A New DG Multiobjective Optimization Method Based on an Improved Evolutionary Algorithm · 2014-04-23 · A New DG Multiobjective Optimization Method Based on ... active power dispatch

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 643791 11 pageshttpdxdoiorg1011552013643791

Research ArticleA New DG Multiobjective Optimization Method Based onan Improved Evolutionary Algorithm

Wanxing Sheng Ke-yan Liu Yongmei Liu Xiaoli Meng and Xiaohui Song

Power Distribution Research Department China Electric Power Research Institute No 15 Xiaoying East RoadQinghe Haidian District Beijing 100192 China

Correspondence should be addressed to Ke-yan Liu liukeyaneprisgcccomcn

Received 16 January 2013 Revised 19 March 2013 Accepted 21 March 2013

Academic Editor Ricardo Perera

Copyright copy 2013 Wanxing Sheng et alThis 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

A distribution generation (DG) multiobjective optimization method based on an improved Pareto evolutionary algorithm isinvestigated in this paperThe improved Pareto evolutionary algorithm which introduces a penalty factor in the objective functionconstraints uses an adaptive crossover and a mutation operator in the evolutionary process and combines a simulated annealingiterative process The proposed algorithm is utilized to the optimize DG injection models to maximize DG utilization whileminimizing system loss and environmental pollution A revised IEEE 33-bus system with multiple DG units was used to test themultiobjective optimization algorithm in a distribution power system The proposed algorithm was implemented and comparedwith the strength Pareto evolutionary algorithm 2 (SPEA2) a particle swarm optimization (PSO) algorithm and nondominatedsorting genetic algorithm II (NGSA-II) The comparison of the results demonstrates the validity and practicality of utilizing DGunits in terms of economic dispatch and optimal operation in a distribution power system

1 Introduction

With the increasing demand for clean and renewable energythe issue of distribution generation (DG) is drawing moreattention worldwide DG provides voltage support to large-scale distribution power systems which results in reliabil-ity improvements and reduction loss in the power gridDG technology has become a hot research topic giventhe increasing global concerns about environmental protec-tion energy conservation and the increasing sophisticationof wind power photovoltaic power generation and otherrenewable energy technologies After DG is connected toa distribution network the distribution networkrsquos structureoperation and control mode will tremendously change andthe distribution system automation and the demand-sidemanagement must consider the coordination between DGand distribution network control Deciding the optimal DGoutput is a challenging research problem especially consid-ering the multiple optimal objectives associated with cases ofmultiple DG unit injections

Traditionally multi-objective DG optimization has beentreated as a single-objective optimization problem using

suitable weighting factors to form a weighted sum of singleobjectives This approach has the disadvantage of findingonly a single solution that does not express the trade-offs with different weighting factors Generating multiplesolutions using this approach requires several runs withdifferent factors which leads to long running times [1 2]Recent studies have treated the objectives simultaneouslyand independently as a true multi-objective optimizationproblem However the optimization problem becomes morecomplicated due to such issues as continuity local optimalinearization and so forth New optimization techniquessuch as particle swarm optimization (PSO) different evolu-tion (DE) and evolutionary programming (EP) have recentlybeen introduced and applied in the field of power systemsand with promising results [3ndash12] In a recent study [3]a differential evolution approach was proposed to solve anoptimal power flow problem with multiple objectives Theactive power dispatch and reactive power dispatch wereconsidered A nonlinear constrained multi-objective opti-mization problemwas formulated A general overviewof evo-lutionary multi-objective optimization was provided in [4]and themost representative algorithmswere discussed In [5]

2 Journal of Applied Mathematics

Pareto-based multi-objective evolutionary algorithms werediscussed and evaluated A nondominated sorting geneticalgorithm (NSGA) a niched Pareto genetic algorithm and astrength Pareto evolutionary algorithm (SPEA) were devel-oped and applied to an environmentaleconomic electricpower dispatch problem A multi-objective formulation forsitting and sizing of DG units was proposed in [6] Themethod involved searching for a compromise between thecost of network upgrades cost of power losses cost of energynot supplied and cost of energy required by the servedcustomers A genetic algorithm was implemented to obtain anoninferior solution set However theirmethod cannot guar-antee a solution to be optimal solution In [7] an improvedswarm optimization (IPSO) method was presented to solvethe multi-objective optimal power flow problem The multi-objective optimal power flow considered the cost loss volt-age stability and emission impacts as the objective functionsA fuzzy decision-based mechanism is used to select the bestcompromise Pareto set solution obtained by the proposedalgorithm In [8] a new penalty parameter-less constraint-handling scheme was employed to improve the performanceof the evolutionary algorithmThe experiments in that paperrevealed that PSOperforms better in terms of solution qualityand consistency and DE performs better in terms of meancomputation time An improved Cai andWangrsquos method hasbeen proposed to combinemulti-objective optimization withdifferential evolution to address constrained optimizationproblems in [9] The method provided a novel infeasiblesolution replacement mechanism for differential evolutionin theory In [10] a robust DE algorithm was proposedfor the control of selective harmonic distortion and totalharmonic distortion A fuzzy optimization technique andDEoptimization method are described

The literature includes several DG output studies thatexamined multiple objectives and applied evolutionary opti-mization techniques From the perspective of mathemat-ical optimization DG unit injection is a complex multi-objective optimization problem that presents a challengeto the optimization analysis of a distribution power sys-tem The objectives include optimal energy consumptionthe minimum power consumerrsquos electricity purchasing costand the minimum power loss based on the constraints ofpower grid security and DG power output Multi-objectiveeconomicemission dispatch algorithms were investigated in[11 12] In the optimization methods literature the simulatedannealing technique has been applied to optimize the pro-posed multi-objective model of DG planning [13]Themulti-objective problems were solved by converting the originalmodel into an equivalent model through calibration of theweighted factors method In [14] a multi-objective Tabusearch- (TS-) based method was utilized to optimize a DGallocation problem In that paper the TS-based approach wasprovided to find the optimal Pareto set Fuzzy optimizationwas also used to solve the multi-objective optimization ofDG allocation in [15] Voltage drop reduction short circuitcapacity augmentation decrease operation cost and systemloss reduction were considered objectives for formulatingfuzzy optimization

In this paper a DG multi-objective optimization methodbased on an improved evolutionary algorithm was investi-gated for a distribution power system Adaptive crossoverand a mutation operator were used in the evolutionaryprocess and simulated annealing was combined in theiterative process A fuzzy clustering algorithm was appliedto manage the size of the Pareto set The rest of the paper isorganized as follows In Section 2 the formulation of the DGmultiple-objective optimization for distributionmanagementis presented The Pareto-based algorithm and some basicconcepts are introduced in Section 3 The improved Paretoevolution algorithm is described in Section 4 Section 5 pro-vides the numerical results and comparison analysis with theproposed approachusing the revised IEEE 33-bus systemTheconclusion and future work are provided in Section 6

2 Problem Formulation

21 Objective Functions Three objectives are considered inthe optimization model which includes the fuel cost andthe pollutant emission penalty reducing consumer costs onelectricity bills when DG units are injected into the distri-bution network and reducing transmission line losses Thefirst optimization objective is minimum energy consumptionand a pollutant emission model which is mainly basedon government requirements There will be more penaltiesif the system emits more pollutants and exhibits greaterfuel consumption The second objective is consumer relatedwhere the consumer uses DG to maximize savings on theirbills The third objective is to lower system line losses whichis the demand objective of the power supply provider Thethree objectives involve perspectives based on governmentrequirements consumer needs and power supply enterpriseneeds and the objectives can conflict For example whena consumer utilizes a micro gas turbine to maximize theirsavings on their energy bill there is a subsequent increase infuel cost and pollutant emission In addition the extra powerfrom the micro gas turbine will increase or decrease the linelosses depending on the size and placement location of themicro gas turbine

211 Fuel Cost andPollutant EmissionMinimization Thefirstobjective is to minimize the fuel cost and the pollutant emis-sion penalty which reflects the impact of energy utilizationon the environment It can be expressed as follows

1198651(119909) =

119879

sum

119905=1

[119862119877+ 119862119882] (1)

where 119862119877is the energy consumption cost and 119862

119882is the

pollutant emission penaltyThe fuel cost 119862

119877normally can be further expressed as

follows

119862119877(119875DG) =

119873DG

sum

119894=1

(119888119894+ 119887119894119875DG119894 + 119886119894119875

2

DG119894) (2)

Note that 119888119894 119887119894 and 119886

119894are the quadratic cost coefficients of

the 119894th DG and119873DG is the number of distributed generators

Journal of Applied Mathematics 3

119875DG119894 is the real power output of the 119894th generator 119875DG isthe vector of real power outputs of generators and defined asfollows

