Research Article A Hybrid Algorithm of Particle Swarm ...downloads.hindawi.com › journals › mpe...

Research ArticleA Hybrid Algorithm of Particle Swarm Optimization and TabuSearch for Distribution Network Reconfiguration

Sidun Fang12 and Xiaochen Zhang12

1School of Electronic Information and Electrical Engineering Shanghai Jiao Tong University Shanghai 200240 China2Department of Computer Science and Electrical Engineering Georgia Institute of Technology Atlanta GA 30314 USA

Correspondence should be addressed to Sidun Fang fangstonsjtueducn

Received 15 January 2016 Accepted 19 July 2016

Academic Editor Mauro Pontani

Copyright copy 2016 S Fang and X Zhang This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

This paper deals with the distribution network reconfiguration problem A hybrid algorithm of particle swarm optimization (PSO)and tabu search (TS) is proposed as the searching algorithm The new algorithm shares the advantages of PSO and TS which hasa fast computation speed and a strong ability to avoid local optimal solution After a thorough comparison network random key(NRK) is introduced as the corresponding coding strategy among various tree representation strategies NRK could completelyavoid the generation of infeasible solutions during the searching process and has a good locality property which allows the newhybrid algorithm to perform to its fullest potential The proposed algorithm has been validated through an IEEE 33 bus test caseCompared with other algorithms the proposed method is both accurate and computationally efficient Furthermore a test to solveanother problem also proves the robustness of the proposed algorithm for a different problem

1 Introduction

Distribution system is usually designed with loops while run-ning in a radial structure Distribution network reconfigura-tion (DNR) is a process of altering the topological structureof distribution feeders by changing the openclosed status ofthe sectionalizing and tie switches [1] DNR is not limitedto fault isolation from time to time network reconfigurationis performed to achieve various goals such as system lossreduction overloads relieving [2] load aggregation [3 4] andsystem reliability improvement [5]

The performance and efficiency of any DNR algorithmlargely rely on a wise combination of a smart topological cod-ing strategy and an efficient searching algorithm As a non-deterministic polynomial hard (NP-hard) problem DNR hasbeen heavily studied with various searching algorithms fromstep-by-step heuristics such as branch-exchangemethod [6]to metaheuristics based algorithms such as tabu search (TS)[4] simulated annealing [7] genetic algorithm (GA) [8ndash10] and particle swarm optimization (PSO) [11] Apart fromthe choice of searching algorithms the distribution networkrepresentation or coding strategy is equally important due

to the topological nature of DNR Various coding strategieshave been studied in DNR problem including binary stringrepresentation [5] and Prufer number representation [3 12]

After a brief comparison of different coding strategiesand existing searching algorithms this paper proposes anew DNR algorithm The new method adopts a hybridoptimization of PSO and TS as the searching algorithm andnetwork random keys (NRK) as the corresponding codingstrategy To boost up the overall searching efficiency a directmethod for distribution system power flow analysis [13] isintroduced which has been proved to be both robust andtime-efficient

In recent years PSO has been successfully applied tosolving different kinds of problems ranging frommultimodaland topological mathematical problems [14 15] to aerospace[11 16ndash20] and chemical engineering [21 22] It is famous forits easy realization and fast convergence while suffering fromthe possibility of early convergence to local optimums Inthe proposed hybrid algorithm whenever early convergenceoccurs the original particle swarm would be separated intothree groups of swarms Swarm 1 continues performingthe basic PSO algorithm swarm 2 is replaced with newly

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 7410293 7 pageshttpdxdoiorg10115520167410293

2 Mathematical Problems in Engineering

generated random particles and swarm 3 begins to performTS on each particle With the integration of TS the hybridPSO algorithm could effectively avoid local optimum byaccepting worse solution under certain condition during thesearching process

NRK which is originally used for GAs in this paper hasbeen applied to PSO algorithm As a topological coding strat-egy NRK could completely avoid the possibility of generatingunfeasible solutionswhen using heuristic algorithms in graphoptimization problems It also transforms the original dis-crete DNR problem into a continuous optimization problemWhen applied inGAsNRK is nomore than a coding strategywhich possesses little physical meaning during the solutionsearching process However when used in PSO the codingstrategy has a physical meaning The value of the ldquokeyrdquo inthe NRK can be interpreted as an importance index for eachbranch in the graph The whole searching process could beinterpreted as a process of adjusting the importance index ofeach branch and choosing the most important branches toform the optimal tree structure

The remainder of this paper is structured as followsIn Section 2 the DNR problem is formulated as an opti-

mization problem In Section 3 different network topologyrepresentation schemes are discussed and NRK is intro-duced In Section 4 the hybrid algorithm of PSO and TSis proposed and explained in detail In Section 5 the newalgorithm is tested on an IEEE 33 bus system with numericalresults The conclusion is drawn in Section 6

