119873par 4

Open circuit voltage 119881oc 642 VShort circuit current 119868sc 596AVoltage at maximum power 119881

119898547 V

Current at maximum power 119868119898

558AMaximum power (plusmn5) 119875

119898305W

Maximum system voltage IEC UL 1000V 600VReference cell temperature 119879ref 25∘CReference irradiation 119878ref 1000Wm2

Table 18 DCDC boost converter specifications

Parameters Variable ValueInput filter capacitance 119862if 2mFBoost inductance 119871 001HOutput filter capacitance 119862of 2mFLoad resistance 119877

119871500Ω

Table 19 Fuzzy logic control rule base

Change in error Δ119864(119899)NB NS ZE PS PB

Error119864(119899)

NB ZE ZE PB PB PBNS ZE ZE PS PS PSZE PS ZE ZE ZE NSPS NS NS NS ZE ZEPB NB NB NB ZE ZE

is utilized to validate the optimized fuzzy logic maximumpower point tracking controller All updates and transfer ofdata are executed as set in the model During the generationprocess the parameters of the fuzzy logic controllers and thescaling factors are updated These parameters are retaineduntil a new global fitness is obtained during the optimizationprocess At the end of each generation the parameters inthe fuzzy logic and the scaling factors are updated based onthe obtained global fitness until a convergence is made forthe best solution found so far by the swarm The Appendixpresents the solar PV array specifications (also see Table 17)the parameters used for DCDC boost converter [68ndash72]and the rule base of the fuzzy logic controller (Table 19)Figure 23 displays the optimal best inputs and output widthofmembership functions obtained by the swarmTheoptimalfuzzy logic solution yields symmetric triangular membershipfunctions for 119864(119899) Δ119864(119899) and Δ119863(119899) The chaos-enhancedPSO with adaptive parameters causes the maximum powerpoint tracking of a PV system to converge toward the bestfitness that has been obtained by the swarm Figure 24 shows


Fuzzy logic controller

PWM generator

DCDC converter(boost)

Resistive load

Currentvoltage sensor

Solar PVarray PWM

Chaos-enhanced PSO with adaptive parameters

Scaling factor (K3)

Scaling factor (K2)

Scaling factor (K1)

Insolation

Temperature

Maximum power point tracking

Vm Vo

+

minus

+

minus

Im

E(n)

ΔE(n)

D(n) = ΔD(n) + D(n minus 1)

ddt

Figure 21 Block diagram of maximum power point tracking for standalone PV system

0 1 2 3 4 5 6 7 8 9 10 11300400500600700800900

Irra

dian

ce

Time (sec)

(a)

0 1 2 3 4 5 6 7 8 9 10 1125

30

35

40

Tem

pera

ture

Time (sec)

(b)

Figure 22 Staircase change of solar irradiation (300 to 900Wm2) and temperature (25∘C to 40∘C)

the output powerThe obtained optimal fuzzy logic controllerismore robust than and outperforms othermaximumpowerpoint tracking algorithms

5 Conclusion

This paper presents a novel technique with promising newfeatures to enhance the performance and robustness ofthe standard PSO for solving optimization problems Theimproved technique incorporates chaos searching to avoidthe risk of stagnation premature convergence and trappingin a local optimum it incorporates type 110158401015840 constriction toimprove the quality of convergence and adaptive cognitiveand social learning coefficients to improve the exploitationand exploration search characteristics of the algorithm Theproposed chaos-enhanced PSOwith adaptive parameters wasexperimentally tested in high-dimensional problem space

using a four benchmark functions to verify its effectivenessThe advantages of the chaos-enhanced PSO with adap-tive parameters over the population-based algorithms areverified and the numerical results demonstrate that theproposed technique offers a faster convergence with nearprecise results better reliability and lower computationalburden avoids stagnation and premature convergence andcan escape from local minimum and low CPU time andspeed requirements A complete stand-alone PV model wasdeveloped in which maximum power point tracking controlwith fuzzy logic is utilized to evaluate the performance of theproposed algorithm in real-world engineering optimizationapplications

It is envisaged that the proposed chaos-enhanced PSOwith adaptive parameters can be applied to a wider classof complex scientific and engineering problems such aselectric power system optimization (eg minimization ofnonconvex fuel cost and power losses) robust design


0

05

1Membership function plots

NB NS ZE PS PB

43210minus1minus2minus3minus4

181Plot points

Input variable ldquoE(n)rdquo

(a)

0

05


NB NS ZE PS PB

4 620minus2minus4minus6

Input variable ldquoΔE(n)rdquo

181Plot points

(b)

NB NS ZE PS PBMembership function plots 181Plot points

0

05

1

minus005 minus004 minus003 minus002 001 0 001 002 003 004 005

Output variable ldquoΔD(n)rdquo

(c)

Figure 23 Graphical representation of obtained optimal fuzzy logic membership function for inputs (a) 119864(119899) and (b) Δ119864(119899) and output (c)Δ119863(119899) linguistic variables

0 1 2 3 4 65 9 10 1187Time (sec)

Time offset 0

OFLCIncCondFLC

Output power (Watts)

0100200300400500600700

Figure 24 Output power response of optimal tuned fuzzy logic(OFLC) incremental conductance (IncCond) and fuzzy logic (FLC)under variable irradiance and temperature

of nonlinear plant control system under the presence ofparametric uncertainties and forecasting of wind farm out-put power using the artificial neural network where theconnection weights and thresholds are needed to be adjustedoptimally

Appendix

This appendix presents the solar PV array specifications [73]DCDCboost converter parameters and fuzzy logic rule baseof the five membership functions that are used in the DCDCboost converter

Themaximumpower for a single PV array (watts) is givenas follows

119875array = 119875119898times 119873ser times 119873par (A1)

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the Ministry of Science andTechnology of the Republic of China Taiwan for financiallysupporting this research under Contract MOST 104-2221-E-033-029

References

[1] R Eberhart and J Kennedy ldquoNew optimizer using particleswarm theoryrdquo in Proceedings of the 6th International Sympo-sium on Micro Machine and Human Science (MHS rsquo95) pp 39ndash43 Nagoya Japan October 1995

[2] J Kennedy and R C Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks vol 4 pp 1942ndash1948 Perth Wash USA December1995

[3] F Marini and B Walczak ldquoParticle swarm optimization (PSO)A tutorialrdquo Chemometrics and Intelligent Laboratory Systemsvol 149 pp 153ndash165 2015

[4] S Bouallegue J Haggege and M Benrejeb ldquoParticle swarmoptimization-based fixed-structure H

infincontrol designrdquo Inter-

national Journal of Control Automation and Systems vol 9 no2 pp 258ndash266 2011

[5] R C Eberhart and Y Shi ldquoParticle swarm optimization devel-opments applications and resourcesrdquo in Proceedings of theIEEE Congress on Evolutionary Computation pp 81ndash86 SeoulRepublic of Korea May 2001

[6] F Van den Bergh An analysis of particle swarm optimizers[PhD thesis] University of Pretoria Pretoria South Africa2006


[7] E P Ruben and B Kamran ldquoParticle swarm optimization instructural designrdquo in Swarm Intelligence Focus on Ant andParticle Swarm Optimization F T S Chan and M K TiwariEds pp 373ndash394 I-Tech Education and Publication ViennaAustria 2007

[8] K E Parsopoulos andMN Vrahatis Particle SwarmOptimiza-tion and Intelligence Advances and Applications IGI GlobalHershey Pa USA 2010

[9] R Poli J Kennedy and T Blackwell ldquoParticle swarm optimiza-tion an overviewrdquo Swarm Intelligence vol 1 no 1 pp 33ndash572007

[10] H-Q Li and L Li ldquoA novel hybrid particle swarm optimizationalgorithm combinedwith harmony search for high dimensionaloptimization problemsrdquo in Proceedings of the InternationalConference on Intelligent Pervasive Computing (IPC rsquo07) pp 94ndash97 IEEE Jeju South Korea October 2007

[11] A Khare and S Rangnekar ldquoA review of particle swarmoptimization and its applications in solar photovoltaic systemrdquoApplied Soft Computing Journal vol 13 no 5 pp 2997ndash30062013

[12] B Samanta and C Nataraj ldquoUse of particle swarm optimizationfor machinery fault detectionrdquo Engineering Applications ofArtificial Intelligence vol 22 no 2 pp 308ndash316 2009

[13] L Liu W Liu and D A Cartes ldquoParticle swarm optimization-based parameter identification applied to permanent magnetsynchronous motorsrdquo Engineering Applications of ArtificialIntelligence vol 21 no 7 pp 1092ndash1100 2008

[14] HModares A Alfi andM-M Fateh ldquoParameter identificationof chaotic dynamic systems through an improved particleswarm optimizationrdquo Expert Systems with Applications vol 37no 5 pp 3714ndash3720 2010

[15] Y del Valle G K Venayagamoorthy S Mohagheghi J-CHernandez and R G Harley ldquoParticle swarm optimizationbasic concepts variants and applications in power systemsrdquoIEEE Transactions on Evolutionary Computation vol 12 no 2pp 171ndash195 2008

[16] W Jiekang Z Jianquan C Guotong and Z Hongliang ldquoAhybrid method for optimal scheduling of short-term electricpower generation of cascaded hydroelectric plants based onparticle swarm optimization and chance-constrained program-mingrdquo IEEE Transactions on Power Systems vol 23 no 4 pp1570ndash1579 2008

[17] Y-Y Hong F-J Lin Y-C Lin and F-Y Hsu ldquoChaotic PSO-based VAR control considering renewables using fast proba-bilistic power flowrdquo IEEE Transactions on Power Delivery vol29 no 4 pp 1666ndash1674 2014

[18] Y Marinakis M Marinaki and G Dounias ldquoA hybrid particleswarm optimization algorithm for the vehicle routing problemrdquoEngineering Applications of Artificial Intelligence vol 23 no 4pp 463ndash472 2010