119875DG = [119875DG1 119875DG2 119875DG119899]119879

(3)

The pollutant emission quantity can be obtained basedon DG output Then based on the penalty standard theenvironmental penalty for pollutant emission is calculated asfollows

119862119882=

119875

sum

119895=1

119884119895119863119895 (4)

where 119884119895is pollutant 119895rsquos emission quantity and 119863

119895is the

penalty standard of pollutant 119895

212 Maximization of Cost Savings Using DG The secondobjective is to maximize the cost savings on electricity userbills when the DG is injected into the distribution networkThe savings in electricity which should have been purchasedfrom the power supply enterprise are the total power outputof the DG units Utilizing DG output and time-of-use (TOU)rate consumer electricity purchasing costs could be reducedas follows

maximize 1198652(119909) =

1198792

sum

119905=1198791

1198621198891119875DG119905 +

1198791

sum

119905=0

1198621198892119875DG119905

+

24

sum

119905=1198792

1198621198892119875DG119905

(5)

where 1198621198891

is peak price from 1198791to 1198792 1198621198892

is off-peak priceand 119875DG119905 is the DG total power output at moment 119905

213 Minimization of Line Losses The third objective isto minimize the system line losses after DG injection intothe distribution network This objective can be expressed asfollows

1198653(119909) = 119875loss =

119898

sum

119894=0

(119875[119894]2+ 119876[119894]

2

119880[119894]2

)119877 [119894] (6)

where 119875[119894] is the active power 119876[119894] is the reactive power atbranch 119894 119880[119894] is the voltage at branch 119894 after DG injection119877[119894] is the resistance of branch 119894 and 119898 is the number ofbranches in the distribution network

In the previous three optimization models the fuel costand the pollutant emission penalty function 119865

1(119909) and the

system loss function 1198653(119909) should be minimized whereas the

cost-saving function 1198652(119909) should be maximized

22 Constraints Three constraint conditions are consideredin the optimization model which includes constraints ofpower flow equations nodal voltage and DG capacity

221 Equality Constraints The constraint of power flowequations is described as follows

119875DG119894 minus 119875119889119894 = 119881119894

119873

sum

119895=1

119881119895(119866119894119895cos 120579119894119895+ 119861119894119895sin 120579119894119895)

119876DG119894 minus 119876119889119894 = 119881119894

119873

sum

119895=1

119881119895(119866119894119895sin 120579119894119895minus 119861119894119895cos 120579119894119895)

(7)

where 119875DG119894 and 119876DG119894 are active and reactive generationoutputs whereas 119875

119889119894and 119876

119889119894are the active and reactive load

at bus 119894 respectively 119866119894119895and 119861

119894119895are the transfer conductance

and susceptance between bus 119894 and 119895 respectively and 119873 isthe number of buses

222 Inequality Constraints

Generation limits119875minDG119894 le 119875DG119894 le 119875

maxDG119894

119876minDG119894 le 119876DG119894 le 119876

maxDG119894

(8)

Load bus voltage constraints119881119894min le 119881119894 le 119881119894max (9)

Thermal limits10038161003816100381610038161003816119878119894119895

10038161003816100381610038161003816=100381610038161003816100381610038161198812

119894119866119894119895minus 119881119894119881119895(119866119894119895cos 120579119894119895+ 119861119894119895sin 120579119894119895)10038161003816100381610038161003816

le 119878max119894119895

(10)

In the inequality constraints 119875minDG119894 119875

maxDG119894 119876

minDG119894 and 119876

maxDG119894

are the lowerupper active and reactive generating unit limitsof DG respectively 119878max

119894119895is the apparent power thermal limit

of the circuit between buses 119894 and 119895There is always a limit on penetration of DG for a distri-

bution power system to ensure reliability Different countrieshave different penetration factor values The penetrationfactor indicates the aggregatedDGrating on an electric powersystem (EPS) feeder divided by the peak EPS feeder load Ifwe assume that the maximum DG penetration factor is 25then themaximum injected DG capacity should be limited to25 of the maximum total load in the distribution networkwhich can be described as follows

119899

sum

119894=1

119875DG119894 le 025119878max (119894 isin Φ

119878) (11)

where 119875DG119894 is the DG access capacity at node 119894 and 119878max is themaximum load capacity of distribution network

23 Overview Formulation Aggregating the objectives andconstraints the problem can be formulated as a nonlinearprogramming problem as follows

minimize [1198651(119909) 119865

2(119909) 119865

119896(119909)]

subject to ℎ119894(119909) = 0 119894 = 1 119901

119892119894(119909) le 0 119894 = 1 119899

(12)


where 119896 is the number of objectives and 119909 is the vector ofdependent variables consisting of slack bus power output andDG active power out 119875DG119894 load bus voltage119881119894 and generatorreactive power outputs 119876DG119894 Thus 119909 can be expresses asfollows

119909 = [119875DG1 119875DG119873119866 1198811198711 119881119871119873119871 1198761198631198661 119876DG119873119866]119879

(13)

where 119899 is the number of inequality constraints 119901 is numberof equation constraints 119896 is the number of objectives and119873119866is the number of DG units

3 A Pareto-Based Algorithmand Additional Concepts

31 Concepts of Dominated Nondominated and Pareto SetMulti-objective optimization can be expressed as

min119891119894(119909) 119894 = 1 2 119898 119909 isin 120594 (14)

where 119891119894(119909) denotes the 119894th objective function 119898 is the

number of objectives and 120594 represents the feasible searchspace

Definition 1 A solution 1199091is said to dominate 119909

2(denoted by

1199091≺ 1199092) if and only if

forall119894 isin 1 2 119898 119891119894(1199091) le 119891119894(1199092)

and exist119895 isin 1 2 119898 119891119895(1199091) lt 119891119895(1199092)

(15)

Definition 2 For 119878 = 119909119894 119894 = 1 119899 solution 119909 is said to be

a nondominated solution (Pareto solution) of set 119878 if 119909 isin 119878and there is no solution 1199091015840 isin 119878 for which 1199091015840 dominates 119909

Definition 3 Assume that set 119875 contains all the nondomi-nated solutions of 119878 then PF = V | V = [119891

1(119909) 1198912(119909)

119891119898(119909)]119879 119909 isin 119875 is a Pareto front of set 119878

32 Basic Pareto-Based Evolutionary Algorithm The tra-ditional Pareto-based evolutionary algorithm is shown inFigure 1 The detailed algorithm procedure is explained in[16] The main improvements on the Pareto-based algorithmcan be generalized as follows The penalty function is estab-lished to constrain the solution of the objective functionThe adaptive crossover and mutation are adopted in theevolution process which improves the probability of globaloptimization The simulated annealing algorithm is addedto the iterative process so that the algorithm is able to seekthe optimal solution globally and rapidly converges to theoptimal solution

4 Proposed Improved ParetoEvolutionary Algorithm

41 Overview To solve the difficulties in traditional opti-mization techniques a new evolutionary population-based

searching technique is proposed to solve the multiobjectiveoptimization problem based on SPEA2 [17 18]

42 Initialization In the improved SPEA2 an individual119894 at generation 119866 is a multidimensional vector 119909119866

119894=

(1199091198941 119909

119894119863) The population is initialized by randomly

generating individuals as

119909119866

119894119896= 119909119896min

+ rand [0 1] times (119909119896maxminus 119909119896min)

119894 isin [1119873119901] 119896 isin [1 119863]

(16)

where 119873119901is the population size and 119863 is the number of

control variables Each variable 119896 in a solution vector 119894 in thegeneration 119866 initialized within its boundaries 119909

119896minand 119909

119896max

43 Fitness Evaluation The objective of each solution1198651(119909) 1198652(119909) 119865

119896(119909) will be computed The individual

fitness values in both the population-based set 119875119900119901 andnondominated archive set119873119863119878119890119905will be calculated based on(17) The mismatch of each constraint value is multiplied bya large value and added to all objectives to remove infeasiblesolutions The methodology is to evaluate the feasible solu-tions according to the value of objective function and removethe infeasible solutions according to the constraints

The individualrsquos fitness 119872(119894) will be obtained from thesum of the primary fitness value 119877(119894) and the density 119863(119894) asfollows

119872(119894) = 119877 (119894) + 119863 (119894) (17)

where 119877(119894) = sum119895isin119875119900119901+119873119863119878119890119905119895≻119894

119878(119895) 119878(119895) is the objective evalu-ation for the individual 119895 (≻ indicates a dominated relation119909119894≻ 119909119895indicates 119909