2 Problem Representation

DNR is originally used in planned outages for maintenancepurpose or fault isolation to restore service A Merlinand H Back [14] were the first to come to the idea thatreconfiguration may lead to a system total loss reduction andthey tried to search for such an optimal configuration usingthe branch-and-bound method Since then loss reductionhas been considered as a common objective for the study ofnew DNR optimization algorithms

A noticeable characteristic of DNR is the repeated anal-ysis of power flow during the solution searching processIn order to improve searching efficiency several refinedor approximate algorithms for power flow analysis havebeen studied such as decoupled method [15] hashing tablemethod [16] and perturbation method [2] A direct method[13] is adopted in this paper which has been proved to behighly efficient in distribution network power flow analysis

Another characteristic of DNR is the topological con-straints whichmeans any feasible solution of a DNR problemshould represent a tree structure with every node beingconnected Configuration space is the set of allowed systemconfigurations over which the optimal system configurationis to be searched for [7] In DNR only solutions that belongto the configuration space are considered feasible

Assume that a distribution network has 119899 branches ADNR problem for system losses reduction can be formulatedas

min 119891 =119899

sum119887=1

1199091198871199031198871198752119887 + 119876

2119887

1198812119887

+ 120572119875 (119881119887) + 120573119875 (119868119887) (1)

where 119909 is an 119899-dimensional vector If branch 119887 is closed119909119887 = 1 otherwise 119909119887 = 0 119875119887 and 119876119887 represent the active andreactive power flow on branch 119887 120572 and 120573 are penalty factorswhile 119875(119881119887) and 119875(119868119887) are penalty functions for node voltageconstraint and branch current constraint

3 Network Topology Representation

The process of searching the optimal DNR solution involvesthe graph theory of optimal spanning tree Let graph 119866(119881 119864)represent the topology of a distribution network where 119881stands for vertices and 119864 stands for edges Each potentialsolution is a spanning tree of 119866 All spanning trees ofthe graph 119866 make up the configuration space A goodnetwork topology representation strategy should have fourcharacteristics

(1) Being Easy to Encode and Decode A less complicatedcoding strategy would cost less time to encode anddecode thus leading to a boost in computationalefficiency

(2) Being Compatible with Other Optimization Algo-rithms Many metaheuristic algorithms have theirown limitations in dealing with different types ofoptimization problems For example GAs require abinary string representation and PSO requires con-tinuous variables A good coding strategy should becompatible with corresponding searching algorithms

(3) Avoiding Infeasible Solutions Topological constraint isone of the thorniest issues in DNR especially when itcomes to the utilization of metaheuristic algorithmsWhenever an infeasible solution is generated theoriginal searching process will be interrupted Agood coding strategy should effectively rule out thepossibility of generating infeasible solutions whichwould greatly improve the computational efficiencyand avoid the tedious topological checking process

(4) Having a Good Locality Property A good localityproperty means that the objective function valueis relatively continuous and smooth rather thanirregular jumps within a local area in the searchingspace Most metaheuristic algorithms determine thebest searching direction based on current objectivefunction values Then the algorithms will lead thesearching process towards the most promising direc-tion In other words a coding strategy with bad local-ity properties will greatly restrict the effectivenessof the searching algorithm The configuration spacegenerated by a good coding strategy should alwayskeep a high locality

There are many different ways to represent the distribu-tion network topology and each of them has its merits andflaws

Binary string representation is the most intuitive andstraightforward way to represent the network topology byassigning a binary string 119909 The dimension of 119909 is the totalnumber of switches The elements in 119909 are set to be 0 or

Mathematical Problems in Engineering 3

1 representing the open and closed status of each switchHowever binary string representation is usually blamedfor the high probability of generating infeasible solutionswhen applied by many searching algorithms such as SAor PSO Genetic operators such as crossover or mutationalmost always generate infeasible solution which forces thealgorithm to stop

In order to reduce the probability of generating infeasiblesolutions homeomorphism [12] and fundamental loop [17]representation method are widely adopted The graph theoryof homeomorphism simplifies the original graph by smooth-ing out unnecessary vertices from the original graph Afterthe simplification each branch in the new graph representsa group of branches in the original graph According to thegraph theory one and only one branch could be opened ineach branch group in order to form a tree structure Similarlyfundamental loop representation avoids infeasible solutionsby introducing fundamental loop tables Only one branchshould be opened in each fundamental loopThese twometh-ods help to reduce the probability of generating infeasiblesolutions and keep the searching process from interruptionHowever none of the methods above could completely avoidinfeasible solutions and additional checking rules are stillnecessary

Random key (RK) is an efficient method for encodingand scheduling problems Rothlauf et al [9] proposes a treerepresentation for GAs using RK by the name of networkrandom keys Queiroz and Lyra [3] are the first to introducethe combination of NRK and GAs in the DNR problem

Taking a 5-node system as an example see Figure 1 TheNRK coding and decoding process goes as follows