[19] G K Venayagamoorthy S C Smith and G Singhal ldquoParticleswarm-based optimal partitioning algorithm for combinationalCMOS circuitsrdquo Engineering Applications of Artificial Intelli-gence vol 20 no 2 pp 177ndash184 2007

[20] S Bouallegue J Haggege M Ayadi and M Benrejeb ldquoPID-type fuzzy logic controller tuning based on particle swarmoptimizationrdquo Engineering Applications of Artificial Intelligencevol 25 no 3 pp 484ndash493 2012

[21] Y-Y Hong F-J Lin S-Y Chen Y-C Lin and F-Y Hsu ldquoANovel adaptive elite-based particle swarm optimization appliedto VAR optimization in electric power systemsrdquo Mathematical

Problems in Engineering vol 2014 Article ID 761403 14 pages2014

[22] S Saini D R B A RambliMN B Zakaria and S B SulaimanldquoA review on particle swarm optimization algorithm and itsvariants to human motion trackingrdquoMathematical Problems inEngineering vol 2014 Article ID 704861 16 pages 2014

[23] R Mendes J Kennedy and J Neves ldquoThe fully informedparticle swarm simpler maybe betterrdquo IEEE Transactions onEvolutionary Computation vol 8 no 3 pp 204ndash210 2004

[24] J J Liang A K Qin P N Suganthan and S Baskar ldquoCompre-hensive learning particle swarm optimizer for global optimiza-tion of multimodal functionsrdquo IEEE Transactions on Evolution-ary Computation vol 10 no 3 pp 281ndash295 2006

[25] H Wang C Li Y Liu and S Zeng ldquoA hybrid particle swarmalgorithm with cauchy mutationrdquo in Proceedings of the IEEESwarm Intelligence Symposium (SIS rsquo07) pp 356ndash360 IEEEHonolulu Hawaii USA April 2007

[26] H Wang Y Liu S Zeng H Li and C Li ldquoOpposition-basedparticle swarm algorithmwith cauchymutationrdquo in Proceedingsof the IEEE Congress on Evolutionary Computation (CEC rsquo07)pp 4750ndash4756 IEEE Singapore September 2007

[27] M Pant T Radha andV P Singh ldquoParticle swarmoptimizationusing gaussian inertia weightrdquo in Proceedings of the Interna-tional Conference on Computational Intelligence andMultimediaApplications vol 1 pp 97ndash102 Sivakasi India December 2007

[28] T Xiang K-W Wong and X Liao ldquoA novel particle swarmoptimizer with time-delayrdquo Applied Mathematics and Compu-tation vol 186 no 1 pp 789ndash793 2007

[29] Z Cui X Cai J Zeng andG Sun ldquoParticle swarmoptimizationwith FUSS and RWS for high dimensional functionsrdquo AppliedMathematics and Computation vol 205 no 1 pp 98ndash108 2008

[30] M A Montes de Oca T Stutzle M Birattari and M DorigoldquoFrankensteinrsquos PSO a composite particle swarm optimizationalgorithmrdquo IEEE Transactions on Evolutionary Computationvol 13 no 5 pp 1120ndash1132 2009

[31] M Clerc and J Kennedy ldquoThe particle swarm-explosion sta-bility and convergence in a multidimensional complex spacerdquoIEEE Transactions on Evolutionary Computation vol 6 no 1pp 58ndash73 2002

[32] R Eberhart and Y Shi ldquoParameter selection in particle swarmoptimizationrdquo in Proceedings of the 7th International Conferenceon Evolutionary Programming (EP rsquo98) pp 591ndash600 San DiegoCalif USA March 1998

[33] R C Eberhart and Y Shi ldquoComparing inertia weights and con-striction factors in particle swarm optimizationrdquo in Proceedingsof the Congress on Evolutionary Computation (CEC rsquo00) vol 1pp 84ndash88 La Jolla Calif USA July 2000

[34] S Janson and M Middendorf ldquoA hierarchical particle swarmoptimizer and its adaptive variantrdquo IEEE Transactions onSystems Man and Cybernetics Part B Cybernetics vol 35 no6 pp 1272ndash1282 2005

[35] Z-H Zhan J Zhang Y Li and H S-H Chung ldquoAdaptiveparticle swarm optimizationrdquo IEEE Transactions on SystemsMan and Cybernetics Part B Cybernetics vol 39 no 6 pp1362ndash1381 2009

[36] Y-T Juang S-L Tung andH-C Chiu ldquoAdaptive fuzzy particleswarm optimization for global optimization of multimodalfunctionsrdquo Information Sciences vol 181 no 20 pp 4539ndash45492011

[37] P S Shelokar P Siarry V K Jayaraman and B D KulkarnildquoParticle swarm and ant colony algorithms hybridized for


improved continuous optimizationrdquo Applied Mathematics andComputation vol 188 no 1 pp 129ndash142 2007

[38] B Xin J Chen J Zhang H Fang and Z-H Peng ldquoHybridizingdifferential evolution and particle swarmoptimization to designpowerful optimizers a review and taxonomyrdquo IEEE Transac-tions on Systems Man and Cybernetics Part C Applications andReviews vol 42 no 5 pp 744ndash767 2012

[39] A Kaveh and V R Mahdavi ldquoA hybrid CBO-PSO algorithmfor optimal design of truss structureswith dynamic constraintsrdquoApplied Soft Computing vol 34 pp 260ndash273 2015

[40] M S Innocente and J Sienz ldquoParticle swarm optimization withinertia weight and constriction factorrdquo in Proceedings of theInternational Joint Conference on Swarm Intelligence (ICSI rsquo11)pp 1ndash11 EISTI Cergy France June 2011

[41] Y-H Liu S-C Huang J-W Huang and W-C Liang ldquoAparticle swarm optimization-based maximum power pointtracking algorithm for PV systems operating under partiallyshaded conditionsrdquo IEEE Transactions on Energy Conversionvol 27 no 4 pp 1027ndash1035 2012

[42] E Ott C Grebogi and J A Yorke ldquoControlling chaosrdquo PhysicalReview Letters vol 64 no 11 pp 1196ndash1199 1990

[43] L Liu Z Han and Z Fu ldquoNon-fragile sliding mode controlof uncertain chaotic systemsrdquo Journal of Control Science andEngineering vol 2011 Article ID 859159 6 pages 2011

[44] E N Lorenz ldquoDeterministic non periodic flowrdquo Journal of theAtmospheric Sciences vol 20 no 11 pp 131ndash141 1963

[45] V Pediroda L Parussini C Poloni S Parashar N Fateh andM Poian ldquoEfficient stochastic optimization using chaos collo-cationmethod with modefrontierrdquo SAE International Journal ofMaterials and Manufacturing vol 1 no 1 pp 747ndash753 2009

[46] Y Huang P Zhang andW Zhao ldquoNovel gridmultiwing butter-fly chaotic attractors and their circuit designrdquo IEEETransactionson Circuits and Systems II Express Briefs vol 62 no 5 pp 496ndash500 2015

[47] X H Mai D Q Wei B Zhang and X S Luo ldquoControllingchaos in complex motor networks by environmentrdquo IEEETransactions on Circuits and Systems II Express Briefs vol 62no 6 pp 603ndash607 2015

[48] Z Wang J Chen M Cheng and K T Chau ldquoField-orientedcontrol and direct torque control for paralleled VSIs Fed PMSMdrives with variable switching frequenciesrdquo IEEE Transactionson Power Electronics vol 31 no 3 pp 2417ndash2428 2016

[49] J D Morcillo D Burbano and F Angulo ldquoAdaptive ramptechnique for controlling chaos and subharmonic oscillationsin DC-DC power convertersrdquo IEEE Transactions on PowerElectronics vol 31 no 7 pp 5330ndash5343 2016

[50] G Kaddoum and F Shokraneh ldquoAnalog network coding formulti-user multi-carrier differential chaos shift keying commu-nication systemrdquo IEEE Transactions on Wireless Communica-tions vol 14 no 3 pp 1492ndash1505 2015

[51] J M Valenzuela ldquoAdaptive anti control of chaos for robotmanipulators with experimental evaluationsrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 18 no 1 pp1ndash11 2013

[52] Y Feng G-F Teng A-XWang andY-M Yao ldquoChaotic inertiaweight in particle swarmoptimizationrdquo inProceedings of the 2ndInternational Conference on Innovative Computing Informationand Control (ICICIC rsquo07) p 475 Kumamoto Japan September2007

[53] M Ausloos and M Dirickx The Logistic Map and the Routeto Chaos Understanding Complex Systems Springer BerlinGermany 2006

[54] A M Arasomwan and A O Adewumi ldquoAn investigation intothe performance of particle swarm optimization with variouschaotic mapsrdquoMathematical Problems in Engineering vol 2014Article ID 178959 17 pages 2014

[55] X-S Yang ldquoFirefly algorithms for multimodal optimizationrdquoin Stochastic Algorithms Foundations and Applications 5thInternational Symposium SAGA 2009 Sapporo Japan October26ndash28 2009 Proceedings vol 5792 of Lecture Notes in ComputerScience pp 169ndash178 Springer Berlin Germany 2009

[56] X S Yang Engineering Optimisation An Introduction withMetaheuristic Applications JohnWiley and SonsNewYorkNYUSA 2010

[57] MDorigo VManiezzo andA Colorni ldquoAnt system optimiza-tion by a colony of cooperating agentsrdquo IEEE Transactions onSystems Man and Cybernetics Part B Cybernetics vol 26 no1 pp 29ndash41 1996

[58] M Dorigo and L M Gambardella ldquoAnt colony system a coop-erative learning approach to the traveling salesman problemrdquoIEEETransactions on Evolutionary Computation vol 1 no 1 pp53ndash66 1997