119894dominates 119909

119895 119909119894is nondominated and

119909119895is dominated)119863(119894) = 1(120590

119896

119894+ 2) 119896 = radic119873 +119873 120590119896

119894represents the

objective space distance between individual 119894 in 119875119900119901 and the119896th nearest neighbor individual in the 119873119863119878119890119905 It is the K-nearest neighbor (KNN) method and the distance betweenthe individual 119894 in119875119900119901 and the other individuals in the119873119863119878119890119905need to be computed and then the distance value can besorted

44 Adaptive Crossover and Mutation Probability The selec-tion of crossover probability 119875

119888and mutation probability

119875119898

dominates the solution process 119875119888and 119875

119898determine

the generation speed and the probability of new individualsrespectively If 119875

119888exceeds the threshold the generation speed

of the new population will be quicker whichmeans that thereis a stronger capability to explore new space If119875

119888is extremely

small the search process will be quite slow If 119875119898is too large

the search process will bemore randomThe adaptive value of


InitializationCreate population and initialize gene

Objectives evaluation1198651(119909) 1198652(119909) 119865119896(119909)

Find the Pareto set

Check stopping criteria

Perform mutation

Perform crossover

Selection using dominance criteria

Find the bestcompromise solution

Stop

Yes

No

Figure 1 Flowchart of traditional Pareto-based approach

119875119888and119875119898is obtained from the following evaluated equations

119875119888=

1198751198881(119872avg minus119872

1015840) + 1198751198882(1198721015840minus119872min)

Mavg minusMmin 119872

1015840lt 119872avg

1198751198882(119872max minus119872

1015840) + 1198751198883(1198721015840minus119872avg)

119872max minus119872avg 1198721015840ge 119872avg

119875119898=

1198751198981(119872avg minus119872) + 1198751198982 (119872 minus119872min)

119872avg minus119872min 119872 lt 119872avg

1198751198982(119872max minus119872) + 1198751198983 (119872 minus119872avg)

119872max minus119872avg 119872 ge 119872avg

(18)

where 119872avg is the average fitness value 119872max is the biggestfitness value 1198721015840 is the bigger fitness value of two genes inthe crossover process119872 is the mutating individualrsquos fitnessvalue and the constants 119875

1198881 1198751198882 1198751198883 1198751198981 1198751198982 1198751198983

isin [0 1]1198751198881gt 1198751198882gt 1198751198883 1198751198981gt 1198751198982gt 1198751198983

45 Pareto Optimal Selection Using Fuzzy Set Theory In thispaper fuzzy set theory is used to select the optimal solutionset among the obtained multiobjective solution sets Fuzzysets are sets whose elements have degrees of membershipFuzzy set theory permits the gradual assessment of the mem-bership of elements in a set This membership is describedwith the aid of a membership function valued in the real unitinterval [0 1]

First define a linearmembership function 120591119894as theweight

of target 119894 in a solution

120591i =

1 119865119894= 119865

min119894

119865max119894

minus 119865119894

119865max119894

minus 119865min119894

119865min119894

lt 119865119894lt 119865

max119894

0 119865119894le 119865

max119894

(19)

where 119865max119894

is the maximum of 119894th objective function 119865min119894

isthe minimum of 119894th objective function and 119865

119894is the solution

of 119894th objective The previous equation provides a measureof the degree of satisfaction for each objective function fora particular solution


The dominant function 120591119896for each nondominated solu-

tion 119896 in Pareto solution set is calculated as follows

120591119896=

sum1198730

119894=1120591119894

119896

sum119906

119895=1sum1198730

119894=1120591119894

119895

(20)

where 119906 is the number of the Pareto solution set and1198730is the

number of the optimization objectivesBecause the value of 120591

119896determines the capability of

the solution the solution with maximum 120591119896will be Pareto

optimal Moreover the feasible priority sequence can beobtained by the value of 120591

119896 in descending order

The best Pareto optimal solution is the one achieving themaximummembership function 120591

119896 as shown in (20)

46 Simulated Annealing in Population-Based IndividualSelection Simulated annealing (SA) is a generic probabilisticmetaheuristic for the global optimization problem of locatinga good approximation to the global optimum of a givenfunction in a large search space It is often used when thesearch space is discrete Here SA is utilized in the individualselection

Based on the individuals after selection crossover andmutation steps the simulated annealing operation is per-formed on the individuals of the population The two genesin each individual will be selected and disturbed randomlyThen the new individual will be evaluated to formnewfitnessvalues If the fitness value of a new individual is larger than theold value then the old individual will be replaced by the newindividual If the fitness value of the new individual is smallerthan the old value the new individual can also be acceptedusing the following probability

119901 (119879119896+1) =

1 (119872119896+1

gt 119872119896)

exp(minus119872119896+1

minus119872119896

119879119896+1

) (119872119896+1

le 119872119896)

119879119896+1

= 120572119879119896

(21)

where 119872119896+1

and 119872119896are the fitness of the new individual

and old individual respectively 119901(119879119896+1) is the acceptance

probability at 119879119896+1

temperature and 120572 is the temperaturedescending coefficient

47 Convergence Condition The iterative procedure can beterminated when any of the following conditions are met (1)the true Pareto front is obtained and (2) the iteration numberof the algorithm reaches the predefinedmaximumnumber ofiterations However the true Pareto front will not be knownin advance in most practical multiobjective problems so theconvergence condition is to iterate to a predefined maximaliteration number

48 Flowchart of Proposed Algorithm The flow chart of theproposed algorithm is illustrated in Figure 2 As shown inFigure 2 the steps of the proposed evolutionary algorithm aredescribed as follows

Step 1 Generate an initial set 119875119900119901 randomly and an emptyarchive set 119873119863119878119890119905 over the problem space initialize theparameters of the population size 119873 nondominated archivesize119873 and maximum generationrsquos number 119879

Step 2 Establish the penalty function to constrain eachobjective function and then form new objective functions

Step 3 Compute the fitness of individual in both thepopulation-based set 119875119900119901 and the nondominated archive setNDSet The objective of each solution 119865

1(119909) 1198652(119909) 119865

119896(119909)

will be computed

Step 4 Duplicate the nondominated individuals in both thepopulation and nondominated archive set to a new archiveset 119873119863119878119890119905 119899119890119908 if the size of 119873119863119878119890119905 119899119890119908 exceeds 119873 thenreduce 119873119863119878119890119905 119899119890119908 by means of the truncation operatorotherwise fill119873119863119878119890119905 119899119890119908with dominated individuals in119875119900119901and119873119863119878119890119905

Step 5 Evaluate if the nondominated set119873119863119878119890119905 119899119890119908 exceedsthe predefined size119873 If the size of119873119863119878119890119905 119899119890119908 is larger than119873 then truncate the nondominated individuals otherwisecontinue to Step 6

Step 6 Copy the superior dominated individual to119873119863119878119890119905 119899119890119908

Step 7 Evaluate the convergence criteria If the iterationnumber 119905 ge 119879 terminate the iteration to obtain the Paretooptimal solution and output the best solution otherwise set119905 = 119905 + 1 and continue to Step 8

Step 8 Perform adaptive crossover and mutation operationon the individuals of 119875119900119901

Step 9 Perform a simulated annealing operation and then goto Step 3

5 Experiments and Results

To demonstrate the effectiveness of the proposedmethod thealgorithm in Section 4 was implemented to obtain solutionsfor optimal active power dispatch of DG The IEEE 33bus distribution system was examined and three objectiveswere considered in this study These objectives were fuelcostpollutant emission transmission line loss and costsavings on bills using DG Photovoltaic (PV) panels dieselturbine and wind turbine distribution are injected into bus 7bus 17 bus 21 and bus 32 respectively as shown in Figure 3In this paper 119875

119888and 119875

119898are defined as follows 119875

1198881= 04

1198751198882= 03 and 119875

1198883= 02 and 119875

1198981= 02 119875

1198982= 01 and

1198751198983

= 005 The maximum iteration number is set to 200The proposed algorithm was coded in C++ and run on anIntel i5-3210M 25GHz notebook with 4GB RAM

51 Energy Utilization Cost Among the four DG units onlythe two diesel turbine DG units have fuel cost Because itwould be difficult for market players to acceptimplement


Generate an initial population set withan empty non-dominated set randomly

population set and archive set

Copy all the non-dominated individualsfrom population set and archive set to

new archive set

Non-dominatedindividuals exceed the

size of archive set

Truncate non-dominatedindividuals

Copy the excellent dominatedindividuals to the archive set

Yes

Yes

No

No

Performmutation

Performcrossover

Check theconvergence

condition

Obtain the best solution set

Establish the penalty function toconstrain each objective function

Performsimulatedannealing

1198651(119909) 1198652(119909) 119865119896(119909)

Calculate the fitness value of both

Figure 2 Flowchart of improved Pareto-based evolutionary algorithm

a central cost-based dispatch in the distribution systemincludingDGunits the cost of fossil-fuel consumed bymicrodiesel turbine is calculated as follows