Step 1 Generate a ldquokeyrdquo vector with the dimension of 6denoted as119909 And each element in119909 stands for a branch in thenetwork The value of each key is a random number isin [0 1]

Step 2 Rank the elements in ldquokeyrdquo vector according to theirvalue in descending order denoted as 1199091015840

Step 3 Let 119865 stand for a branch set which contains thebranches chosen to form the tree Starting from the firstelement in 1199091015840 add one branch into 119865 at a time Whenever aloop is formed abandon the latest added branch and continueadding branches with the next element until all 4 brancheshave been chosen

The ordering of the branches in 1199091015840 would change accord-ing to the variation of values in the ldquokeyrdquo vector Conse-quently the final tree structure would also change Branchwith a higher ldquokeyrdquo value will rank in the front and is morelikely to be chosen to form the final tree structure Similarlybranches with lower ldquokeyrdquo values are more unlikely to bechosen In other words a higher ldquokeyrdquo value means a higherimportance of that branch

To begin with as a tree representation strategy NRKcould guarantee that any ldquokeyrdquo vector could generate one andonly one feasible solution making the topological checkingprocess completely unnecessary Moreover by using RKswhich are continuous variables NRK transforms the originaldiscreteDNRproblem into a continuous problemAs a result

algorithms such as basic PSO could be applied to NRKdirectly Finally since the construction of the tree is basedon the relative order of the branches the locality of NRK isvery high which is a good prerequisite for other optimizationalgorithms to perform to their fullest potential

4 Hybrid Algorithm

Various algorithms have been implemented inDNRproblemincluding GAs TS and SA In this paper a hybrid algorithmof PSO and TS has been introduced The new algorithmshares the advantages of both PSO and TS

Tabu search is a local search algorithm that can be usedfor solving combinatorial optimization problems It usessome memory structures such as tabu list or frequency listto force the searching process to cover new searching areaand prevent early convergence to the local optimal solutionThe advantage of TS lies in a strong local searching abilityand the ability to jump out of local optimum In [4] tabusearch is successfully applied to DNR with some necessarymodificationsMeanwhile the disadvantage of TS is its strongdependence on a proper initial solution and relatively lowsearching efficiency compared with other metaheuristics

PSO is a stochastic optimization technique developedby Kennedy and Eberhart [23] The algorithm introduces anumber of particles to form a swarm Each particle travels inthe searching space to search for the global optimum usingthe experiences of other particles [16] It has the featuresof parallel computing and high computational efficiencyHowever PSO also suffers from the probability of earlyconvergence With improper parameters PSO may easilyfall into local optimum Since the original PSO is designedfor continuous variable optimization binary particle swarmoptimization (BPSO) a modified PSO algorithm is intro-duced to deal with DNR problem [11]

In the basic PSO algorithm particle updates its velocityand position with the following equations

V119894119889 = V119894119889 + 11988811199031 (119901119894119889 minus 119909119894119889) + 11988821199032 (119901119892119889 minus 119909119894119889)

119909119894119889 = 119909119894119889 + V119894119889(2)

where119909119894119889 and V119894119889 stand for the position and velocity of the119889thdimension of particle 119894 119901119894119889 and 119901119892119889 stand for the particle bestand global best position 1198881 and 1198882 are nonnegative constantsand 1199031 and 1199032 are two random numbers isin [0 1]

In order to overcome the various shortcomings of PSOand TS a hybrid algorithm is proposed On the one handPSO could greatly increase the searching efficiency on theother hand TS would help to avoid local optimal solu-tion Moreover TSrsquos strong local searching ability may evenincrease PSOrsquos accuracy when the true global optimum is notfar away The flow chart of the new algorithm is shown inFigure 2 In the hybrid algorithm the original particle swarmsare partitioned into three swarms swarm 1 swarm 2 andswarm 3 At the beginning the new algorithm performs justas the basic PSO algorithm When the algorithm stoppedupdating 119901119894119889 and 119901119892119889 within certain period of time whichmight be a sign of early convergence the new algorithm


1 2 3 4

5Original network

1 2 3 4

5

b1 b2 b3

b1 b2 b3

b4b5

b5

b6

Decoded network

Step 1

Branch 1 2 3 4 5 6

1 02 09 01 07 04

Step 2

Branch 1 2 3 4 5 6

1 09 07 04 02 01

Step 3

F 1 3 5 2

x(key)

x998400

Figure 1 NRK coding and decoding process

Start

Initialization

Basic PSO process

pBest and gBest stopped updating

No

Swarm 1 Swarm 2 Swarm 3

Yes

Reset position and velocity

Basic PSO process Basic TS process

Output PSO result

Yes

Stop criteriaStop criteria

Stop criteria

Output TS result

Yes

Choose the better result as new global best

Output global best solution

End

Yes

No

No No

Figure 2 Hybrid algorithm flow chart

would begin to perform differently Swarm 1 would continueperforming basic PSO algorithm using the equation aboveSwarm 2 would be reset with new random positions andvelocities Swarm 3 would perform TS based on the currentoptimal solution