[59] R Storn ldquoOn the usage of differential evolution for functionoptimizationrdquo in Proceedings of the Biennial Conference of theNorth American Fuzzy Information Processing Society (NAFIPSrsquo96) pp 519ndash523 Berkeley Calif USA June 1996

[60] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997

[61] J-K Shiau M-Y Lee Y-C Wei and B-C Chen ldquoCircuit sim-ulation for solar power maximum power point tracking withdifferent buck-boost converter topologiesrdquo Energies vol 7 no8 pp 5027ndash5046 2014

[62] J-K Shiau Y-C Wei and B-C Chen ldquoA study on the fuzzy-logic-based solar powerMPPT algorithms using different fuzzyinput variablesrdquo Algorithms vol 8 no 2 pp 100ndash127 2015

[63] P-C Cheng B-R Peng Y-H Liu Y-S Cheng and J-WHuang ldquoOptimization of a fuzzy-logic-control-based MPPTalgorithm using the particle swarm optimization techniquerdquoEnergies vol 8 no 6 pp 5338ndash5360 2015

[64] A Mellit A Messai A Guessoum and S A Kalogirou ldquoMax-imum power point tracking using a GA optimized fuzzy logiccontroller and its FPGA implementationrdquo Solar Energy vol 85no 2 pp 265ndash277 2011

[65] C Larbes S M Aıt Cheikh T Obeidi and A ZerguerrasldquoGenetic algorithms optimized fuzzy logic control for the max-imum power point tracking in photovoltaic systemrdquo RenewableEnergy vol 34 no 10 pp 2093ndash2100 2009

[66] L K Letting J L Munda and Y Hamam ldquoOptimization ofa fuzzy logic controller for PV grid inverter control using S-function based PSOrdquo Solar Energy vol 86 no 6 pp 1689ndash17002012

[67] R Ramaprabha M Balaji and B L Mathur ldquoMaximumpower point tracking of partially shaded solar PV systemusing modified Fibonacci search method with fuzzy controllerrdquoInternational Journal of Electrical Power andEnergy Systems vol43 no 1 pp 754ndash765 2012

[68] K C Wu Pulse Width Modulated DC-DC Converters SpringerNew York NY USA 1997

[69] F L Luo and H Ye Advanced DCDC Converters CRC Press2003


[70] H Sira-Ramirez and R Silva-Ortigoza Control Design Tech-niques in Power Electronics Devices Springer London UK2006

[71] M K Kazimierczuk Pulse-Width Modulated DC-DC PowerConverters John Wiley amp Sons 2008

[72] M H Rashid Electric Renewable Energy Systems AcademicPress Cambridge Mass USA 2016

[73] SunPower Corporation SunPower 305 Solar Panel SunPowerCorporation San Jose Calif USA 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

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Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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OptimizationJournal of


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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


with multiple solutions and optimization in dynamic envi-ronments [5ndash8] Some of the advantages of particle swarmoptimization are the following (a) computational efficiency[6] (b) effective convergence and parameter selection [7] (c)simplicity flexibility robustness and ease of implementations[9] (d) ability to hybridize with other algorithms [10] andmany others PSO has few parameters to adjust and a smallmemory requirement and uses few CPU resources making itcomputationally efficient Unlike simulated annealing whichcan work only with discrete variables PSO can work forboth discrete and analog variables without ADC or DACconversion Also genetic algorithm optimization requirescrossover selection and mutation operators whereas PSOutilizes only the exchange of information among individualssearching the problem space repeatedly [11] In recent yearsthe use of particle swarm optimization has been investigatedwith focus on its use to solve a wide range of scientific andengineering problems such as fault detection [12] parameteridentification [13 14] power systems [15ndash17] transportation[18] electronic circuit design [19] and plant control design[20] Most relevant research focuses on either constrained orunconstrained optimization problems

The particle swarm optimization was developed to opti-mally search for the local best and the global best thesesearches are frequently known as the exploitation and explo-ration of the problem space respectively Hong et al [21]stated that exploitation involves an intense search of particlesin a local region while exploration is a long term searchwhose main objective is to find the global optimum ofthe fitness function Although particle swarm optimizationrapidly searches the solution of many complex optimizationproblems it suffers from premature convergence trappingat a local minimum the slowing down of convergence nearthe global optimum and stagnation in a particular region ofthe problem space especially in a multimodal functions andhigh-dimensional problem space If a particle is located atthe position of the global best and the preceding velocity andweight inertia are non-zero then the particle is moving awayfrom that particular point [16 22] Premature convergencehappens if no particle moves and the previous velocities arenear to zero Stagnation thus occurs if themajority of particlesare concentrated at the best position that is disclosed bythe neighbors or the swarm This fact has in recent yearsmotivated various investigations by several researchers onvariants of the particle swarm optimization in an attemptto improve the performance of exploitation and explorationand to eliminate the aforementioned problems The variousmethods of particle swarm optimization have been used forseveral purposes including scheduling classification featureselection and optimization

Mendes et al [23] presented fully informed particleswarm optimization in which during the optimizationsearch particles are influenced by the best particles intheir neighborhood and information is evenly distributedduring the generations of the algorithm Liang et al [24]proposed a comprehensive learning PSO in which eachparticle learns from the other neighborhood personal bestat different dimensions Accordingly particles update theirvelocity based on the history of the their own personal bests

Wang et al [25 26] developed the opposition-based PSOwith Cauchy mutation Their technique uses an oppositionlearning scheme in which the Cauchy mutation operatorhelps the particles move to the best positions Pant etal [27] modified the inertia weight to exhibit a Gaussiandistribution Xiang et al [28] applied the time delay conceptPSO to enable the processing of information by particlesto find the global best Cui et al [29] presented the fitnessuniform selection strategy (FUSS) and the random walkstrategy (RWS) to enhance the exploitation and explorationcapabilities of PSO Montes de Oca et al [30] developedFrankensteinrsquos PSO which incorporates several variants ofPSO in the literature such as constriction [31] the time-varying inertia weight optimizer [32 33] the fully informedparticle swarm optimizer [23] and the adaptive hierarchicalPSO [34] The adaptive PSO that was proposed by Zhan etal [35] utilized the information that was obtained from thepopulation distribution and fitness of particles to determinethe status of the swarm and an elitist learning strategy tosearch for the global optimum Juang et al [36] presented theuse of fuzzy set theory to tune automatically the accelerationcoefficients in the standard PSO A quadratic interpolationand crossover operator is also incorporated to improve theglobal search ability The literature includes hybridizationof particle swarm optimization with other stochastic orevolutionary techniques [10 37ndash39] to realize all of theirstrengths

Every modification of the particle swarm optimizationuses a different method to solve optimization problems Theinvestigations cited above therefore elucidate some improve-ments of the standard particle swarm optimization Howevervariants of particle swarm optimization generally exhibit thefollowing limitations (a) The particles may be positionedin a region that has a lower quality index than previouslyleading to a risk of premature convergence trapping in localoptima and the impossibility of further improvement of thebest positions of the particles because the inertia weightcognitive factors and social learning factors in the algorithmare not adaptive or self-organizing (b) The inclusion of themutation operator may improve the speed of convergenceNevertheless global convergence is not guaranteed becausethe method is likely to become trapped in the local optimumduring local searches of several functions (c)The probabilityin the algorithm may improve the updated positions ofparticles However the changes in the new positions ofparticles consistent with the probabilistic calculations canmove the particles into the worst positions (d) Improv-ing information sharing and the particle learning processcapability of the algorithm can provide several benefits butdoing so often increases CPU times for computing the globaloptimum (e) Integrating particle swarm optimization withother evolutionary or stochastic algorithms may increasethe number of required generations the complexity of thealgorithm and the number of required calculations

This paper proposes a novel particle swarm optimizationframework The primary merits of the proposed variant ofparticle swarm optimization are as follows (a) A modifiedsine chaos inertia weight operator is introduced overcomingthe drawback of trapping in a local minimum which is






119905

best1 119901119905

best2 119901119905

best3 119901119905

bestpop)


(1198921198941 1198921198942 1198921198943 119892



119901(119905) = (119909

1198941 1199091198942 1199091198943 119909

119894119889)

and V119901(119905) = (V

1198941 V1198942 V1198943 V



V119905+1119901


best minus 119909119905

119901) + 1198882times 1199032

times (119892best minus 119909119905

119901)

119909119905+1

119901= 119909119905

119901+ V119905+1119901

(1)











= 120583119911119905(1 minus 119911

119905) [53 54] where 120583 = 4





1

08

06

04

02

0

Popu

latio

n siz

e



) at interval [0 1]


119911119905+1

= 120573 sin (120587119911119905) (2)

where 120573 gt 0 119911119905 119911119905+1




119911119905+1

=

10038161003816100381610038161003816100381610038161003816

sin(120587119911119905

rand (sdot))

10038161003816100381610038161003816100381610038161003816

(3)

where 120573 = 1 and 119911119905 119911119905+1


119905+1isin


120596119905





120594 =2


(5)



0 100 200 300 400 5000

02

04

06

08

1

Chao

tic v

alue

Generations




1

08

06

04

02

0

Popu

latio

n siz

e



) at interval [0 1]


119888119905

1= 1198881119891

minus119905

MAXITR(1198881119891

minus 1198881119894)

119888119905

2= 1198882119894+

119905

MAXITR(1198882119891

minus 1198882119894)

(6)

where 1198881119894 1198882119894 1198881119891

and 1198882119891






Time 12669 14928 79243 73712 12056


Time 12845 14131 80510 73559 12117


Time 12849 14669 80826 74529 11936




Time 12669 13332 79243 72014 11381


Time 13084 14928 82801 74778 12438


V119905+1119901

= 120594 times 120596119905

chaos times V119905119901+ 119888119905

1times (119901119905

best minus 119909119905

119901) + 119888119905

2

times (119892119905

best minus 119909119905

119901)

119909119905+1

119901= 119909119905

119901+ V119905+1119901

(7)













