Genetic Algorithm for Supply Restoration and Optimal Daniel

8/8/2019 Genetic Algorithm for Supply Restoration and Optimal Daniel

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chromosome is an integer representing one controllableswitch. Graph theory is employed to decide the final statusof each switch according to the radiality constraint ofdistribution networks. The GA objective function includesall the objectives and constraints required for a practicalsupply restoration scheme. A special gene 0 s incorporatedin the chromosome, indicating the optimal load sheddingpoint for the system, for cases where a network cannotsupply all the demands. Software implementing theproposed algorithm has been developed, and has beentested on a practical system.2 Problem formulationNetwork reconfiguration after a line removal for supplyrestoration aims to transfer de-energised loads in the out-of-service areas to other supporting distribution feederswithout violating the operating or engineering constraints.When the system is not able to supply all of its loads, loadwith low priority should be shed. The solution to th smultiobjective constrained optimisation problem shouldmeet the following requirements:

As much load as possible should be supplied in theresulting network withn an allowable time frame; optimalload shedding (in terms of load priority and magnitude)should take place when the demand is greater than. supplycapabilityThe switch operation costs occurring during reconfigura-tion should be minimised. Th s would take into accountfactors of switch operation time and cost, such as whetherthe switch is manually operated or remote controllable, and(for manually operated switches) the time needed to accessthe switchThe energy loss in the resulting network should beminimised

To achieve these objectives, the following constraints shouldbe observed in the resulting network:The radial structure of the distribution network should beretainedNo violation of busbar voltage limitsNo current overloading in any line

3 Genetic algorithmsInspired by the theory of evolution, GAS are adaptivesearch techniques that derive their models from the geneticprocesses of biological organisms. A GA starts with anumber of solutions to a problem, encoded as strings ofsymbols. The string that encodes each solution is achromosome and the set of solutions is called thepopulation. The position of a symbol in the string isnamed the allele. The symbol or value that an allele cancontain is called the gene. The initial population can begenerated randomly, or may consist of a number of knownsolutions, or a combination of both. The GA goes througha number of steps in whch the population at the beginningof each step is replaced with another population, w hch it ishoped will include better solutions to the problem. Aprocess called reproduction, in which the chromosomes ofthe old population are combined to create new ones, isapplied to define each new generation. Reproduction worksby repeatedly applying three operations to the currentgeneration: selection, crossover and mutation, until therequired number of chromosomes are available in the newpopulation.146

A GA requires an evaluation function that assigns afitness value to each potential solution (Chromosome). AGA works with only the symbol strings and has no inherentknowledge about the problem. Problem specific informationis provided by the objective or evaluation function. It is theevaluation function that guides the GA to evolve towardsbetter solutions. The fitter the chromosome, the higher theprobability that it will be retained and selected to generate anew candidate solution.The GA evolution process is a powerful global searchmechanism, whose computational code is very simple.4load sheddingA GA is suitable for the supply restoration algorithmbecause it is very easy to change constraints or objectives, orapply new ones.

GA for supply restoration and optimal

4. I String encoding4. I. I lnteger permutation encoding scheme: Be-fore a GA can be applied to an optimisation problem, anencoding scheme that maps all possible solutions of theproblem into symbol strings (chromosomes) must beintroduced. It is also helpful if the encoding maps as manychromosomes as possible to feasible solutions. The idealencoding would give only feasible solutions, so that the GAwould not need to test for feasibility.Since the topology of a ,distribution network can beuniquely defined by the status of all available switches, asolution to the supply restoration problem can be encodedas a function of the controllable switch states of thenetwork. The most natural coding method is to have abinary string with length equal to the number of switches inthe network. Each switch state is represented by one bitwith a value 1 or 0 corresponding to closed or open,respectively. However, this encoding is a poor choicebecause it does not ensure that the chromosome willgenerate a feasible solution.Instead, an integer permutation encoding scheme isproposed here. A similar approach has been widely appliedto the wellknown travelling salesman problem. Theproblem is to find the minimum cost path through a graphor network visiting all the nodes exactly once. If every nodeis assigned a unique integer, then the ordering of theintegers in the chromosome defines the order in which thenodes are visited.In an integer permutation encoding, the genes are theintegers from one up to the number of positions, andthe solution that the chromosome represents depends on therelative ordering of the integers.For the supply restoration problem, each integer in thepermutation will be an index into a list of switches that canbe used for restoring the supply. The integers therefore takevalues from one to the number of switches whch need to beinvestigated, i.e. if there are k switches in the study, eachchromosome will be composed of k unique integers withvalues from 1 to k.The sequence of these integers represents an orderedoperation sequence of switches to be followed to generate avalid network topology. For instance, for a system with sixswitches (initially all open) considered for supply restora-tion, the string of length six:

(3 2 6 5 14 )defines an ordered switch sequence: switch 3, switch 2:switch 6, switch 5, switch 1 and finally switch 4.

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4.1.2 String interpretation for supply restorationproblem: The string shown above is only a permutationofswitches, so that the string can be meaningful and map avalid network topology, these switches must be assignedstatus values either as closed or open. Here, a schemebased on graph theory is proposed whch decides the statusof these switches by ensuring that the radiality constraint ofthe distribution network is met.Before the GA begins, the actual distribution network ismapped to a graph.The branches of the graph correspondto switches that can be opeFdted for supply restoration,whde the nodes of the graph represent all the connectedelements of the network that do not include any of theseswitches. Each node therefore will represent an island of thenetwork that does not need to be further subdivided duringsupply restoration analysis.Initially all switches are assumed to be open, and startingwith the first gene (integer) in the chromosome, itscorresponding switch is closed. The process is repeated foreach gene until the end of the chromosome is reached. Ifclosing a switch would violate the radiality constraint forthe distribution network by creating loops or connectingsources together, then that switch operation is abandonedand the next gene in the chromosome is considered.To test for loop or source connection violations, eachnode of the graph is initially assigned a unique colourvalue (represented as a positive integer), excepting thosenodes that contain generators or infeeds (considered assource nodes) which are assigned a particular colour valueof 0. All the branches are removed to represent a networkin which all switches are open. Each branch is visited in theorder determined by the chromosome. If a visited branchconnects two nodes of different colours, then it is insertedinto the graph and all nodes having the higher colour of thetwo will be changed to have the lower colour.Loops are avoided by preventing two nodes of the samecolour being connected, and assigning the same colourvalue 0 to all source nodes ensures that two sources willnever be connected together. At the end of this procedurethe switches in the network that have their correspondingbranches inserted in the graph will be closed and the otherswitches will remain open, thus generating the complete setof switch states corresponding to this chromosome.If there are k switches in the network under study, thenthe chromosome takes the formwhere 1 I , K and I, # 4 for i#j (Idenotes an integer).Suppose there are N nodes in the graph of which N , aresource nodes. The problem is to connect each of the (N-N,)nonsource nodes (with uniquely assigned colour value from1 to N-N,) to any of the N , source nodes (each havingcolour value 0)ccording to the sequence order defined bythe chromosome.The switch operation status decided by the abovementioned branch insertion scheme guarantees that eachchromosome will be mapped to a valid network configura-tion which is normally a radial network consisting of N ,spanning trees rooted at the N , source nodes of thenetwork.If a prefix symbol + denotes the switch state is open,while - denotes the switch state to be closed, the initialstring can be denoted by

(+TI + 2 + 3 . .+ k )where all switches are in open states. The permutationfinally obtained will be of the form

(-11 (+-)12(+-)I3 . . (+-)&) (3)