5 Case Studies

The proposed hybrid algorithm was realized using MATLABand tested on the IEEE 33-node system [2] see Figure 3 Thesystem consists of 37 branches and 33 nodes The numberof fundamental loops is 37 minus 33 + 1 = 5 The originalsystem losses are 20267 kW with branches 33 34 35 36

and 37 opened Under the optimal network configuration thesystem losses should be 12872 kW with branches 7 9 14 31and 37 opened

In order to compare the performances between theproposed hybrid algorithm and other basic algorithms threedifference cases are studied

In case 1 binary string representation is adopted as thetree representation method BPSO is performed to deal withthe discrete optimization problem The logical transforma-tion function of BPSO is established as the sigmoid functionin [18] Both the BPSO parameters 1198881 and 1198882 are set tobe 4 As binary string representation allows the generationof infeasible solutions whenever an infeasible solution is


0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

b1

b2

b3

b4

b5

b6

b7

b8

b9

b10

b11

b12

b13

b14

b15

b16b17

b22

b23

b24

b25

b26

b27

b28

b29

b30

b31

b32

b36

b37

b34

22

23

24

18

19

20

21

31

32

25

26

27

28

29

30

1617

b18

b19

b20

b21

b35

b33

Figure 3 IEEE 33 bus case

generated a feasible particle is picked to replace the infeasibleone

In case 2 NRK is chosen as the tree representationmethod As continuous variables ldquokeysrdquo enable basic PSOalgorithm to perform directly into DNR problem The basicPSO parameters are set as follows 1198881 = 1198882 = 4 Each ldquokeyrdquovalue in NRK vector is limited within [0 1]

In case 3 the new hybrid algorithm of PSO and TS isadopted with NRK being the tree representation methodThe PSO parameters are the same as case 2 The length oftabu list is 3 the length of frequency list is 10 The frequencypenalty factor is set to be 12The detailed aspiration criterionis explained in [19]

To get a population large enough to study the differencesamong the three cases each case is performed 100 times Theparticle numbers in all cases are set to 10 and the maximumiteration time is set as 50 The average computation timeand losses achieved by three cases in 100 trails are shownin Table 1 The standard deviation (STDV) of losses enablesevaluating the algorithmsrsquo robustness

By comparing case 1 with case 2 we can see that thebad locality property of binary string greatly limited theperformance of PSO and the high probability of generating ofinfeasible particles significantly slowed the entire searchingprocess By comparing case 2 with case 3 we can see thatthe integration of tabu search could effectively increase thelocal search ability and help PSO jump out of local optimalsolution

Table 1 System losses and computation time

Case ID Optimal configuration (kW) Mean computationtime (seconds)Mean of losses STDV

Case 1 13503 21 2936Case 2 13419 291 1664Case 3 13033 223 1712

Table 2 Histogram of system losses

Case ID Percentage among 100 trialsLosses = 12872 Losses lt 13400 Losses lt 13900

Case 1 2 11 69Case 2 26 41 90Case 3 89 86 97The system losses for the true optimal solution are 12872 kW

Table 3 The results of PSO

Parameter Percentage among 100 trials1198882 = 2 1198882 = 3 1198882 = 4 1198882 = 5

1198881 = 2 17 16 18 201198881 = 3 19 22 22 231198881 = 4 20 23 26 251198881 = 5 21 23 26 26

Table 2 shows the percentage of solutions with a systemloss less than certain criteria within 100 trials for each case Itfurther illustrates that both NRK and the hybrid algorithmsignificantly increase the probability of finding the globaloptimal solution 69 out of 100 trails have found the globaloptimal solution in the new algorithm which is much higherthan other two cases

Figure 4 shows the global best solution curve during 50iterations for all cases In the hybrid algorithm when pBestand gBest stopped updating for a period of time a localoptimal solution is found as seen in Figure 4 ldquoHybrid-PSOrdquoThen the original swarm process is forced to stop at the41th iteration Then the original particle swarm is separatedinto three swarms ldquoHybrid-TSrdquo shows how tabu search helpa particle in swarm 3 jump out of the local optimum andincrease the possibility of finding the global optimal networkconfiguration

To test the parametric robustness of the proposed algo-rithm the percentages to get global best among 100 trialsof PSO and hybrid algorithm under different parametercombinations are shown in Tables 3 and 4 respectively

From the results of Tables 3 and 4 the hybrid algorithmhas similar parameter selection region compared to the PSOThe best parameters (1198881 and 1198882) of two algorithms are both 4or 5 That means the hybrid process has no evident impactson the parameter selection The main reason is that the PSOprocess and TS process are independent of each otherThe TSprocess is activated when the solving process is caught into


BPSO

PSO

Hybrid-PSO

Hybrid-TS

50 4515 252010 30 40 5035

Iteration number

125

130

135

140

145

150

155

160

Net

wo

rk l

oss

es (

MW

)