119862119877=

119899

sum

119894=1

119891 (119875119905

119894) 119862119894 (22)

where119862119894is fuel price at power unit 119894 and119891(119875119905

119894) is the required

fuel quantity for power unit 119894 at the moment 119905

52 Penalty on Pollutant Emission As global environmentalpollution is growing optimizing power generation andpollutant emission costs are two conflicting goals Thesegoals present a restrictive and coordinated relationshipEnvironmental cost mainly refers to the fines related topollutant emission Tables 1 and 2 show the pollutantemission data for various DG units and the standard electricpower industry pollution fines respectively In referencereport [19] there are similar pollutant cost coefficientsfor distributed generation Based on the DG unit output


1 2 3 4 5 6Substation

25 26 27 28 29 30 3231

7 8 9 10 11 12 13 14 15 16 17

18 19 20 21

22 23 24

0

Small diesel turbine

PV

Windturbine

Smalldiesel

turbine

Figure 3 Schematic diagram of the IEEE 33-bus system with 4 DGunits

Table 1 The pollutant emission rate of DG units (gkWh)

Pollutantemission

Coalgeneration Diesel engine PV panel Wind

turbineNO119909

646 43314 0 0CO2 1070 2320373 0 0CO 155 23204 0 0SO2 993 04641 0 0

Table 2 Standard pollutant emission penalties ($kg)

SO2 NO119909

CO2 CO075 100 0002875 0125

following multiple-objective optimization the quantity ofpollutant emission can be obtained

53 Optimization Results Using the optimization modeldeveloped in Section 3 the optimized output of four DGunits over 24 h and the power system losses after DG unitinjection are shown in Figures 4 and 5 respectively As shownin Figure 4 the four DG units have different active poweroutputs at different time periods in a day The diesel poweroutput will increase when the solar and wind power outputsare at a low levelWhen the PVoutput andwind power outputincrease to the peak it will stop increasing and stay at thepeak power output and then the diesel power output willgradually decrease

As shown in Figure 5 the line losses greatly decrease DGunit penetration into the distribution system From the hours8 to 17 the total output of the four DG units provides enoughactive power which improves the voltage quality and reducesthe line loss

The forecasted and optimized solar power outputs basedon the computed results are shown in Figure 6 and theforecasting and optimized wind power output are shownin Figure 7 The forecasted PV and wind generation valuesare based on historical distribution system data Because ofthe cooperative optimization the optimal real power valuesof PV and wind generation are smaller than the forecastedvalues in the peak time period

Assuming that the coal consumption from the powerplant is 035 kgkWh and the highest coal price is 0124$kg the cost savings for coal consumption by using cleanenergy is illustrated in Figure 8 which shows that these

24222018161412108642Time period (h)

0

100

200

300

400

500

600

DG

out

put (

kW)

Diesel turbine 1Diesel turbine 2

Photovoltaic generationWind power generation

Figure 4 The optimized output of four DG units in 24 hours

0

50

100

150

200

250Li

ne lo

ss (k

W)

Before the DGs penetrationAfter the DGs penetration

24222018161412108642Time period (h)

Figure 5The power system loss before and after DG unit injection

increases in solar and wind power output results in greatercoal consumption cost savings

A pollutant emission penalty reduction curve wasobtained based on data from Tables 1 and 2 and the hourlypenalty reduction for pollutant emission is shown in Figure 9As there is no pollutant emission for solar and wind powergeneration when the output of new energy power supplyincreases the environment cost will decrease significantly

Assuming that the time-of-use price is 0095 $kWh forpeak time from 600 am to 2200 pm and 0054 $kWh inother period the cost saving for the electrical bills of usersper hour is shown in Figure 10 Because the price is at a highlevel from 600 am to 1800 pm the bill saving increases withthe increase of PV and wind power output The results fromthe case study demonstrates that the system loss is greatlyreduced by 65 so that the users in total can save $1671per day on their electricity bills and power plants can save

Journal of Applied Mathematics 9Ac

tive p

ower

of p

hoto

volta

ic g

ener

atio

n (k

W)

42222018161412108642Time period (h)

0

100

200

300

400

500

600

700

The forecast of PV generationThe optimal computation of PV generation

Figure 6The comparison of forecasted PV value and the computedoptimal PV value

24222018161412108642Time period (h)

0

100

200

300

400

500

600

700

Activ

e pow

er o

f win

d (k

W)

The forecast of wind generationThe optimal computation of wind generation

Figure 7 The comparison of forecasted wind values and thecomputed optimal value

$870 and $9906 on their coal costs and pollutant emissionpenalties per day respectively

54 Comparison of Different Algorithms The proposed algo-rithm was compared with the SPEA2 [17] and the particleswarm optimization method [20] and NSGA-II [21 22] TheIEEE 33-bus system with four DG units was utilized as anexample for this comparison The load data at hour 11 isselected as the basic load data The convergence conditionwas that the iteration number exceeded the preset maximumiteration number which was set to 200 In PSO the cognitiveratio and social ratio are all equal to 20Thenumber of swarmparticles is 100 In NSGA-II the crossover ratio is set to 08and the mutation ratio is set to 02 The size of population isset to 100

24222018161412108642Time period (h)

30

31

32

33

34

35

36

37

38

39

40

Save

d co

st of

coal

usin

g D

Gs (

$)

Figure 8 Hourly cost savings on coal consumption

24222018161412108642Time period (h)

0

200

400

600

800

1000

1200

1400

1600

Save

d co

st of

envi

ronm

enta

l pol

lutio

n fin

es ($

)

Figure 9 Hourly penalty reduction for pollutant emission

Table 3 shows that the proposed algorithm performsbetter than the SPEA2 and the PSO algorithms with respectto calculating the multiple objective objectives in the samelimited iterations and the proposed algorithm has betterconvergence speed than SPEA2 and PSO because simulatedannealing is added in addition to the adaptive crossover andmutation operations Compared withNSGA-II the proposedalgorithm has approximate speed in searching the Paretofront

6 Conclusion

This paper presented an improved Pareto-based evolutionaryalgorithm which increases the global optimization abilitywith a simulated annealing iterative process and fuzzy settheory to solve the multiobjective optimization problemfor a distribution power system The proposed algorithmwas utilized to optimize a model of DG unit injectionwith objectives of maximizing the utilization of DG whileminimizing the system loss and environmental pollutionTheresults indicate that the proposed optimization is applicableto practical multiobjective optimization problems that take


242220181614121086420

10

20

30

40

50

60

70

80

90

100

Time period (h)

Pow

er p

urch

ase c

ost (

$)

Figure 10 Hourly bill saving for consumers with DG unit injection

Table 3 Comparison of different algorithms

Iterations Time Min1198651(119909)

Max1198652(119909)

Min1198653(119909)

Proposedalgorithm 200 34 s $15353 $860 1145 kW

SPEA2 200 42 s $15685 $845 1290 kWPSO 200 39 s $15896 $826 1312 kWNSGA-II 200 36 s $15739 $832 1252 kW

into considering the requirements from utilities consumersand the environment

With respect to the state of the art the improvementsfrom this new multiobjective optimization method can belisted as follows (1) the ability to search an entire set ofPareto optimal solutions is enhanced by using SA which isproven by the comparison experiments and (2) the Paretofront converges to better optimum set of solutions using theproposed algorithm Future work will be focused on proba-bilistic evaluation and optimization that considers multipleDG units and load profile in distribution systems

References

[1] E Zitzler and L Thiele ldquoMultiobjective evolutionary algo-rithms a comparative case study and the strength Paretoapproachrdquo IEEETransactions onEvolutionaryComputation vol3 no 4 pp 257ndash271 1999

[2] M A Abido and N A Al-Ali ldquoMulti-objective optimal powerflow using differential evolutionrdquo Arabian Journal for Scienceand Engineering vol 37 no 4 pp 991ndash1005 2012

[3] M Varadarajan andK S Swarup ldquoSolvingmulti-objective opti-mal power flow using differential evolutionrdquo IET GenerationTransmission and Distribution vol 2 no 5 pp 720ndash730 2008

[4] C A Coello Coello ldquoEvolutionary multi-objective optimiza-tion a historical view of the fieldrdquo IEEE Computational Intel-ligence Magazine vol 1 no 1 pp 28ndash36 2006

[5] M A Abido ldquoMultiobjective evolutionary algorithms for elec-tric power dispatch problemrdquo IEEE Transactions on Evolution-ary Computation vol 10 no 3 pp 315ndash329 2006

[6] G Celli E Ghiani S Mocci and F Pilo ldquoA multiobjectiveevolutionary algorithm for the sizing and siting of distributedgenerationrdquo IEEE Transactions on Power Systems vol 20 no 2pp 750ndash757 2005