1(119909) = sum

119889

119894=11199092

119894 minus100 lt 119909

119894lt 100 where

119889 = 20 and 119909 = (1199091 1199092 119909


1(119909)


1= 1199092= 1199093= sdot sdot sdot = 119909



1(119909)





1(119909) using all


1(119909) in the


1(119909)






(7)

End of criterion


and (5)

Initialization

No


Return

Start


(6)


No

YesEnd

Yes

No


loop

Yes

t = 0

i = 1

Set Pbest and Gbest

t = 1

i = 1


Gbest

t = t + 1

i = i + 1

i gt pop

i = i + 1

i gt pop

















10 20 30 40 50 60 70 800

05

1

15

2

25

3

35

4

Generations

Fitn

ess v

alue

times104

ACODEFA

PSOProposed



1(119909)



sum1198894

119894=1[(1199094119894minus3

+ 101199094119894minus2

)2+ 5(119909


minus 21199094119894minus1

)4

10 20 30 40 50 60 70 80Generations

ACODEFA

PSOProposed

0

1

2

3

4

5

6

Fitn

ess v

alue

times104


+ 10(1199094119894minus3

minus 1199094119894)4] minus4 lt 119909

119894lt 5 where 119889 = 20 and

119909 = (1199091 1199092 119909



1= 1199092

= 1199093

=



2(119909)






Time 17946 17312 117101 83304 16328


Time 17969 18037 116532 82524 16128


Time 18766 18609 117172 81815 15889




Time 17684 17275 115836 81192 15722


Time 18766 18609 117172 83445 16441

10 20 30 40 50 60 70 800

1

2

3

4

5

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed

times104









3(119909) = sum

119889

119894=1(1199092


119889

119894=1cos(119909119894


1

10 20 30 40 50 60 70 800

1

2

3

4

5

Generations

Fitn

ess v

alue

times104

ACODEFA

PSOProposed


1199092 119909







3(119909)

























Time 15032 15415 87746 76647 13218


Time 14964 15460 89015 76075 13114


Time 15037 15000 89016 75366 13135




Time 14636 15000 87636 73618 12926


Time 15142 15975 90928 78357 13246
















20 40 60 80 100 120 1400

2000

4000

6000

8000

10000

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed


50 100 150 2000

2000

4000

6000

8000

10000

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed




3(119909)



tion1198914(119909) = minus20119890


119894=11199092

119894 minus119890(1119889)sum

119889

119894=1cos(2120587times119909119894)+20+1198901


1 1199092 119909

20) is a


















20 40 60 80 100 120 1400

1000

2000

3000

4000

5000

6000

7000

8000

Generations

Fitn

ess v

alue

ACODE

(a)

2 4 6 8 100

2000

4000

6000

8000

10000

Generations

Fitn

ess v

alue

FAPSOProposed

(b)


20 40 60 80 100 120 1400

5000

10000

15000

Generations

Fitn

ess v

alue

ACODE

(a)

2 4 6 8 100

2000

4000

6000

8000

Generations

Fitn

ess v

alue

FAPSOProposed

(b)





Time 14763 16495 95650 76734 14923


Time 14848 16477 95221 76975 14897


Time 15081 16460 95146 77872 14923





Time 14625 15487 94085 75984 14536


Time 14996 17014 97459 79017 16048









20 40 60 80 1000

50

100

150

200

250

300

350

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed



sum119899

119894=1(120583(119863119894) minus119863119894)sum119899



20 40 60 80 1000

50

100

150

200

250

300

350

400

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed











20 40 60 80 1000

50

100

150

200

250

300

350

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed


20 40 60 80 1000

50

100

150

200

250

300

350

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed


50 100 150 2000

5

10

15

20

25

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed


50 100 150 2000

5

10

15

20

25

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed



50 100 150 200 2500

5

10

15

20

25

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed


50 100 150 200 2500

5

10

15

20

25

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed



infin

0|119901(119905)ref minus








119898547 V



119898305W




119871500Ω



Error119864(119899)





PWM generator


Resistive load


Solar PVarray PWM


Scaling factor (K3)

Scaling factor (K2)

Scaling factor (K1)

Insolation

Temperature


Vm Vo

+

minus

+

minus

Im

E(n)

ΔE(n)


ddt


0 1 2 3 4 5 6 7 8 9 10 11300400500600700800900

Irra

dian

ce

Time (sec)

(a)

0 1 2 3 4 5 6 7 8 9 10 1125

30

35

40

Tem

pera

ture

Time (sec)

(b)



5 Conclusion





0

05


NB NS ZE PS PB


181Plot points


(a)

0

05


NB NS ZE PS PB



181Plot points

(b)


0

05

1



(c)


0 1 2 3 4 65 9 10 1187Time (sec)

Time offset 0

OFLCIncCondFLC


0100200300400500600700



Appendix




Competing Interests


Acknowledgments


References
























































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














119905

best1 119901119905

best2 119901119905

best3 119901119905

bestpop)


(1198921198941 1198921198942 1198921198943 119892



119901(119905) = (119909

1198941 1199091198942 1199091198943 119909

119894119889)

and V119901(119905) = (V

1198941 V1198942 V1198943 V



V119905+1119901


best minus 119909119905

119901) + 1198882times 1199032

times (119892best minus 119909119905

119901)

119909119905+1

119901= 119909119905

119901+ V119905+1119901

(1)











= 120583119911119905(1 minus 119911

119905) [53 54] where 120583 = 4





1

08

06

04

02

0

Popu

latio

n siz

e



) at interval [0 1]


119911119905+1

= 120573 sin (120587119911119905) (2)

where 120573 gt 0 119911119905 119911119905+1




119911119905+1

=

10038161003816100381610038161003816100381610038161003816

sin(120587119911119905

rand (sdot))

10038161003816100381610038161003816100381610038161003816

(3)

where 120573 = 1 and 119911119905 119911119905+1


119905+1isin


120596119905





120594 =2


(5)



0 100 200 300 400 5000

02

04

06

08

1

Chao

tic v

alue

Generations




1

08

06

04

02

0

Popu

latio

n siz

e



) at interval [0 1]


119888119905

1= 1198881119891

minus119905

MAXITR(1198881119891

minus 1198881119894)

119888119905

2= 1198882119894+

119905

MAXITR(1198882119891

minus 1198882119894)

(6)

where 1198881119894 1198882119894 1198881119891

and 1198882119891






Time 12669 14928 79243 73712 12056


Time 12845 14131 80510 73559 12117


Time 12849 14669 80826 74529 11936




Time 12669 13332 79243 72014 11381


Time 13084 14928 82801 74778 12438


V119905+1119901

= 120594 times 120596119905

chaos times V119905119901+ 119888119905

1times (119901119905

best minus 119909119905

119901) + 119888119905

2

times (119892119905

best minus 119909119905

119901)

119909119905+1

119901= 119909119905

119901+ V119905+1119901

(7)













1(119909) = sum

119889

119894=11199092

119894 minus100 lt 119909

119894lt 100 where

119889 = 20 and 119909 = (1199091 1199092 119909


1(119909)


1= 1199092= 1199093= sdot sdot sdot = 119909



1(119909)





1(119909) using all


1(119909) in the


1(119909)






(7)

End of criterion


and (5)

Initialization

No


Return

Start


(6)


No

YesEnd

Yes

No


loop

Yes

t = 0

i = 1

Set Pbest and Gbest

t = 1

i = 1


Gbest

t = t + 1

i = i + 1

i gt pop

i = i + 1

i gt pop

















10 20 30 40 50 60 70 800

05

1

15

2

25

3

35

4

Generations

Fitn

ess v

alue

times104

ACODEFA

PSOProposed



1(119909)



sum1198894

119894=1[(1199094119894minus3

+ 101199094119894minus2

)2+ 5(119909


minus 21199094119894minus1

)4

10 20 30 40 50 60 70 80Generations

ACODEFA

PSOProposed

0

1

2

3

4

5

6

Fitn

ess v

alue

times104


+ 10(1199094119894minus3

minus 1199094119894)4] minus4 lt 119909

119894lt 5 where 119889 = 20 and

119909 = (1199091 1199092 119909



1= 1199092

= 1199093

=



2(119909)






Time 17946 17312 117101 83304 16328


Time 17969 18037 116532 82524 16128


Time 18766 18609 117172 81815 15889




Time 17684 17275 115836 81192 15722


Time 18766 18609 117172 83445 16441

10 20 30 40 50 60 70 800

1

2

3

4

5

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed

times104









3(119909) = sum

119889

119894=1(1199092


119889

119894=1cos(119909119894


1

10 20 30 40 50 60 70 800

1

2

3

4

5

Generations

Fitn

ess v

alue

times104

ACODEFA

PSOProposed


1199092 119909







3(119909)

























Time 15032 15415 87746 76647 13218


Time 14964 15460 89015 76075 13114


Time 15037 15000 89016 75366 13135




Time 14636 15000 87636 73618 12926


Time 15142 15975 90928 78357 13246
















20 40 60 80 100 120 1400

2000

4000

6000

8000

10000

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed


50 100 150 2000

2000

4000

6000

8000

10000

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed




3(119909)



tion1198914(119909) = minus20119890


119894=11199092

119894 minus119890(1119889)sum

119889

119894=1cos(2120587times119909119894)+20+1198901


1 1199092 119909

20) is a


















20 40 60 80 100 120 1400

1000

2000

3000

4000

5000

6000

7000

8000

Generations

Fitn

ess v

alue

ACODE

(a)

2 4 6 8 100

2000

4000

6000

8000

10000

Generations

Fitn

ess v

alue

FAPSOProposed

(b)


20 40 60 80 100 120 1400

5000

10000

15000

Generations

Fitn

ess v

alue

ACODE

(a)

2 4 6 8 100

2000

4000

6000

8000

Generations

Fitn

ess v

alue

FAPSOProposed

(b)