Figure 4 Global best solution during the searching process

Table 4 The results of proposed method


1198881 = 2 73 75 79 821198881 = 3 78 80 83 851198881 = 4 80 85 89 891198881 = 5 81 87 90 91

Table 5 The comparison between PSO and proposed hybridalgorithm

Method Percentage among 100 trialsLosses = 11216 Losses lt 115 Losses lt 118

PSO 5 14 55Hybrid 31 67 100

local best Thus the parameter tuning problems do not existin the proposed hybrid process

To prove the robustness of the proposed hybrid algorithmto different problems PSO and the proposed hybrid algo-rithm are both tested in a reactive power dispatchmodel [24]The test system is the IEEE 118 bus system The algorithmicparameters are the same with case 1 and case 2 The resultsare shown in Table 5

From Table 5 the proposed hybrid algorithm also hassuperior characteristics compared to PSO which shows thatproposed method is able to enhance the global searchingability of PSO for different models

6 Conclusion

PSO is a very promising algorithm to large scale optimizationproblems as DNR problem The main contribution of thispaper is presenting a hybrid PSO searching algorithm andintroduces NRK as the tree representation strategy for thenew algorithm

The numerical results drawn from the test system validatethe effectiveness and efficiency of both the hybrid algorithmand the introduction of the new coding strategy

The main drawback of the algorithm lies in the situationthat PSO algorithm converges to a local optimal solutionwhich is located very far away from the true global optimalsolution tabu search may not have the ability to jump out ofsuch a very deep local optimal solution Another drawbackis that a subtle change of ldquokeyrdquo value does not necessarilylead to a change in the tree structure formulation unlessthe change is big enough to change the importance sequenceof the branches In other words NRK is not very sensitiveto PSO algorithm and that is also the very reason whyalgorithmswith strong local search ability like tabu search areindispensable

Further analysis shows that the hybrid process has no evi-dent influences on the parameter selection and the proposedmethod is also able to enhance the global searching ability ofPSO in different problems

Competing Interests

The authors declare that they have no competing interests

References

[1] S Civanlar J J Grainger H Yin and S S H Lee ldquoDistributionfeeder reconfiguration for loss reductionrdquo IEEE Transactions onPower Delivery vol 3 no 3 pp 1217ndash1223 1988

[2] M E Baran and F F Wu ldquoNetwork reconfiguration in distri-bution systems for loss reduction and load balancingrdquo IEEETransactions on Power Delivery vol 4 no 2 pp 1401ndash1407 1992

[3] L M O Queiroz and C Lyra ldquoAdaptive hybrid geneticalgorithm for technical loss reduction in distribution networksunder variable demandsrdquo IEEE Transactions on Power Systemsvol 24 no 1 pp 445ndash453 2009

[4] S F Mekhamer A Y Abdelaziz F M Mohammed andM A L Badr ldquoA new intelligent optimization technique fordistribution systems reconfigurationrdquo in Proceedings of the2008 12th International Middle East Power System Conference(MEPCON rsquo08) pp 397ndash401 IEEE Aswan Egypt March 2008

[5] B Amanulla S Chakrabarti and S N Singh ldquoReconfigurationof power distribution systems considering reliability and powerlossrdquo IEEE Transactions on Power Delivery vol 27 no 2 pp918ndash926 2012

[6] D W Ross M Carson and A I Cohen ldquoDevelopment ofadvanced method for planning electric energy distributionsystemsrdquo USDOE Report ET-8-c-03-1845 1980

[7] H D Chiang and R Jean-Jumeau ldquoOptimal network recon-figurations in distribution systems II Solution algorithms andnumerical resultsrdquo IEEE Transactions on Power Delivery vol 5no 3 pp 1568ndash1574 1990

[8] K Nara A Shiose M Kitagawa and T Ishihara ldquoImplementa-tion of genetic algorithm for distribution systems lossminimumre-configurationrdquo IEEE Transactions on Power Systems vol 7no 3 pp 1044ndash1051 1992

[9] F Rothlauf D E Goldberg and A Heinzl ldquoNetwork randomkeysmdasha tree representation scheme for genetic and evolution-ary algorithmsrdquoEvolutionary Computation vol 10 no 1 pp 75ndash97 2002


[10] E M Carreno N Moreira and R Romero ldquoDistribution net-work reconfiguration using an efficient evolutionary algorithmrdquoin Proceedings of the IEEE Power Engineering Society GeneralMeeting (PES rsquo07) pp 1ndash6 Tampa Fla USA June 2007

[11] P C Fourie and A A Groenwold ldquoThe particle swarm opti-mization algorithm in size and shape optimizationrdquo Structuraland Multidisciplinary Optimization vol 23 no 4 pp 259ndash2672002