[7] T Niknam M R Narimani J Aghaei et al ldquoImproved particleswarm optimisation for multi-objective optimal power flowconsidering the cost loss emission and voltage stability indexrdquoIET Generation Transmission and Distribution vol 6 no 6 pp515ndash527 2012

[8] P S Manoharan P S Kannan S Baskar andMW Iruthayara-jan ldquoPenalty parameter-less constraint handling scheme basedevolutionary algorithm solutions to economic dispatchrdquo IETGeneration Transmission andDistribution vol 2 no 4 pp 478ndash490 2008

[9] Y Wang and Z Cai ldquoCombining multiobjective optimizationwith differential evolution to solve constrained optimizationproblemsrdquo IEEE Transactions on Evolutionary Computationvol 16 no 1 pp 117ndash134 2012

[10] A Darvishi A Alimardani and S H Hosseinian ldquoFuzzymulti-objective technique integrated with differential evolutionmethod to optimise power factor and total harmonic distor-tionrdquo IET Generation Transmission and Distribution vol 5 no9 pp 921ndash929 2011

[11] A Bhattacharya and P K Chattopadhyay ldquoHybrid differentialevolutionwith biogeography-based optimization for solution ofeconomic load dispatchrdquo IEEE Transactions on Power Systemsvol 25 no 4 pp 1955ndash1964 2010

[12] T Niknam and H Doagou-Mojarrad ldquoMultiobjective eco-nomicemission dispatch by multiobjective 120579-particle swarmoptimisationrdquo IET Generation Transmission and Distributionvol 6 no 5 pp 363ndash377 2012

[13] A I Aly Y G Hegazy and M A Alsharkawy ldquoA simulatedannealing algorithm for multi-objective distributed generationplanningrdquo in IEEE Power and Energy Society General Meeting(PES rsquo10) pp 1ndash7 July 2010

[14] R S Maciel and A Padilha-Feltrin ldquoDistributed generationimpact evaluation using a multi-objective tabu searchrdquo in Pro-ceedings of the 15th International Conference on Intelligent SystemApplications to Power Systems (ISAP rsquo09) pp 1ndash5 November2009

[15] E B Cano ldquoUtilizing fuzzy optimization for distributed gen-eration allocationrdquo in Proceedings of the 10 Annual InternationalConference (TENCON rsquo07) pp 1ndash4 Taipei ChinaOctober 2007

[16] C A Coello Coello G B Lamont and D A Van VeldhuizenEvolutionary Algorithms for Solving Multi-Objective ProblemsGenetic and Evolutionary Computation Series Springer NewYork NY USA 2nd edition 2007

[17] Z Eckart M Laumanns and L Thiele ldquoSPEA2 improving theStrength Pareto evolutionary algorithmrdquo TIK Report 103 SwissFederal Institute of Technology (ETH) Zurich Switzerland2001

[18] W Sheng Y Liu X Meng and T Zhang ldquoAn ImprovedStrength Pareto Evolutionary Algorithm 2 with application tothe optimization of distributed generationsrdquo Computers andMathematics with Applications vol 64 no 5 pp 944ndash955 2012

[19] Y H Wan and S Adelman Distributed Utility Technology CostPerformance and Environmental Characteristic United StatesNational Renewable Energy Laboratory 1995

[20] M AlHajri Sizing and placement of distributed generation inelectrical distribution systems using conventional and heuristicoptimization methods [PhD Dissertation] Dalhousie Univer-sity Halifax Canada 2009


[21] K Deb A Pratap S Agarwal and T Meyarivan ldquoA fastand elitist multiobjective genetic algorithm nSGA-IIrdquo IEEETransactions on Evolutionary Computation vol 6 no 2 pp 182ndash197 2002

[22] httpwwwtikeeethzchsoppisaselectorsnsga2page=nsga2php

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014


Stochastic AnalysisInternational Journal of


Pareto-based multi-objective evolutionary algorithms werediscussed and evaluated A nondominated sorting geneticalgorithm (NSGA) a niched Pareto genetic algorithm and astrength Pareto evolutionary algorithm (SPEA) were devel-oped and applied to an environmentaleconomic electricpower dispatch problem A multi-objective formulation forsitting and sizing of DG units was proposed in [6] Themethod involved searching for a compromise between thecost of network upgrades cost of power losses cost of energynot supplied and cost of energy required by the servedcustomers A genetic algorithm was implemented to obtain anoninferior solution set However theirmethod cannot guar-antee a solution to be optimal solution In [7] an improvedswarm optimization (IPSO) method was presented to solvethe multi-objective optimal power flow problem The multi-objective optimal power flow considered the cost loss volt-age stability and emission impacts as the objective functionsA fuzzy decision-based mechanism is used to select the bestcompromise Pareto set solution obtained by the proposedalgorithm In [8] a new penalty parameter-less constraint-handling scheme was employed to improve the performanceof the evolutionary algorithmThe experiments in that paperrevealed that PSOperforms better in terms of solution qualityand consistency and DE performs better in terms of meancomputation time An improved Cai andWangrsquos method hasbeen proposed to combinemulti-objective optimization withdifferential evolution to address constrained optimizationproblems in [9] The method provided a novel infeasiblesolution replacement mechanism for differential evolutionin theory In [10] a robust DE algorithm was proposedfor the control of selective harmonic distortion and totalharmonic distortion A fuzzy optimization technique andDEoptimization method are described

The literature includes several DG output studies thatexamined multiple objectives and applied evolutionary opti-mization techniques From the perspective of mathemat-ical optimization DG unit injection is a complex multi-objective optimization problem that presents a challengeto the optimization analysis of a distribution power sys-tem The objectives include optimal energy consumptionthe minimum power consumerrsquos electricity purchasing costand the minimum power loss based on the constraints ofpower grid security and DG power output Multi-objectiveeconomicemission dispatch algorithms were investigated in[11 12] In the optimization methods literature the simulatedannealing technique has been applied to optimize the pro-posed multi-objective model of DG planning [13]Themulti-objective problems were solved by converting the originalmodel into an equivalent model through calibration of theweighted factors method In [14] a multi-objective Tabusearch- (TS-) based method was utilized to optimize a DGallocation problem In that paper the TS-based approach wasprovided to find the optimal Pareto set Fuzzy optimizationwas also used to solve the multi-objective optimization ofDG allocation in [15] Voltage drop reduction short circuitcapacity augmentation decrease operation cost and systemloss reduction were considered objectives for formulatingfuzzy optimization

In this paper a DG multi-objective optimization methodbased on an improved evolutionary algorithm was investi-gated for a distribution power system Adaptive crossoverand a mutation operator were used in the evolutionaryprocess and simulated annealing was combined in theiterative process A fuzzy clustering algorithm was appliedto manage the size of the Pareto set The rest of the paper isorganized as follows In Section 2 the formulation of the DGmultiple-objective optimization for distributionmanagementis presented The Pareto-based algorithm and some basicconcepts are introduced in Section 3 The improved Paretoevolution algorithm is described in Section 4 Section 5 pro-vides the numerical results and comparison analysis with theproposed approachusing the revised IEEE 33-bus systemTheconclusion and future work are provided in Section 6

2 Problem Formulation

21 Objective Functions Three objectives are considered inthe optimization model which includes the fuel cost andthe pollutant emission penalty reducing consumer costs onelectricity bills when DG units are injected into the distri-bution network and reducing transmission line losses Thefirst optimization objective is minimum energy consumptionand a pollutant emission model which is mainly basedon government requirements There will be more penaltiesif the system emits more pollutants and exhibits greaterfuel consumption The second objective is consumer relatedwhere the consumer uses DG to maximize savings on theirbills The third objective is to lower system line losses whichis the demand objective of the power supply provider Thethree objectives involve perspectives based on governmentrequirements consumer needs and power supply enterpriseneeds and the objectives can conflict For example whena consumer utilizes a micro gas turbine to maximize theirsavings on their energy bill there is a subsequent increase infuel cost and pollutant emission In addition the extra powerfrom the micro gas turbine will increase or decrease the linelosses depending on the size and placement location of themicro gas turbine

211 Fuel Cost andPollutant EmissionMinimization Thefirstobjective is to minimize the fuel cost and the pollutant emis-sion penalty which reflects the impact of energy utilizationon the environment It can be expressed as follows

1198651(119909) =

119879

sum

119905=1

[119862119877+ 119862119882] (1)

where 119862119877is the energy consumption cost and 119862

119882is the

pollutant emission penaltyThe fuel cost 119862

119877normally can be further expressed as

follows

119862119877(119875DG) =

119873DG

sum

119894=1

(119888119894+ 119887119894119875DG119894 + 119886119894119875

2

DG119894) (2)