Time 14763 16495 95650 76734 14923


Time 14848 16477 95221 76975 14897


Time 15081 16460 95146 77872 14923





Time 14625 15487 94085 75984 14536


Time 14996 17014 97459 79017 16048









20 40 60 80 1000

50

100

150

200

250

300

350

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed



sum119899

119894=1(120583(119863119894) minus119863119894)sum119899



20 40 60 80 1000

50

100

150

200

250

300

350

400

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed











20 40 60 80 1000

50

100

150

200

250

300

350

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed


20 40 60 80 1000

50

100

150

200

250

300

350

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed


50 100 150 2000

5

10

15

20

25

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed


50 100 150 2000

5

10

15

20

25

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed



50 100 150 200 2500

5

10

15

20

25

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed


50 100 150 200 2500

5

10

15

20

25

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed



infin

0|119901(119905)ref minus








119898547 V



119898305W




119871500Ω



Error119864(119899)





PWM generator


Resistive load


Solar PVarray PWM


Scaling factor (K3)

Scaling factor (K2)

Scaling factor (K1)

Insolation

Temperature


Vm Vo

+

minus

+

minus

Im

E(n)

ΔE(n)


ddt


0 1 2 3 4 5 6 7 8 9 10 11300400500600700800900

Irra

dian

ce

Time (sec)

(a)

0 1 2 3 4 5 6 7 8 9 10 1125

30

35

40

Tem

pera

ture

Time (sec)

(b)



5 Conclusion





0

05


NB NS ZE PS PB


181Plot points


(a)

0

05


NB NS ZE PS PB



181Plot points

(b)


0

05

1



(c)


0 1 2 3 4 65 9 10 1187Time (sec)

Time offset 0

OFLCIncCondFLC


0100200300400500600700



Appendix




Competing Interests


Acknowledgments


References
























































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1

08

06

04

02

0

Popu

latio

n siz

e



) at interval [0 1]


119911119905+1

= 120573 sin (120587119911119905) (2)

where 120573 gt 0 119911119905 119911119905+1




119911119905+1

=

10038161003816100381610038161003816100381610038161003816

sin(120587119911119905

rand (sdot))

10038161003816100381610038161003816100381610038161003816

(3)

where 120573 = 1 and 119911119905 119911119905+1


119905+1isin


120596119905





120594 =2


(5)



0 100 200 300 400 5000

02

04

06

08

1

Chao

tic v

alue

Generations




1

08

06

04

02

0

Popu

latio

n siz

e



) at interval [0 1]


119888119905

1= 1198881119891

minus119905

MAXITR(1198881119891

minus 1198881119894)

119888119905

2= 1198882119894+

119905

MAXITR(1198882119891

minus 1198882119894)

(6)

where 1198881119894 1198882119894 1198881119891

and 1198882119891






Time 12669 14928 79243 73712 12056


Time 12845 14131 80510 73559 12117


Time 12849 14669 80826 74529 11936




Time 12669 13332 79243 72014 11381


Time 13084 14928 82801 74778 12438


V119905+1119901

= 120594 times 120596119905

chaos times V119905119901+ 119888119905

1times (119901119905

best minus 119909119905

119901) + 119888119905

2

times (119892119905

best minus 119909119905

119901)

119909119905+1

119901= 119909119905

119901+ V119905+1119901

(7)













1(119909) = sum

119889

119894=11199092

119894 minus100 lt 119909

119894lt 100 where

119889 = 20 and 119909 = (1199091 1199092 119909


1(119909)


1= 1199092= 1199093= sdot sdot sdot = 119909



1(119909)





1(119909) using all


1(119909) in the


1(119909)






(7)

End of criterion


and (5)

Initialization

No


Return

Start


(6)


No

YesEnd

Yes

No


loop

Yes

t = 0

i = 1

Set Pbest and Gbest

t = 1

i = 1


Gbest

t = t + 1

i = i + 1

i gt pop

i = i + 1

i gt pop

















10 20 30 40 50 60 70 800

05

1

15

2

25

3

35

4

Generations

Fitn

ess v

alue

times104

ACODEFA

PSOProposed



1(119909)



sum1198894

119894=1[(1199094119894minus3

+ 101199094119894minus2

)2+ 5(119909


minus 21199094119894minus1

)4

10 20 30 40 50 60 70 80Generations

ACODEFA

PSOProposed

0

1

2

3

4

5

6

Fitn

ess v

alue

times104


+ 10(1199094119894minus3

minus 1199094119894)4] minus4 lt 119909

119894lt 5 where 119889 = 20 and

119909 = (1199091 1199092 119909



1= 1199092

= 1199093

=



2(119909)






Time 17946 17312 117101 83304 16328


Time 17969 18037 116532 82524 16128


Time 18766 18609 117172 81815 15889




Time 17684 17275 115836 81192 15722


Time 18766 18609 117172 83445 16441

10 20 30 40 50 60 70 800

1

2

3

4

5

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed

times104









3(119909) = sum

119889

119894=1(1199092


119889

119894=1cos(119909119894


1

10 20 30 40 50 60 70 800

1

2

3

4

5

Generations

Fitn

ess v

alue

times104

ACODEFA

PSOProposed


1199092 119909







3(119909)

























Time 15032 15415 87746 76647 13218


Time 14964 15460 89015 76075 13114


Time 15037 15000 89016 75366 13135




Time 14636 15000 87636 73618 12926


Time 15142 15975 90928 78357 13246
















20 40 60 80 100 120 1400

2000

4000

6000

8000

10000

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed


50 100 150 2000

2000

4000

6000

8000

10000

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed




3(119909)



tion1198914(119909) = minus20119890


119894=11199092

119894 minus119890(1119889)sum

119889

119894=1cos(2120587times119909119894)+20+1198901


1 1199092 119909

20) is a


















20 40 60 80 100 120 1400

1000

2000

3000

4000

5000

6000

7000

8000

Generations

Fitn

ess v

alue

ACODE

(a)

2 4 6 8 100

2000

4000

6000

8000

10000

Generations

Fitn

ess v

alue

FAPSOProposed

(b)


20 40 60 80 100 120 1400

5000

10000

15000

Generations

Fitn

ess v

alue

ACODE

(a)

2 4 6 8 100

2000

4000

6000

8000

Generations

Fitn

ess v

alue

FAPSOProposed

(b)





Time 14763 16495 95650 76734 14923


Time 14848 16477 95221 76975 14897


Time 15081 16460 95146 77872 14923





Time 14625 15487 94085 75984 14536


Time 14996 17014 97459 79017 16048









20 40 60 80 1000

50

100

150

200

250

300

350

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed



sum119899

119894=1(120583(119863119894) minus119863119894)sum119899



20 40 60 80 1000

50

100

150

200

250

300

350

400

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed











20 40 60 80 1000

50

100

150

200

250

300

350

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed


20 40 60 80 1000

50

100

150

200

250

300

350

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed


50 100 150 2000

5

10

15

20

25

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed


50 100 150 2000

5

10

15

20

25

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed



50 100 150 200 2500

5

10

15

20

25

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed


50 100 150 200 2500

5

10

15

20

25

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed



infin

0|119901(119905)ref minus








119898547 V



119898305W




119871500Ω



Error119864(119899)





PWM generator


Resistive load


Solar PVarray PWM


Scaling factor (K3)

Scaling factor (K2)

Scaling factor (K1)

Insolation

Temperature


Vm Vo

+

minus

+

minus

Im

E(n)

ΔE(n)


ddt


0 1 2 3 4 5 6 7 8 9 10 11300400500600700800900

Irra

dian

ce

Time (sec)

(a)

0 1 2 3 4 5 6 7 8 9 10 1125

30

35

40

Tem

pera

ture

Time (sec)

(b)



5 Conclusion





0

05


NB NS ZE PS PB


181Plot points


(a)

0

05


NB NS ZE PS PB



181Plot points

(b)


0

05

1



(c)


0 1 2 3 4 65 9 10 1187Time (sec)

Time offset 0

OFLCIncCondFLC


0100200300400500600700



Appendix




Competing Interests


Acknowledgments


References
























































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













Time 12669 14928 79243 73712 12056


Time 12845 14131 80510 73559 12117


Time 12849 14669 80826 74529 11936




Time 12669 13332 79243 72014 11381


Time 13084 14928 82801 74778 12438


V119905+1119901

= 120594 times 120596119905

chaos times V119905119901+ 119888119905

1times (119901119905

best minus 119909119905

119901) + 119888119905

2

times (119892119905

best minus 119909119905

119901)

119909119905+1

119901= 119909119905

119901+ V119905+1119901

(7)













1(119909) = sum

119889

119894=11199092

119894 minus100 lt 119909

119894lt 100 where

119889 = 20 and 119909 = (1199091 1199092 119909


1(119909)


1= 1199092= 1199093= sdot sdot sdot = 119909



1(119909)





1(119909) using all


1(119909) in the


1(119909)






(7)

End of criterion


and (5)

Initialization

No


Return

Start


(6)


No

YesEnd

Yes

No


loop

Yes

t = 0

i = 1

Set Pbest and Gbest

t = 1

i = 1


Gbest

t = t + 1

i = i + 1

i gt pop

i = i + 1

i gt pop

















10 20 30 40 50 60 70 800

05

1

15

2

25

3

35

4

Generations

Fitn

ess v

alue

times104

ACODEFA

PSOProposed



1(119909)



sum1198894

119894=1[(1199094119894minus3

+ 101199094119894minus2

)2+ 5(119909


minus 21199094119894minus1

)4

10 20 30 40 50 60 70 80Generations

ACODEFA

PSOProposed

0

1

2

3

4

5

6

Fitn

ess v

alue

times104


+ 10(1199094119894minus3

minus 1199094119894)4] minus4 lt 119909

119894lt 5 where 119889 = 20 and

119909 = (1199091 1199092 119909



1= 1199092

= 1199093

=



2(119909)