[12] J Kennedy and R C Eberhart ldquoDiscrete binary version ofthe particle swarm algorithmrdquo in Proceedings of the 1997 IEEEInternational Conference on SystemsMan and Cybernetics Part1 (of 5) pp 4104ndash4108 October 1997

[13] G R Raidl and B A Julstrom ldquoEdge sets an effectiveevolutionary coding of spanning treesrdquo IEEE Transactions onEvolutionary Computation vol 7 no 3 pp 225ndash239 2003

[14] K Parsopoulos and M N Vrahatis ldquoOn the computation of allglobal minimizers through particle swarm optimizationrdquo IEEETransactions on Evolutionary Computation vol 8 no 3 pp 211ndash224 2004

[15] P C Fourie and A A Groenwold ldquoParticle swarms in topologyoptimizationrdquo in Proceedings of the 4th World Congress ofStructural and Multidisciplinary Optimization pp 1771ndash1776Liaoning Electronic Press 2001

[16] M Pontani C Martin and B A Conway ldquoNew numeri-cal methods for determining periodic orbits in the circularrestricted three-body problemrdquo in Proceedings of the 61st Inter-national Astronautical Congress (IAC rsquo10) pp 499ndash509 PragueCzech Republic October 2010

[17] M Pontani and B A Conway ldquoParticle swarm optimizationapplied to space trajectoriesrdquo Journal of Guidance Control andDynamics vol 33 no 5 pp 1429ndash1441 2010

[18] C R Bessette and D B Spencer ldquoOptimal space trajectorydesign a heuristic-based approachrdquo in Proceedings of theSpaceflight Mechanics 2006-AASAIAA Space Flight MechnaicsMeeting vol 124 of AAS paper 06-197 pp 1611ndash1628 San DiegoCalif USA January 2006

[19] K-J Zhu J-F Li and H-X Baoyin ldquoSatellite scheduling con-sidering maximum observation coverage time and minimumorbital transfer fuel costrdquo Acta Astronautica vol 66 no 1-2 pp220ndash229 2010

[20] K Zhu F Jiang J Li and H Baoyin ldquoTrajectory optimizationof multi-asteroids exploration with low thrustrdquo Transactions ofthe Japan Society for Aeronautical and Space Sciences vol 52 no175 pp 47ndash54 2009

[21] A R Cockshott and G R Sullivan ldquoImproving the fermenta-tion medium for Echinocandin B production part II particleswarm optimizationrdquo Process Biochemistry vol 36 no 7 pp661ndash669 2001

[22] C O Ourique E C Biscaia Jr and J C Pinto ldquoThe use ofparticle swarm optimization for dynamical analysis in chemicalprocessesrdquo Computers amp Chemical Engineering vol 26 no 12pp 1783ndash1793 2002

[23] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 Perth Australia December 1995

[24] S Fang H Cheng Y Song et al ldquoStochastic optimal reactivepower dispatch method based on point estimation consideringload marginrdquo in Proceedings of the IEEE PES General MeetingmdashConference amp Exposition pp 1ndash5 National Harbor Md USAJuly 2014

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generated random particles and swarm 3 begins to performTS on each particle With the integration of TS the hybridPSO algorithm could effectively avoid local optimum byaccepting worse solution under certain condition during thesearching process

NRK which is originally used for GAs in this paper hasbeen applied to PSO algorithm As a topological coding strat-egy NRK could completely avoid the possibility of generatingunfeasible solutionswhen using heuristic algorithms in graphoptimization problems It also transforms the original dis-crete DNR problem into a continuous optimization problemWhen applied inGAsNRK is nomore than a coding strategywhich possesses little physical meaning during the solutionsearching process However when used in PSO the codingstrategy has a physical meaning The value of the ldquokeyrdquo inthe NRK can be interpreted as an importance index for eachbranch in the graph The whole searching process could beinterpreted as a process of adjusting the importance index ofeach branch and choosing the most important branches toform the optimal tree structure

The remainder of this paper is structured as followsIn Section 2 the DNR problem is formulated as an opti-

mization problem In Section 3 different network topologyrepresentation schemes are discussed and NRK is intro-duced In Section 4 the hybrid algorithm of PSO and TSis proposed and explained in detail In Section 5 the newalgorithm is tested on an IEEE 33 bus system with numericalresults The conclusion is drawn in Section 6

2 Problem Representation

DNR is originally used in planned outages for maintenancepurpose or fault isolation to restore service A Merlinand H Back [14] were the first to come to the idea thatreconfiguration may lead to a system total loss reduction andthey tried to search for such an optimal configuration usingthe branch-and-bound method Since then loss reductionhas been considered as a common objective for the study ofnew DNR optimization algorithms

A noticeable characteristic of DNR is the repeated anal-ysis of power flow during the solution searching processIn order to improve searching efficiency several refinedor approximate algorithms for power flow analysis havebeen studied such as decoupled method [15] hashing tablemethod [16] and perturbation method [2] A direct method[13] is adopted in this paper which has been proved to behighly efficient in distribution network power flow analysis