Note that 119888119894 119887119894 and 119886

119894are the quadratic cost coefficients of

the 119894th DG and119873DG is the number of distributed generators



119875DG = [119875DG1 119875DG2 119875DG119899]119879

(3)


119862119882=

119875

sum

119895=1

119884119895119863119895 (4)


119895is the



maximize 1198652(119909) =

1198792

sum

119905=1198791

1198621198891119875DG119905 +

1198791

sum

119905=0

1198621198892119875DG119905

+

24

sum

119905=1198792

1198621198892119875DG119905

(5)

where 1198621198891




1198653(119909) = 119875loss =

119898

sum

119894=0

(119875[119894]2+ 119876[119894]

2

119880[119894]2

)119877 [119894] (6)



1(119909) and the





119875DG119894 minus 119875119889119894 = 119881119894

119873

sum

119895=1

119881119895(119866119894119895cos 120579119894119895+ 119861119894119895sin 120579119894119895)

119876DG119894 minus 119876119889119894 = 119881119894

119873

sum

119895=1

119881119895(119866119894119895sin 120579119894119895minus 119861119894119895cos 120579119894119895)

(7)


119889119894and 119876







maxDG119894

119876minDG119894 le 119876DG119894 le 119876

maxDG119894

(8)


Thermal limits10038161003816100381610038161003816119878119894119895

10038161003816100381610038161003816=100381610038161003816100381610038161198812

119894119866119894119895minus 119881119894119881119895(119866119894119895cos 120579119894119895+ 119861119894119895sin 120579119894119895)10038161003816100381610038161003816

le 119878max119894119895

(10)


maxDG119894 119876

minDG119894 and 119876

maxDG119894





119899

sum

119894=1

119875DG119894 le 025119878max (119894 isin Φ

119878) (11)



minimize [1198651(119909) 119865

2(119909) 119865

119896(119909)]

subject to ℎ119894(119909) = 0 119894 = 1 119901

119892119894(119909) le 0 119894 = 1 119899

(12)



119909 = [119875DG1 119875DG119873119866 1198811198711 119881119871119873119871 1198761198631198661 119876DG119873119866]119879

(13)




min119891119894(119909) 119894 = 1 2 119898 119909 isin 120594 (14)




2(denoted by

1199091≺ 1199092) if and only if

forall119894 isin 1 2 119898 119891119894(1199091) le 119891119894(1199092)

and exist119895 isin 1 2 119898 119891119895(1199091) lt 119891119895(1199092)

(15)




1(119909) 1198912(119909)







119894=

(1199091198941 119909



119909119866

119894119896= 119909119896min


119894 isin [1119873119901] 119896 isin [1 119863]

(16)



119896minand 119909

119896max





119872(119894) = 119877 (119894) + 119863 (119894) (17)

where 119877(119894) = sum119895isin119875119900119901+119873119863119878119890119905119895≻119894




119909119895is dominated)119863(119894) = 1(120590

119896

119894+ 2) 119896 = radic119873 +119873 120590119896





119875119898


119898determine




119888is extremely






Find the Pareto set


Perform mutation

Perform crossover



Stop

Yes

No



119875119888=

1198751198881(119872avg minus119872

1015840) + 1198751198882(1198721015840minus119872min)


1015840lt 119872avg

1198751198882(119872max minus119872

1015840) + 1198751198883(1198721015840minus119872avg)


119875119898=





(18)


1198881 1198751198882 1198751198883 1198751198981 1198751198982 1198751198983

isin [0 1]1198751198881gt 1198751198882gt 1198751198883 1198751198981gt 1198751198982gt 1198751198983




120591i =

1 119865119894= 119865

min119894

119865max119894

minus 119865119894

119865max119894

minus 119865min119894

119865min119894

lt 119865119894lt 119865

max119894

0 119865119894le 119865

max119894

(19)

where 119865max119894








120591119896=

sum1198730

119894=1120591119894

119896

sum119906

119895=1sum1198730

119894=1120591119894

119895

(20)








119896 as shown in (20)



119901 (119879119896+1) =

1 (119872119896+1

gt 119872119896)

exp(minus119872119896+1

minus119872119896

119879119896+1

) (119872119896+1

le 119872119896)

119879119896+1

= 120572119879119896

(21)

where 119872119896+1










1(119909) 1198652(119909) 119865

119896(119909)

will be computed









119888and 119875


1198881= 04

1198751198882= 03 and 119875

1198883= 02 and 119875

1198981= 02 119875

1198982= 01 and

1198751198983







new archive set


size of archive set



Yes

Yes

No

No

Performmutation

Performcrossover


condition




1198651(119909) 1198652(119909) 119865119896(119909)




119862119877=

119899

sum

119894=1

119891 (119875119905

119894) 119862119894 (22)







25 26 27 28 29 30 3231

7 8 9 10 11 12 13 14 15 16 17

18 19 20 21

22 23 24

0


PV

Windturbine

Smalldiesel

turbine



Pollutantemission


turbineNO119909

646 43314 0 0CO2 1070 2320373 0 0CO 155 23204 0 0SO2 993 04641 0 0


SO2 NO119909

CO2 CO075 100 0002875 0125






24222018161412108642Time period (h)

0

100

200

300

400

500

600

DG

out

put (

kW)




0

50

100

150

200

250Li

ne lo

ss (k

W)


24222018161412108642Time period (h)






tive p

ower

of p

hoto

volta

ic g

ener

atio

n (k

W)

42222018161412108642Time period (h)

0

100

200

300

400

500

600

700



24222018161412108642Time period (h)

0

100

200

300

400

500

600

700

Activ

e pow

er o

f win

d (k

W)





24222018161412108642Time period (h)

30

31

32

33

34

35

36

37

38

39

40

Save

d co

st of

coal

usin

g D

Gs (

$)


24222018161412108642Time period (h)

0

200

400

600

800

1000

1200

1400

1600

Save

d co

st of

envi

ronm

enta

l pol

lutio

n fin

es ($

)



6 Conclusion



242220181614121086420

10

20

30

40

50

60

70

80

90

100

Time period (h)

Pow

er p

urch

ase c

ost (

$)




Max1198652(119909)

Min1198653(119909)





References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014





119875DG = [119875DG1 119875DG2 119875DG119899]119879

(3)


119862119882=

119875

sum

119895=1

119884119895119863119895 (4)


119895is the



maximize 1198652(119909) =

1198792

sum

119905=1198791

1198621198891119875DG119905 +

1198791

sum

119905=0

1198621198892119875DG119905

+

24

sum

119905=1198792

1198621198892119875DG119905

(5)

where 1198621198891




1198653(119909) = 119875loss =

119898

sum

119894=0

(119875[119894]2+ 119876[119894]

2

119880[119894]2

)119877 [119894] (6)



1(119909) and the





119875DG119894 minus 119875119889119894 = 119881119894

119873

sum

119895=1

119881119895(119866119894119895cos 120579119894119895+ 119861119894119895sin 120579119894119895)

119876DG119894 minus 119876119889119894 = 119881119894

119873

sum

119895=1

119881119895(119866119894119895sin 120579119894119895minus 119861119894119895cos 120579119894119895)

(7)


119889119894and 119876







maxDG119894

119876minDG119894 le 119876DG119894 le 119876

maxDG119894

(8)


Thermal limits10038161003816100381610038161003816119878119894119895

10038161003816100381610038161003816=100381610038161003816100381610038161198812

119894119866119894119895minus 119881119894119881119895(119866119894119895cos 120579119894119895+ 119861119894119895sin 120579119894119895)10038161003816100381610038161003816

le 119878max119894119895

(10)


maxDG119894 119876

minDG119894 and 119876

maxDG119894





119899

sum

119894=1

119875DG119894 le 025119878max (119894 isin Φ

119878) (11)



minimize [1198651(119909) 119865

2(119909) 119865

119896(119909)]

subject to ℎ119894(119909) = 0 119894 = 1 119901

119892119894(119909) le 0 119894 = 1 119899

(12)



119909 = [119875DG1 119875DG119873119866 1198811198711 119881119871119873119871 1198761198631198661 119876DG119873119866]119879

(13)




min119891119894(119909) 119894 = 1 2 119898 119909 isin 120594 (14)




2(denoted by

1199091≺ 1199092) if and only if

forall119894 isin 1 2 119898 119891119894(1199091) le 119891119894(1199092)

and exist119895 isin 1 2 119898 119891119895(1199091) lt 119891119895(1199092)

(15)




1(119909) 1198912(119909)







119894=

(1199091198941 119909



119909119866

119894119896= 119909119896min


119894 isin [1119873119901] 119896 isin [1 119863]

(16)



119896minand 119909

119896max





119872(119894) = 119877 (119894) + 119863 (119894) (17)

where 119877(119894) = sum119895isin119875119900119901+119873119863119878119890119905119895≻119894