Time 17946 17312 117101 83304 16328


Time 17969 18037 116532 82524 16128


Time 18766 18609 117172 81815 15889




Time 17684 17275 115836 81192 15722


Time 18766 18609 117172 83445 16441

10 20 30 40 50 60 70 800

1

2

3

4

5

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed

times104









3(119909) = sum

119889

119894=1(1199092


119889

119894=1cos(119909119894


1

10 20 30 40 50 60 70 800

1

2

3

4

5

Generations

Fitn

ess v

alue

times104

ACODEFA

PSOProposed


1199092 119909







3(119909)

























Time 15032 15415 87746 76647 13218


Time 14964 15460 89015 76075 13114


Time 15037 15000 89016 75366 13135




Time 14636 15000 87636 73618 12926


Time 15142 15975 90928 78357 13246
















20 40 60 80 100 120 1400

2000

4000

6000

8000

10000

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed


50 100 150 2000

2000

4000

6000

8000

10000

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed




3(119909)



tion1198914(119909) = minus20119890


119894=11199092

119894 minus119890(1119889)sum

119889

119894=1cos(2120587times119909119894)+20+1198901


1 1199092 119909

20) is a


















20 40 60 80 100 120 1400

1000

2000

3000

4000

5000

6000

7000

8000

Generations

Fitn

ess v

alue

ACODE

(a)

2 4 6 8 100

2000

4000

6000

8000

10000

Generations

Fitn

ess v

alue

FAPSOProposed

(b)


20 40 60 80 100 120 1400

5000

10000

15000

Generations

Fitn

ess v

alue

ACODE

(a)

2 4 6 8 100

2000

4000

6000

8000

Generations

Fitn

ess v

alue

FAPSOProposed

(b)





Time 14763 16495 95650 76734 14923


Time 14848 16477 95221 76975 14897


Time 15081 16460 95146 77872 14923





Time 14625 15487 94085 75984 14536


Time 14996 17014 97459 79017 16048









20 40 60 80 1000

50

100

150

200

250

300

350

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed



sum119899

119894=1(120583(119863119894) minus119863119894)sum119899



20 40 60 80 1000

50

100

150

200

250

300

350

400

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed











20 40 60 80 1000

50

100

150

200

250

300

350

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed


20 40 60 80 1000

50

100

150

200

250

300

350

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed


50 100 150 2000

5

10

15

20

25

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed


50 100 150 2000

5

10

15

20

25

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed



50 100 150 200 2500

5

10

15

20

25

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed


50 100 150 200 2500

5

10

15

20

25

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed



infin

0|119901(119905)ref minus








119898547 V



119898305W




119871500Ω



Error119864(119899)





PWM generator


Resistive load


Solar PVarray PWM


Scaling factor (K3)

Scaling factor (K2)

Scaling factor (K1)

Insolation

Temperature


Vm Vo

+

minus

+

minus

Im

E(n)

ΔE(n)


ddt


0 1 2 3 4 5 6 7 8 9 10 11300400500600700800900

Irra

dian

ce

Time (sec)

(a)

0 1 2 3 4 5 6 7 8 9 10 1125

30

35

40

Tem

pera

ture

Time (sec)

(b)



5 Conclusion





0

05


NB NS ZE PS PB


181Plot points


(a)

0

05


NB NS ZE PS PB



181Plot points

(b)


0

05

1



(c)


0 1 2 3 4 65 9 10 1187Time (sec)

Time offset 0

OFLCIncCondFLC


0100200300400500600700



Appendix




Competing Interests


Acknowledgments


References
























































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














(7)

End of criterion


and (5)

Initialization

No


Return

Start


(6)


No

YesEnd

Yes

No


loop

Yes

t = 0

i = 1

Set Pbest and Gbest

t = 1

i = 1


Gbest

t = t + 1

i = i + 1

i gt pop

i = i + 1

i gt pop

















10 20 30 40 50 60 70 800

05

1

15

2

25

3

35

4

Generations

Fitn

ess v

alue

times104

ACODEFA

PSOProposed



1(119909)



sum1198894

119894=1[(1199094119894minus3

+ 101199094119894minus2

)2+ 5(119909


minus 21199094119894minus1

)4

10 20 30 40 50 60 70 80Generations

ACODEFA

PSOProposed

0

1

2

3

4

5

6

Fitn

ess v

alue

times104


+ 10(1199094119894minus3

minus 1199094119894)4] minus4 lt 119909

119894lt 5 where 119889 = 20 and

119909 = (1199091 1199092 119909



1= 1199092

= 1199093

=



2(119909)






Time 17946 17312 117101 83304 16328


Time 17969 18037 116532 82524 16128


Time 18766 18609 117172 81815 15889




Time 17684 17275 115836 81192 15722


Time 18766 18609 117172 83445 16441

10 20 30 40 50 60 70 800

1

2

3

4

5

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed

times104









3(119909) = sum

119889

119894=1(1199092


119889

119894=1cos(119909119894


1

10 20 30 40 50 60 70 800

1

2

3

4

5

Generations

Fitn

ess v

alue

times104

ACODEFA

PSOProposed


1199092 119909







3(119909)

























Time 15032 15415 87746 76647 13218


Time 14964 15460 89015 76075 13114


Time 15037 15000 89016 75366 13135




Time 14636 15000 87636 73618 12926


Time 15142 15975 90928 78357 13246
















20 40 60 80 100 120 1400

2000

4000

6000

8000

10000

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed


50 100 150 2000

2000

4000

6000

8000

10000

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed




3(119909)



tion1198914(119909) = minus20119890


119894=11199092

119894 minus119890(1119889)sum

119889

119894=1cos(2120587times119909119894)+20+1198901


1 1199092 119909

20) is a


















20 40 60 80 100 120 1400

1000

2000

3000

4000

5000

6000

7000

8000

Generations

Fitn

ess v

alue

ACODE

(a)

2 4 6 8 100

2000

4000

6000

8000

10000

Generations

Fitn

ess v

alue

FAPSOProposed

(b)


20 40 60 80 100 120 1400

5000

10000

15000

Generations

Fitn

ess v

alue

ACODE

(a)

2 4 6 8 100

2000

4000

6000

8000

Generations

Fitn

ess v

alue

FAPSOProposed

(b)





Time 14763 16495 95650 76734 14923


Time 14848 16477 95221 76975 14897


Time 15081 16460 95146 77872 14923





Time 14625 15487 94085 75984 14536


Time 14996 17014 97459 79017 16048









20 40 60 80 1000

50

100

150

200

250

300

350

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed



sum119899

119894=1(120583(119863119894) minus119863119894)sum119899



20 40 60 80 1000

50

100

150

200

250

300

350

400

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed











20 40 60 80 1000

50

100

150

200

250

300

350

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed


20 40 60 80 1000

50

100

150

200

250

300

350

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed


50 100 150 2000

5

10

15

20

25

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed


50 100 150 2000

5

10

15

20

25

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed



50 100 150 200 2500

5

10

15

20

25

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed


50 100 150 200 2500

5

10

15

20

25

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed



infin

0|119901(119905)ref minus








119898547 V



119898305W




119871500Ω



Error119864(119899)





PWM generator


Resistive load


Solar PVarray PWM


Scaling factor (K3)

Scaling factor (K2)

Scaling factor (K1)

Insolation

Temperature


Vm Vo

+

minus

+

minus

Im

E(n)

ΔE(n)


ddt


0 1 2 3 4 5 6 7 8 9 10 11300400500600700800900

Irra

dian

ce

Time (sec)

(a)

0 1 2 3 4 5 6 7 8 9 10 1125

30

35

40

Tem

pera

ture

Time (sec)

(b)



5 Conclusion





0

05


NB NS ZE PS PB


181Plot points


(a)

0

05


NB NS ZE PS PB



181Plot points

(b)


0

05

1



(c)


0 1 2 3 4 65 9 10 1187Time (sec)

Time offset 0

OFLCIncCondFLC


0100200300400500600700



Appendix




Competing Interests


Acknowledgments


References
























































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
























10 20 30 40 50 60 70 800

05

1

15

2

25

3

35

4

Generations

Fitn

ess v

alue

times104

ACODEFA

PSOProposed



1(119909)



sum1198894

119894=1[(1199094119894minus3

+ 101199094119894minus2

)2+ 5(119909


minus 21199094119894minus1

)4

10 20 30 40 50 60 70 80Generations

ACODEFA

PSOProposed

0

1

2

3

4

5

6

Fitn

ess v

alue

times104


+ 10(1199094119894minus3

minus 1199094119894)4] minus4 lt 119909

119894lt 5 where 119889 = 20 and

119909 = (1199091 1199092 119909



1= 1199092

= 1199093

=



2(119909)






Time 17946 17312 117101 83304 16328


Time 17969 18037 116532 82524 16128


Time 18766 18609 117172 81815 15889




Time 17684 17275 115836 81192 15722


Time 18766 18609 117172 83445 16441

10 20 30 40 50 60 70 800

1

2

3

4

5

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed

times104









3(119909) = sum

119889

119894=1(1199092


119889

119894=1cos(119909119894


1

10 20 30 40 50 60 70 800

1

2

3

4

5

Generations

Fitn

ess v

alue

times104

ACODEFA

PSOProposed


1199092 119909







3(119909)

























Time 15032 15415 87746 76647 13218


Time 14964 15460 89015 76075 13114


Time 15037 15000 89016 75366 13135




Time 14636 15000 87636 73618 12926


Time 15142 15975 90928 78357 13246
















20 40 60 80 100 120 1400

2000

4000

6000

8000

10000

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed


50 100 150 2000

2000

4000

6000

8000

10000

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed




3(119909)



tion1198914(119909) = minus20119890


119894=11199092

119894 minus119890(1119889)sum

119889

119894=1cos(2120587times119909119894)+20+1198901


1 1199092 119909

20) is a


















20 40 60 80 100 120 1400

1000

2000

3000

4000

5000

6000

7000

8000

Generations

Fitn

ess v

alue

ACODE

(a)