Another characteristic of DNR is the topological con-straints whichmeans any feasible solution of a DNR problemshould represent a tree structure with every node beingconnected Configuration space is the set of allowed systemconfigurations over which the optimal system configurationis to be searched for [7] In DNR only solutions that belongto the configuration space are considered feasible

Assume that a distribution network has 119899 branches ADNR problem for system losses reduction can be formulatedas

min 119891 =119899

sum119887=1

1199091198871199031198871198752119887 + 119876

2119887

1198812119887

+ 120572119875 (119881119887) + 120573119875 (119868119887) (1)

where 119909 is an 119899-dimensional vector If branch 119887 is closed119909119887 = 1 otherwise 119909119887 = 0 119875119887 and 119876119887 represent the active andreactive power flow on branch 119887 120572 and 120573 are penalty factorswhile 119875(119881119887) and 119875(119868119887) are penalty functions for node voltageconstraint and branch current constraint

3 Network Topology Representation

The process of searching the optimal DNR solution involvesthe graph theory of optimal spanning tree Let graph 119866(119881 119864)represent the topology of a distribution network where 119881stands for vertices and 119864 stands for edges Each potentialsolution is a spanning tree of 119866 All spanning trees ofthe graph 119866 make up the configuration space A goodnetwork topology representation strategy should have fourcharacteristics

(1) Being Easy to Encode and Decode A less complicatedcoding strategy would cost less time to encode anddecode thus leading to a boost in computationalefficiency

(2) Being Compatible with Other Optimization Algo-rithms Many metaheuristic algorithms have theirown limitations in dealing with different types ofoptimization problems For example GAs require abinary string representation and PSO requires con-tinuous variables A good coding strategy should becompatible with corresponding searching algorithms

(3) Avoiding Infeasible Solutions Topological constraint isone of the thorniest issues in DNR especially when itcomes to the utilization of metaheuristic algorithmsWhenever an infeasible solution is generated theoriginal searching process will be interrupted Agood coding strategy should effectively rule out thepossibility of generating infeasible solutions whichwould greatly improve the computational efficiencyand avoid the tedious topological checking process

(4) Having a Good Locality Property A good localityproperty means that the objective function valueis relatively continuous and smooth rather thanirregular jumps within a local area in the searchingspace Most metaheuristic algorithms determine thebest searching direction based on current objectivefunction values Then the algorithms will lead thesearching process towards the most promising direc-tion In other words a coding strategy with bad local-ity properties will greatly restrict the effectivenessof the searching algorithm The configuration spacegenerated by a good coding strategy should alwayskeep a high locality

There are many different ways to represent the distribu-tion network topology and each of them has its merits andflaws

Binary string representation is the most intuitive andstraightforward way to represent the network topology byassigning a binary string 119909 The dimension of 119909 is the totalnumber of switches The elements in 119909 are set to be 0 or












4 Hybrid Algorithm





V119894119889 = V119894119889 + 11988811199031 (119901119894119889 minus 119909119894119889) + 11988821199032 (119901119892119889 minus 119909119894119889)

119909119894119889 = 119909119894119889 + V119894119889(2)




1 2 3 4

5Original network

1 2 3 4

5

b1 b2 b3

b1 b2 b3

b4b5

b5

b6

Decoded network

Step 1

Branch 1 2 3 4 5 6

1 02 09 01 07 04

Step 2

Branch 1 2 3 4 5 6

1 09 07 04 02 01

Step 3

F 1 3 5 2

x(key)

x998400


Start

Initialization

Basic PSO process


No


Yes



Output PSO result

Yes


Stop criteria

Output TS result

Yes



End

Yes

No

No No



5 Case Studies






0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

b1

b2

b3

b4

b5

b6

b7

b8

b9

b10

b11

b12

b13

b14

b15

b16b17

b22

b23

b24

b25

b26

b27

b28

b29

b30

b31

b32

b36

b37

b34

22

23

24

18

19

20

21

31

32

25

26

27

28

29

30

1617

b18

b19

b20

b21

b35

b33









Case 1 13503 21 2936Case 2 13419 291 1664Case 3 13033 223 1712






1198881 = 2 17 16 18 201198881 = 3 19 22 22 231198881 = 4 20 23 26 251198881 = 5 21 23 26 26






BPSO

PSO

Hybrid-PSO

Hybrid-TS

50 4515 252010 30 40 5035

Iteration number

125

130

135

140

145

150

155

160

Net

wo

rk l

oss

es (

MW

)




1198881 = 2 73 75 79 821198881 = 3 78 80 83 851198881 = 4 80 85 89 891198881 = 5 81 87 90 91



PSO 5 14 55Hybrid 31 67 100




6 Conclusion





Competing Interests


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra




















4 Hybrid Algorithm





V119894119889 = V119894119889 + 11988811199031 (119901119894119889 minus 119909119894119889) + 11988821199032 (119901119892119889 minus 119909119894119889)