119909119895is dominated)119863(119894) = 1(120590

119896

119894+ 2) 119896 = radic119873 +119873 120590119896





119875119898


119898determine




119888is extremely






Find the Pareto set


Perform mutation

Perform crossover



Stop

Yes

No



119875119888=

1198751198881(119872avg minus119872

1015840) + 1198751198882(1198721015840minus119872min)


1015840lt 119872avg

1198751198882(119872max minus119872

1015840) + 1198751198883(1198721015840minus119872avg)


119875119898=





(18)


1198881 1198751198882 1198751198883 1198751198981 1198751198982 1198751198983

isin [0 1]1198751198881gt 1198751198882gt 1198751198883 1198751198981gt 1198751198982gt 1198751198983




120591i =

1 119865119894= 119865

min119894

119865max119894

minus 119865119894

119865max119894

minus 119865min119894

119865min119894

lt 119865119894lt 119865

max119894

0 119865119894le 119865

max119894

(19)

where 119865max119894








120591119896=

sum1198730

119894=1120591119894

119896

sum119906

119895=1sum1198730

119894=1120591119894

119895

(20)








119896 as shown in (20)



119901 (119879119896+1) =

1 (119872119896+1

gt 119872119896)

exp(minus119872119896+1

minus119872119896

119879119896+1

) (119872119896+1

le 119872119896)

119879119896+1

= 120572119879119896

(21)

where 119872119896+1










1(119909) 1198652(119909) 119865

119896(119909)

will be computed









119888and 119875


1198881= 04

1198751198882= 03 and 119875

1198883= 02 and 119875

1198981= 02 119875

1198982= 01 and

1198751198983







new archive set


size of archive set



Yes

Yes

No

No

Performmutation

Performcrossover


condition




1198651(119909) 1198652(119909) 119865119896(119909)




119862119877=

119899

sum

119894=1

119891 (119875119905

119894) 119862119894 (22)







25 26 27 28 29 30 3231

7 8 9 10 11 12 13 14 15 16 17

18 19 20 21

22 23 24

0


PV

Windturbine

Smalldiesel

turbine



Pollutantemission


turbineNO119909

646 43314 0 0CO2 1070 2320373 0 0CO 155 23204 0 0SO2 993 04641 0 0


SO2 NO119909

CO2 CO075 100 0002875 0125






24222018161412108642Time period (h)

0

100

200

300

400

500

600

DG

out

put (

kW)




0

50

100

150

200

250Li

ne lo

ss (k

W)


24222018161412108642Time period (h)






tive p

ower

of p

hoto

volta

ic g

ener

atio

n (k

W)

42222018161412108642Time period (h)

0

100

200

300

400

500

600

700



24222018161412108642Time period (h)

0

100

200

300

400

500

600

700

Activ

e pow

er o

f win

d (k

W)





24222018161412108642Time period (h)

30

31

32

33

34

35

36

37

38

39

40

Save

d co

st of

coal

usin

g D

Gs (

$)


24222018161412108642Time period (h)

0

200

400

600

800

1000

1200

1400

1600

Save

d co

st of

envi

ronm

enta

l pol

lutio

n fin

es ($

)



6 Conclusion



242220181614121086420

10

20

30

40

50

60

70

80

90

100

Time period (h)

Pow

er p

urch

ase c

ost (

$)




Max1198652(119909)

Min1198653(119909)





References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014





119909 = [119875DG1 119875DG119873119866 1198811198711 119881119871119873119871 1198761198631198661 119876DG119873119866]119879

(13)




min119891119894(119909) 119894 = 1 2 119898 119909 isin 120594 (14)




2(denoted by

1199091≺ 1199092) if and only if

forall119894 isin 1 2 119898 119891119894(1199091) le 119891119894(1199092)

and exist119895 isin 1 2 119898 119891119895(1199091) lt 119891119895(1199092)

(15)




1(119909) 1198912(119909)







119894=

(1199091198941 119909



119909119866

119894119896= 119909119896min


119894 isin [1119873119901] 119896 isin [1 119863]

(16)



119896minand 119909

119896max





119872(119894) = 119877 (119894) + 119863 (119894) (17)

where 119877(119894) = sum119895isin119875119900119901+119873119863119878119890119905119895≻119894




119909119895is dominated)119863(119894) = 1(120590

119896

119894+ 2) 119896 = radic119873 +119873 120590119896





119875119898


119898determine




119888is extremely






Find the Pareto set


Perform mutation

Perform crossover



Stop

Yes

No



119875119888=

1198751198881(119872avg minus119872

1015840) + 1198751198882(1198721015840minus119872min)


1015840lt 119872avg

1198751198882(119872max minus119872

1015840) + 1198751198883(1198721015840minus119872avg)


119875119898=





(18)


1198881 1198751198882 1198751198883 1198751198981 1198751198982 1198751198983

isin [0 1]1198751198881gt 1198751198882gt 1198751198883 1198751198981gt 1198751198982gt 1198751198983




120591i =

1 119865119894= 119865

min119894

119865max119894

minus 119865119894

119865max119894

minus 119865min119894

119865min119894

lt 119865119894lt 119865

max119894

0 119865119894le 119865

max119894

(19)

where 119865max119894








120591119896=

sum1198730

119894=1120591119894

119896

sum119906

119895=1sum1198730

119894=1120591119894

119895

(20)








119896 as shown in (20)



119901 (119879119896+1) =

1 (119872119896+1

gt 119872119896)

exp(minus119872119896+1

minus119872119896

119879119896+1

) (119872119896+1

le 119872119896)

119879119896+1

= 120572119879119896

(21)

where 119872119896+1










1(119909) 1198652(119909) 119865

119896(119909)

will be computed









119888and 119875


1198881= 04

1198751198882= 03 and 119875

1198883= 02 and 119875

1198981= 02 119875

1198982= 01 and

1198751198983







new archive set


size of archive set



Yes

Yes

No

No

Performmutation

Performcrossover


condition




1198651(119909) 1198652(119909) 119865119896(119909)




119862119877=

119899

sum

119894=1

119891 (119875119905

119894) 119862119894 (22)







25 26 27 28 29 30 3231

7 8 9 10 11 12 13 14 15 16 17

18 19 20 21

22 23 24

0


PV

Windturbine

Smalldiesel

turbine



Pollutantemission


turbineNO119909

646 43314 0 0CO2 1070 2320373 0 0CO 155 23204 0 0SO2 993 04641 0 0


SO2 NO119909

CO2 CO075 100 0002875 0125






24222018161412108642Time period (h)

0

100

200

300

400

500

600

DG

out

put (

kW)




0

50

100

150

200

250Li

ne lo

ss (k

W)


24222018161412108642Time period (h)






tive p

ower

of p

hoto

volta

ic g

ener

atio

n (k

W)

42222018161412108642Time period (h)

0

100

200

300

400

500

600

700



24222018161412108642Time period (h)

0

100

200

300

400

500

600

700

Activ

e pow

er o

f win

d (k

W)





24222018161412108642Time period (h)

30

31

32

33

34

35

36

37

38

39

40

Save

d co

st of

coal

usin

g D

Gs (

$)


24222018161412108642Time period (h)

0

200

400

600

800

1000

1200

1400

1600

Save

d co

st of

envi

ronm

enta

l pol

lutio

n fin

es ($

)



6 Conclusion



242220181614121086420

10

20

30

40

50

60

70

80

90

100

Time period (h)

Pow

er p

urch

ase c

ost (

$)




Max1198652(119909)

Min1198653(119909)





References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014






Find the Pareto set


Perform mutation

Perform crossover



Stop

Yes

No



119875119888=

1198751198881(119872avg minus119872

1015840) + 1198751198882(1198721015840minus119872min)


1015840lt 119872avg

1198751198882(119872max minus119872

1015840) + 1198751198883(1198721015840minus119872avg)


119875119898=





(18)


1198881 1198751198882 1198751198883 1198751198981 1198751198982 1198751198983

isin [0 1]1198751198881gt 1198751198882gt 1198751198883 1198751198981gt 1198751198982gt 1198751198983




120591i =

1 119865119894= 119865

min119894

119865max119894

minus 119865119894

119865max119894

minus 119865min119894

119865min119894

lt 119865119894lt 119865

max119894

0 119865119894le 119865

max119894

(19)

where 119865max119894








120591119896=

sum1198730

119894=1120591119894

119896

sum119906

119895=1sum1198730

119894=1120591119894

119895

(20)








119896 as shown in (20)



119901 (119879119896+1) =

1 (119872119896+1

gt 119872119896)

exp(minus119872119896+1

minus119872119896

119879119896+1

) (119872119896+1

le 119872119896)