2 4 6 8 100

2000

4000

6000

8000

10000

Generations

Fitn

ess v

alue

FAPSOProposed

(b)


20 40 60 80 100 120 1400

5000

10000

15000

Generations

Fitn

ess v

alue

ACODE

(a)

2 4 6 8 100

2000

4000

6000

8000

Generations

Fitn

ess v

alue

FAPSOProposed

(b)





Time 14763 16495 95650 76734 14923


Time 14848 16477 95221 76975 14897


Time 15081 16460 95146 77872 14923





Time 14625 15487 94085 75984 14536


Time 14996 17014 97459 79017 16048









20 40 60 80 1000

50

100

150

200

250

300

350

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed



sum119899

119894=1(120583(119863119894) minus119863119894)sum119899



20 40 60 80 1000

50

100

150

200

250

300

350

400

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed











20 40 60 80 1000

50

100

150

200

250

300

350

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed


20 40 60 80 1000

50

100

150

200

250

300

350

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed


50 100 150 2000

5

10

15

20

25

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed


50 100 150 2000

5

10

15

20

25

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed



50 100 150 200 2500

5

10

15

20

25

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed


50 100 150 200 2500

5

10

15

20

25

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed



infin

0|119901(119905)ref minus








119898547 V



119898305W




119871500Ω



Error119864(119899)





PWM generator


Resistive load


Solar PVarray PWM


Scaling factor (K3)

Scaling factor (K2)

Scaling factor (K1)

Insolation

Temperature


Vm Vo

+

minus

+

minus

Im

E(n)

ΔE(n)


ddt


0 1 2 3 4 5 6 7 8 9 10 11300400500600700800900

Irra

dian

ce

Time (sec)

(a)

0 1 2 3 4 5 6 7 8 9 10 1125

30

35

40

Tem

pera

ture

Time (sec)

(b)



5 Conclusion





0

05


NB NS ZE PS PB


181Plot points


(a)

0

05


NB NS ZE PS PB



181Plot points

(b)


0

05

1



(c)


0 1 2 3 4 65 9 10 1187Time (sec)

Time offset 0

OFLCIncCondFLC


0100200300400500600700



Appendix




Competing Interests


Acknowledgments


References
























































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













Time 17946 17312 117101 83304 16328


Time 17969 18037 116532 82524 16128


Time 18766 18609 117172 81815 15889




Time 17684 17275 115836 81192 15722


Time 18766 18609 117172 83445 16441

10 20 30 40 50 60 70 800

1

2

3

4

5

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed

times104









3(119909) = sum

119889

119894=1(1199092


119889

119894=1cos(119909119894


1

10 20 30 40 50 60 70 800

1

2

3

4

5

Generations

Fitn

ess v

alue

times104

ACODEFA

PSOProposed


1199092 119909







3(119909)

























Time 15032 15415 87746 76647 13218


Time 14964 15460 89015 76075 13114


Time 15037 15000 89016 75366 13135




Time 14636 15000 87636 73618 12926


Time 15142 15975 90928 78357 13246
















20 40 60 80 100 120 1400

2000

4000

6000

8000

10000

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed


50 100 150 2000

2000

4000

6000

8000

10000

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed




3(119909)



tion1198914(119909) = minus20119890


119894=11199092

119894 minus119890(1119889)sum

119889

119894=1cos(2120587times119909119894)+20+1198901


1 1199092 119909

20) is a


















20 40 60 80 100 120 1400

1000

2000

3000

4000

5000

6000

7000

8000

Generations

Fitn

ess v

alue

ACODE

(a)

2 4 6 8 100

2000

4000

6000

8000

10000

Generations

Fitn

ess v

alue

FAPSOProposed

(b)


20 40 60 80 100 120 1400

5000

10000

15000

Generations

Fitn

ess v

alue

ACODE

(a)

2 4 6 8 100

2000

4000

6000

8000

Generations

Fitn

ess v

alue

FAPSOProposed

(b)





Time 14763 16495 95650 76734 14923


Time 14848 16477 95221 76975 14897


Time 15081 16460 95146 77872 14923





Time 14625 15487 94085 75984 14536


Time 14996 17014 97459 79017 16048









20 40 60 80 1000

50

100

150

200

250

300

350

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed



sum119899

119894=1(120583(119863119894) minus119863119894)sum119899



20 40 60 80 1000

50

100

150

200

250

300

350

400

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed











20 40 60 80 1000

50

100

150

200

250

300

350

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed


20 40 60 80 1000

50

100

150

200

250

300

350

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed


50 100 150 2000

5

10

15

20

25

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed


50 100 150 2000

5

10

15

20

25

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed



50 100 150 200 2500

5

10

15

20

25

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed


50 100 150 200 2500

5

10

15

20

25

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed



infin

0|119901(119905)ref minus








119898547 V



119898305W




119871500Ω



Error119864(119899)





PWM generator


Resistive load


Solar PVarray PWM


Scaling factor (K3)

Scaling factor (K2)

Scaling factor (K1)

Insolation

Temperature


Vm Vo

+

minus

+

minus

Im

E(n)

ΔE(n)


ddt


0 1 2 3 4 5 6 7 8 9 10 11300400500600700800900

Irra

dian

ce

Time (sec)

(a)

0 1 2 3 4 5 6 7 8 9 10 1125

30

35

40

Tem

pera

ture

Time (sec)

(b)



5 Conclusion





0

05


NB NS ZE PS PB


181Plot points


(a)

0

05


NB NS ZE PS PB



181Plot points

(b)


0

05

1



(c)


0 1 2 3 4 65 9 10 1187Time (sec)

Time offset 0

OFLCIncCondFLC


0100200300400500600700



Appendix




Competing Interests


Acknowledgments


References
























































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



























Time 15032 15415 87746 76647 13218


Time 14964 15460 89015 76075 13114


Time 15037 15000 89016 75366 13135




Time 14636 15000 87636 73618 12926


Time 15142 15975 90928 78357 13246
















20 40 60 80 100 120 1400

2000

4000

6000

8000

10000

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed


50 100 150 2000

2000

4000

6000

8000

10000

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed




3(119909)



tion1198914(119909) = minus20119890


119894=11199092

119894 minus119890(1119889)sum

119889

119894=1cos(2120587times119909119894)+20+1198901


1 1199092 119909

20) is a


















20 40 60 80 100 120 1400

1000

2000

3000

4000

5000

6000

7000

8000

Generations

Fitn

ess v

alue

ACODE

(a)

2 4 6 8 100

2000

4000

6000

8000

10000

Generations

Fitn

ess v

alue

FAPSOProposed

(b)


20 40 60 80 100 120 1400

5000

10000

15000

Generations

Fitn

ess v

alue

ACODE

(a)

2 4 6 8 100

2000

4000

6000

8000

Generations

Fitn

ess v

alue

FAPSOProposed

(b)





Time 14763 16495 95650 76734 14923


Time 14848 16477 95221 76975 14897


Time 15081 16460 95146 77872 14923





Time 14625 15487 94085 75984 14536


Time 14996 17014 97459 79017 16048









20 40 60 80 1000

50

100

150

200

250

300

350

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed



sum119899

119894=1(120583(119863119894) minus119863119894)sum119899



20 40 60 80 1000

50

100

150

200

250

300

350

400

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed











20 40 60 80 1000

50

100

150

200

250

300

350

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed


20 40 60 80 1000

50

100

150

200

250

300

350

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed


50 100 150 2000

5

10

15

20

25

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed


50 100 150 2000

5

10

15

20

25

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed



50 100 150 200 2500

5

10

15

20

25

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed


50 100 150 200 2500

5

10

15

20

25

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed



infin

0|119901(119905)ref minus








119898547 V



119898305W




119871500Ω



Error119864(119899)





PWM generator


Resistive load


Solar PVarray PWM


Scaling factor (K3)

Scaling factor (K2)

Scaling factor (K1)

Insolation

Temperature


Vm Vo

+

minus

+

minus

Im

E(n)

ΔE(n)


ddt


0 1 2 3 4 5 6 7 8 9 10 11300400500600700800900

Irra

dian

ce

Time (sec)

(a)

0 1 2 3 4 5 6 7 8 9 10 1125

30

35

40

Tem

pera

ture

Time (sec)

(b)



5 Conclusion





0

05


NB NS ZE PS PB


181Plot points


(a)

0

05


NB NS ZE PS PB



181Plot points

(b)


0

05

1



(c)


0 1 2 3 4 65 9 10 1187Time (sec)

Time offset 0

OFLCIncCondFLC


0100200300400500600700



Appendix




Competing Interests


Acknowledgments


References
























































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















20 40 60 80 100 120 1400

2000

4000

6000

8000

10000

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed


50 100 150 2000

2000

4000

6000

8000

10000

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed




3(119909)



tion1198914(119909) = minus20119890


119894=11199092

119894 minus119890(1119889)sum

119889

119894=1cos(2120587times119909119894)+20+1198901


1 1199092 119909

20) is a


















20 40 60 80 100 120 1400

1000

2000

3000

4000

5000

6000

7000

8000

Generations

Fitn

ess v

alue

ACODE

(a)

2 4 6 8 100

2000

4000

6000

8000

10000

Generations

Fitn

ess v

alue

FAPSOProposed

(b)


20 40 60 80 100 120 1400

5000

10000

15000

Generations

Fitn

ess v

alue

ACODE

(a)

2 4 6 8 100

2000

4000

6000

8000

Generations

Fitn

ess v

alue

FAPSOProposed

(b)