119909119894119889 = 119909119894119889 + V119894119889(2)




1 2 3 4

5Original network

1 2 3 4

5

b1 b2 b3

b1 b2 b3

b4b5

b5

b6

Decoded network

Step 1

Branch 1 2 3 4 5 6

1 02 09 01 07 04

Step 2

Branch 1 2 3 4 5 6

1 09 07 04 02 01

Step 3

F 1 3 5 2

x(key)

x998400


Start

Initialization

Basic PSO process


No


Yes



Output PSO result

Yes


Stop criteria

Output TS result

Yes



End

Yes

No

No No



5 Case Studies






0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

b1

b2

b3

b4

b5

b6

b7

b8

b9

b10

b11

b12

b13

b14

b15

b16b17

b22

b23

b24

b25

b26

b27

b28

b29

b30

b31

b32

b36

b37

b34

22

23

24

18

19

20

21

31

32

25

26

27

28

29

30

1617

b18

b19

b20

b21

b35

b33









Case 1 13503 21 2936Case 2 13419 291 1664Case 3 13033 223 1712






1198881 = 2 17 16 18 201198881 = 3 19 22 22 231198881 = 4 20 23 26 251198881 = 5 21 23 26 26






BPSO

PSO

Hybrid-PSO

Hybrid-TS

50 4515 252010 30 40 5035

Iteration number

125

130

135

140

145

150

155

160

Net

wo

rk l

oss

es (

MW

)




1198881 = 2 73 75 79 821198881 = 3 78 80 83 851198881 = 4 80 85 89 891198881 = 5 81 87 90 91



PSO 5 14 55Hybrid 31 67 100




6 Conclusion





Competing Interests


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1 2 3 4

5Original network

1 2 3 4

5

b1 b2 b3

b1 b2 b3

b4b5

b5

b6

Decoded network

Step 1

Branch 1 2 3 4 5 6

1 02 09 01 07 04

Step 2

Branch 1 2 3 4 5 6

1 09 07 04 02 01

Step 3

F 1 3 5 2

x(key)

x998400


Start

Initialization

Basic PSO process


No


Yes



Output PSO result

Yes


Stop criteria

Output TS result

Yes



End

Yes

No

No No



5 Case Studies






0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

b1

b2

b3

b4

b5

b6

b7

b8

b9

b10

b11

b12

b13

b14

b15

b16b17

b22

b23

b24

b25

b26

b27

b28

b29

b30

b31

b32

b36

b37

b34

22

23

24

18

19

20

21

31

32

25

26

27

28

29

30

1617

b18

b19

b20

b21

b35

b33









Case 1 13503 21 2936Case 2 13419 291 1664Case 3 13033 223 1712






1198881 = 2 17 16 18 201198881 = 3 19 22 22 231198881 = 4 20 23 26 251198881 = 5 21 23 26 26






BPSO

PSO

Hybrid-PSO

Hybrid-TS

50 4515 252010 30 40 5035

Iteration number

125

130

135

140

145

150

155

160

Net

wo

rk l

oss

es (

MW

)




1198881 = 2 73 75 79 821198881 = 3 78 80 83 851198881 = 4 80 85 89 891198881 = 5 81 87 90 91



PSO 5 14 55Hybrid 31 67 100




6 Conclusion





Competing Interests


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

b1

b2

b3

b4

b5

b6

b7

b8

b9

b10

b11

b12

b13

b14

b15

b16b17

b22

b23

b24

b25

b26

b27

b28

b29

b30

b31

b32

b36

b37

b34

22

23

24

18

19

20

21

31

32

25

26

27

28

29

30

1617

b18

b19

b20

b21

b35

b33









Case 1 13503 21 2936Case 2 13419 291 1664Case 3 13033 223 1712






1198881 = 2 17 16 18 201198881 = 3 19 22 22 231198881 = 4 20 23 26 251198881 = 5 21 23 26 26






BPSO

PSO

Hybrid-PSO

Hybrid-TS

50 4515 252010 30 40 5035

Iteration number

125

130

135

140

145

150

155

160

Net

wo

rk l

oss

es (

MW

)




1198881 = 2 73 75 79 821198881 = 3 78 80 83 851198881 = 4 80 85 89 891198881 = 5 81 87 90 91



PSO 5 14 55Hybrid 31 67 100




6 Conclusion





Competing Interests


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










BPSO

PSO

Hybrid-PSO

Hybrid-TS

50 4515 252010 30 40 5035

Iteration number

125

130

135

140

145

150

155

160

Net

wo

rk l

oss

es (

MW

)




1198881 = 2 73 75 79 821198881 = 3 78 80 83 851198881 = 4 80 85 89 891198881 = 5 81 87 90 91



PSO 5 14 55Hybrid 31 67 100




6 Conclusion





Competing Interests


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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