119879119896+1

= 120572119879119896

(21)

where 119872119896+1










1(119909) 1198652(119909) 119865

119896(119909)

will be computed









119888and 119875


1198881= 04

1198751198882= 03 and 119875

1198883= 02 and 119875

1198981= 02 119875

1198982= 01 and

1198751198983







new archive set


size of archive set



Yes

Yes

No

No

Performmutation

Performcrossover


condition




1198651(119909) 1198652(119909) 119865119896(119909)




119862119877=

119899

sum

119894=1

119891 (119875119905

119894) 119862119894 (22)







25 26 27 28 29 30 3231

7 8 9 10 11 12 13 14 15 16 17

18 19 20 21

22 23 24

0


PV

Windturbine

Smalldiesel

turbine



Pollutantemission


turbineNO119909

646 43314 0 0CO2 1070 2320373 0 0CO 155 23204 0 0SO2 993 04641 0 0


SO2 NO119909

CO2 CO075 100 0002875 0125






24222018161412108642Time period (h)

0

100

200

300

400

500

600

DG

out

put (

kW)




0

50

100

150

200

250Li

ne lo

ss (k

W)


24222018161412108642Time period (h)






tive p

ower

of p

hoto

volta

ic g

ener

atio

n (k

W)

42222018161412108642Time period (h)

0

100

200

300

400

500

600

700



24222018161412108642Time period (h)

0

100

200

300

400

500

600

700

Activ

e pow

er o

f win

d (k

W)





24222018161412108642Time period (h)

30

31

32

33

34

35

36

37

38

39

40

Save

d co

st of

coal

usin

g D

Gs (

$)


24222018161412108642Time period (h)

0

200

400

600

800

1000

1200

1400

1600

Save

d co

st of

envi

ronm

enta

l pol

lutio

n fin

es ($

)



6 Conclusion



242220181614121086420

10

20

30

40

50

60

70

80

90

100

Time period (h)

Pow

er p

urch

ase c

ost (

$)




Max1198652(119909)

Min1198653(119909)





References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014






120591119896=

sum1198730

119894=1120591119894

119896

sum119906

119895=1sum1198730

119894=1120591119894

119895

(20)








119896 as shown in (20)



119901 (119879119896+1) =

1 (119872119896+1

gt 119872119896)

exp(minus119872119896+1

minus119872119896

119879119896+1

) (119872119896+1

le 119872119896)

119879119896+1

= 120572119879119896

(21)

where 119872119896+1










1(119909) 1198652(119909) 119865

119896(119909)

will be computed









119888and 119875


1198881= 04

1198751198882= 03 and 119875

1198883= 02 and 119875

1198981= 02 119875

1198982= 01 and

1198751198983







new archive set


size of archive set



Yes

Yes

No

No

Performmutation

Performcrossover


condition




1198651(119909) 1198652(119909) 119865119896(119909)




119862119877=

119899

sum

119894=1

119891 (119875119905

119894) 119862119894 (22)







25 26 27 28 29 30 3231

7 8 9 10 11 12 13 14 15 16 17

18 19 20 21

22 23 24

0


PV

Windturbine

Smalldiesel

turbine



Pollutantemission


turbineNO119909

646 43314 0 0CO2 1070 2320373 0 0CO 155 23204 0 0SO2 993 04641 0 0


SO2 NO119909

CO2 CO075 100 0002875 0125






24222018161412108642Time period (h)

0

100

200

300

400

500

600

DG

out

put (

kW)




0

50

100

150

200

250Li

ne lo

ss (k

W)


24222018161412108642Time period (h)






tive p

ower

of p

hoto

volta

ic g

ener

atio

n (k

W)

42222018161412108642Time period (h)

0

100

200

300

400

500

600

700



24222018161412108642Time period (h)

0

100

200

300

400

500

600

700

Activ

e pow

er o

f win

d (k

W)





24222018161412108642Time period (h)

30

31

32

33

34

35

36

37

38

39

40

Save

d co

st of

coal

usin

g D

Gs (

$)


24222018161412108642Time period (h)

0

200

400

600

800

1000

1200

1400

1600

Save

d co

st of

envi

ronm

enta

l pol

lutio

n fin

es ($

)



6 Conclusion



242220181614121086420

10

20

30

40

50

60

70

80

90

100

Time period (h)

Pow

er p

urch

ase c

ost (

$)




Max1198652(119909)

Min1198653(119909)





References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014







new archive set


size of archive set



Yes

Yes

No

No

Performmutation

Performcrossover


condition




1198651(119909) 1198652(119909) 119865119896(119909)




119862119877=

119899

sum

119894=1

119891 (119875119905

119894) 119862119894 (22)







25 26 27 28 29 30 3231

7 8 9 10 11 12 13 14 15 16 17

18 19 20 21

22 23 24

0


PV

Windturbine

Smalldiesel

turbine



Pollutantemission


turbineNO119909

646 43314 0 0CO2 1070 2320373 0 0CO 155 23204 0 0SO2 993 04641 0 0


SO2 NO119909

CO2 CO075 100 0002875 0125






24222018161412108642Time period (h)

0

100

200

300

400

500

600

DG

out

put (

kW)




0

50

100

150

200

250Li

ne lo

ss (k

W)


24222018161412108642Time period (h)






tive p

ower

of p

hoto

volta

ic g

ener

atio

n (k

W)

42222018161412108642Time period (h)

0

100

200

300

400

500

600

700



24222018161412108642Time period (h)

0

100

200

300

400

500

600

700

Activ

e pow

er o

f win

d (k

W)





24222018161412108642Time period (h)

30

31

32

33

34

35

36

37

38

39

40

Save

d co

st of

coal

usin

g D

Gs (

$)


24222018161412108642Time period (h)

0

200

400

600

800

1000

1200

1400

1600

Save

d co

st of

envi

ronm

enta

l pol

lutio

n fin

es ($

)



6 Conclusion



242220181614121086420

10

20

30

40

50

60

70

80

90

100

Time period (h)

Pow

er p

urch

ase c

ost (

$)




Max1198652(119909)

Min1198653(119909)





References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014





25 26 27 28 29 30 3231

7 8 9 10 11 12 13 14 15 16 17

18 19 20 21

22 23 24

0


PV

Windturbine

Smalldiesel

turbine



Pollutantemission


turbineNO119909

646 43314 0 0CO2 1070 2320373 0 0CO 155 23204 0 0SO2 993 04641 0 0


SO2 NO119909

CO2 CO075 100 0002875 0125






24222018161412108642Time period (h)

0

100

200

300

400

500

600

DG

out

put (

kW)




0

50

100

150

200

250Li

ne lo

ss (k

W)


24222018161412108642Time period (h)






tive p

ower

of p

hoto

volta

ic g

ener

atio

n (k

W)

42222018161412108642Time period (h)

0

100

200

300

400

500

600

700



24222018161412108642Time period (h)

0

100

200

300

400

500

600

700

Activ

e pow

er o

f win

d (k

W)





24222018161412108642Time period (h)

30

31

32

33

34

35

36

37

38

39

40

Save

d co

st of

coal

usin

g D

Gs (

$)


24222018161412108642Time period (h)

0

200

400

600

800

1000

1200

1400

1600

Save

d co

st of

envi

ronm

enta

l pol

lutio

n fin

es ($

)



6 Conclusion



242220181614121086420

10

20

30

40

50

60

70

80

90

100

Time period (h)

Pow

er p

urch

ase c

ost (

$)




Max1198652(119909)

Min1198653(119909)





References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014




tive p

ower

of p

hoto

volta

ic g

ener

atio

n (k

W)

42222018161412108642Time period (h)

0

100

200

300

400

500

600

700



24222018161412108642Time period (h)

0

100

200

300

400

500

600

700

Activ

e pow

er o

f win

d (k

W)





24222018161412108642Time period (h)

30

31

32

33

34

35

36

37

38

39

40

Save

d co

st of

coal

usin

g D

Gs (

$)


24222018161412108642Time period (h)

0

200

400

600

800

1000

1200

1400

1600

Save

d co

st of

envi

ronm

enta

l pol

lutio

n fin

es ($

)



6 Conclusion



242220181614121086420

10

20

30

40

50

60

70

80

90

100

Time period (h)

Pow

er p

urch

ase c

ost (

$)




Max1198652(119909)

Min1198653(119909)





References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014




242220181614121086420

10

20

30

40

50

60

70

80

90

100

Time period (h)

Pow

er p

urch

ase c

ost (

$)




Max1198652(119909)

Min1198653(119909)





References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014













Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014










Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014



Documents

A New DG Multiobjective Optimization Method Based on an Improved Evolutionary Algorithm · 2014-04-23 · A New DG Multiobjective Optimization Method Based on ... active power dispatch