Time 14763 16495 95650 76734 14923


Time 14848 16477 95221 76975 14897


Time 15081 16460 95146 77872 14923





Time 14625 15487 94085 75984 14536


Time 14996 17014 97459 79017 16048









20 40 60 80 1000

50

100

150

200

250

300

350

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed



sum119899

119894=1(120583(119863119894) minus119863119894)sum119899



20 40 60 80 1000

50

100

150

200

250

300

350

400

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed











20 40 60 80 1000

50

100

150

200

250

300

350

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed


20 40 60 80 1000

50

100

150

200

250

300

350

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed


50 100 150 2000

5

10

15

20

25

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed


50 100 150 2000

5

10

15

20

25

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed



50 100 150 200 2500

5

10

15

20

25

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed


50 100 150 200 2500

5

10

15

20

25

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed



infin

0|119901(119905)ref minus








119898547 V



119898305W




119871500Ω



Error119864(119899)





PWM generator


Resistive load


Solar PVarray PWM


Scaling factor (K3)

Scaling factor (K2)

Scaling factor (K1)

Insolation

Temperature


Vm Vo

+

minus

+

minus

Im

E(n)

ΔE(n)


ddt


0 1 2 3 4 5 6 7 8 9 10 11300400500600700800900

Irra

dian

ce

Time (sec)

(a)

0 1 2 3 4 5 6 7 8 9 10 1125

30

35

40

Tem

pera

ture

Time (sec)

(b)



5 Conclusion





0

05


NB NS ZE PS PB


181Plot points


(a)

0

05


NB NS ZE PS PB



181Plot points

(b)


0

05

1



(c)


0 1 2 3 4 65 9 10 1187Time (sec)

Time offset 0

OFLCIncCondFLC


0100200300400500600700



Appendix




Competing Interests


Acknowledgments


References
























































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










20 40 60 80 100 120 1400

1000

2000

3000

4000

5000

6000

7000

8000

Generations

Fitn

ess v

alue

ACODE

(a)

2 4 6 8 100

2000

4000

6000

8000

10000

Generations

Fitn

ess v

alue

FAPSOProposed

(b)


20 40 60 80 100 120 1400

5000

10000

15000

Generations

Fitn

ess v

alue

ACODE

(a)

2 4 6 8 100

2000

4000

6000

8000

Generations

Fitn

ess v

alue

FAPSOProposed

(b)





Time 14763 16495 95650 76734 14923


Time 14848 16477 95221 76975 14897


Time 15081 16460 95146 77872 14923





Time 14625 15487 94085 75984 14536


Time 14996 17014 97459 79017 16048









20 40 60 80 1000

50

100

150

200

250

300

350

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed



sum119899

119894=1(120583(119863119894) minus119863119894)sum119899



20 40 60 80 1000

50

100

150

200

250

300

350

400

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed











20 40 60 80 1000

50

100

150

200

250

300

350

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed


20 40 60 80 1000

50

100

150

200

250

300

350

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed


50 100 150 2000

5

10

15

20

25

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed


50 100 150 2000

5

10

15

20

25

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed



50 100 150 200 2500

5

10

15

20

25

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed


50 100 150 200 2500

5

10

15

20

25

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed



infin

0|119901(119905)ref minus








119898547 V



119898305W




119871500Ω



Error119864(119899)





PWM generator


Resistive load


Solar PVarray PWM


Scaling factor (K3)

Scaling factor (K2)

Scaling factor (K1)

Insolation

Temperature


Vm Vo

+

minus

+

minus

Im

E(n)

ΔE(n)


ddt


0 1 2 3 4 5 6 7 8 9 10 11300400500600700800900

Irra

dian

ce

Time (sec)

(a)

0 1 2 3 4 5 6 7 8 9 10 1125

30

35

40

Tem

pera

ture

Time (sec)

(b)



5 Conclusion





0

05


NB NS ZE PS PB


181Plot points


(a)

0

05


NB NS ZE PS PB



181Plot points

(b)


0

05

1



(c)


0 1 2 3 4 65 9 10 1187Time (sec)

Time offset 0

OFLCIncCondFLC


0100200300400500600700



Appendix




Competing Interests


Acknowledgments


References
























































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













Time 14625 15487 94085 75984 14536


Time 14996 17014 97459 79017 16048









20 40 60 80 1000

50

100

150

200

250

300

350

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed



sum119899

119894=1(120583(119863119894) minus119863119894)sum119899



20 40 60 80 1000

50

100

150

200

250

300

350

400

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed











20 40 60 80 1000

50

100

150

200

250

300

350

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed


20 40 60 80 1000

50

100

150

200

250

300

350

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed


50 100 150 2000

5

10

15

20

25

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed


50 100 150 2000

5

10

15

20

25

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed



50 100 150 200 2500

5

10

15

20

25

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed


50 100 150 200 2500

5

10

15

20

25

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed



infin

0|119901(119905)ref minus








119898547 V



119898305W




119871500Ω



Error119864(119899)





PWM generator


Resistive load


Solar PVarray PWM


Scaling factor (K3)

Scaling factor (K2)

Scaling factor (K1)

Insolation

Temperature


Vm Vo

+

minus

+

minus

Im

E(n)

ΔE(n)


ddt


0 1 2 3 4 5 6 7 8 9 10 11300400500600700800900

Irra

dian

ce

Time (sec)

(a)

0 1 2 3 4 5 6 7 8 9 10 1125

30

35

40

Tem

pera

ture

Time (sec)

(b)



5 Conclusion





0

05


NB NS ZE PS PB


181Plot points


(a)

0

05


NB NS ZE PS PB



181Plot points

(b)


0

05

1



(c)


0 1 2 3 4 65 9 10 1187Time (sec)

Time offset 0

OFLCIncCondFLC


0100200300400500600700



Appendix




Competing Interests


Acknowledgments


References
























































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















20 40 60 80 1000

50

100

150

200

250

300

350

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed


20 40 60 80 1000

50

100

150

200

250

300

350

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed


50 100 150 2000

5

10

15

20

25

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed


50 100 150 2000

5

10

15

20

25

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed



50 100 150 200 2500

5

10

15

20

25

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed


50 100 150 200 2500

5

10

15

20

25

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed



infin

0|119901(119905)ref minus








119898547 V



119898305W




119871500Ω



Error119864(119899)





PWM generator


Resistive load


Solar PVarray PWM


Scaling factor (K3)

Scaling factor (K2)

Scaling factor (K1)

Insolation

Temperature


Vm Vo

+

minus

+

minus

Im

E(n)

ΔE(n)


ddt


0 1 2 3 4 5 6 7 8 9 10 11300400500600700800900

Irra

dian

ce

Time (sec)

(a)

0 1 2 3 4 5 6 7 8 9 10 1125

30

35

40

Tem

pera

ture

Time (sec)

(b)



5 Conclusion





0

05


NB NS ZE PS PB


181Plot points


(a)

0

05


NB NS ZE PS PB



181Plot points

(b)


0

05

1



(c)


0 1 2 3 4 65 9 10 1187Time (sec)

Time offset 0

OFLCIncCondFLC


0100200300400500600700



Appendix




Competing Interests


Acknowledgments


References
























































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










50 100 150 200 2500

5

10

15

20

25

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed


50 100 150 200 2500

5

10

15

20

25

Generations

Fitn

ess v

alue

ACODEFA

PSOProposed



infin

0|119901(119905)ref minus








119898547 V



119898305W




119871500Ω



Error119864(119899)





PWM generator


Resistive load


Solar PVarray PWM


Scaling factor (K3)

Scaling factor (K2)

Scaling factor (K1)

Insolation

Temperature


Vm Vo

+

minus

+

minus

Im

E(n)

ΔE(n)


ddt


0 1 2 3 4 5 6 7 8 9 10 11300400500600700800900

Irra

dian

ce

Time (sec)

(a)

0 1 2 3 4 5 6 7 8 9 10 1125

30

35

40

Tem

pera

ture

Time (sec)

(b)



5 Conclusion





0

05


NB NS ZE PS PB


181Plot points


(a)

0

05


NB NS ZE PS PB



181Plot points

(b)


0

05

1



(c)


0 1 2 3 4 65 9 10 1187Time (sec)

Time offset 0

OFLCIncCondFLC


0100200300400500600700



Appendix




Competing Interests


Acknowledgments


References
























































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











PWM generator


Resistive load


Solar PVarray PWM


Scaling factor (K3)

Scaling factor (K2)

Scaling factor (K1)

Insolation

Temperature


Vm Vo

+

minus

+

minus

Im

E(n)

ΔE(n)


ddt


0 1 2 3 4 5 6 7 8 9 10 11300400500600700800900

Irra

dian

ce

Time (sec)

(a)

0 1 2 3 4 5 6 7 8 9 10 1125

30

35

40

Tem

pera

ture

Time (sec)

(b)



5 Conclusion





0

05


NB NS ZE PS PB


181Plot points


(a)

0

05


NB NS ZE PS PB



181Plot points

(b)


0

05

1



(c)


0 1 2 3 4 65 9 10 1187Time (sec)

Time offset 0

OFLCIncCondFLC


0100200300400500600700



Appendix




Competing Interests


Acknowledgments


References
























































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

05


NB NS ZE PS PB


181Plot points


(a)

0

05


NB NS ZE PS PB



181Plot points

(b)


0

05

1



(c)


0 1 2 3 4 65 9 10 1187Time (sec)

Time offset 0

OFLCIncCondFLC


0100200300400500600700



Appendix




Competing Interests


Acknowledgments


References
























































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
























































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





















Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article A Chaos-Enhanced Particle Swarm Optimization …downloads.hindawi.com/journals/mpe/2016/6519678.pdf · 2019-07-30 · algorithms can be found in the literature, such