where (+- ) denotes either the open or closed stateaccording to the graph constraints.It can be seen that the original string expressed in eqn. 1can uniquely lead to a unique sequence of switch statesexpressed as in eqn. 3 whch defines a radial networkconfiguration for supply restoration.An AC loadflow is calculated for each trial networkconfiguration to allow the fitness of the correspondingchromosome to be evaluated. This requires a measuredvalue (or an estimated value) for each load in thedistribution network. In practice, individual loads in adistribution network may be unmeasured. In this case anappropriate load estimation process should be applied [121.4.1 .3 Ogene for optimal load shedding: Accord-ing to the proposed algorithm, the switch operations aredetermined by the integer permutation contained in the GAchromosome. The switch setting algorithm proposedguarantees that a valid network configuration is generated,which is a spanning tree or spanning forest with each loadconnected to a source (if any possible connection exists).The number of spanning trees should be equal to thenumber of supply sources available in the network. If in anyspanning tree, the source cannot supply all of its loads (e.g.the AC load flow is divergent), these loads will remainunsupplied and be treated as lost load.In practice, however, it is expected that as much of thedemand as possible will be supplied. Where demand exceedssupply, some loads with lower priority should be discardedand each source would supply the remaining loads to itsmaximum capability.Based on this consideration, a special gene 0 isintroduced into the chromosome. The string length nowbecomes k+1 if there are k switches in the study. The 0gene is different from the other (strictly positive) genes, inthat instead of denoting one switch it is simply a flag. In theprocess of switch status setting all remaining switches arekept open beyond the 0ene in the chromosome, so as tokeep some parts of the system unconnected (or some loadsunsupplied). In the initial population, the 0genes positionis randomly generated, but it is expected that the 0 genewill evolve to somewhere near the end of the chromosome,after a few generations. Ths is because such chromosomesare likely to be fitter than those in which the 0gene occursat an earlier position. Eventually, the GA should convergeto the optimal network configuration for supply restoration.Where the network has the capability to supply all of theloads, the optimal chromosome should have the 0 gene inthe last position, or a position near the end of thechromosomeso that the switches denoted by the subsequentgenes do not require to be closed. However, when thenetwork cannot supply all of the loads, the 0 gene willoccur at an earlier position preventing one or more selectedloads from being supplied. The switches through whchthese loads would have been supplied would have theirgenes located after the 0 gene in the chromosome. SinceGA is a global search technique, the loads to be shedidentified by the genes occurring after the 0gene, in thefinal chromosome, should indicate the global optimumsolution in terms of the defined objective function.For instance, chromosome (3 2 6 5 0 1 4) will only allowthe status of switches represented by (3 2 6 5) to beevaluated by graph theory, 0 is a stop sign and switches 1and 4 are kept open. Loads connected to sources byswitches 1 and 4 will be shed according to this chromosome.Since the position of the 0 gene in the initial population israndomly generated, the introduction of the 0 gene allowsthe GA effectively to search for an optimum solution from

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a series of strings: (0), (3), (3 2), (3 2 6), (3 2 6 5) , (3 2 6 5 I),(3 2 6 5 1 4). The optimal position for the '0'gene will bedetermined via the evolutionary process.4.2 GA operationsThe order crossover (OX) operator is adopted, which buildsoffspring by choosing a subsequence from one parent andpreserving the relative sequence order from the other parent[13]. Mutation is achieved through exchanging two integerpositions in a chromosome.Since a GA chromosome will uniquely generate a validfeasible network configuration for supply restoration, itsfitness can be evaluated using a standard AC load flow onthe candidate network to provide the quantities required inthe objective function.4.3 GA objective functionMost optimisation problems impose constraints on theacceptable solutions,so it is possible that the solution that achromosome describes would not be feasible. The option ofsimply rejecting every infeasible solution may lead to therejection of some good partial solutions, and is likely to becomputationally inefficient.An objective function can be modified to account forthese constraints by penalising any solution that violates aconstraint. In t hs approach a penalty term, whch dependson the constraint and the extent of its violation, issubtracted from the calculated fitness value. This 'penaltyfunction method' permits new constraint formulations to beadded readily to a GA-based optimisation method.The GA objective function for the supply restorationproblem consists of five terms:

The demand that cannot be supplied because thecorresponding parts of the network are not connected toa source, or the load that has been shed to avoid constraintviolationsThe network power lossesBranch current overloadsBusbar voltage deviationsSwitch operation costsEach factor contributes a penalty term and the objectivefunction for the chromosome is the weighted sum of allthese penalties. Since the smaller the value of the objectivefunction the greater its fitness, searching for maximumfitness corresponds to minimising the total penalty.The objective function is defined as

nf = WLLPLL + K P L + R O Z O + WDVD + wsw Clz= 1

wherePL Lis the ratio of total weighted lost load to total weighteddemand (E,,,PLL,E,W , , ~ L , )WLLs the weight for PLLP L is the ratio of total losses to total power supplied(phs/ pGlWL s the weight for PLIo is the sum of the ratios of current overload to maximumpermitted current for each line (Cf=,Z;'- mat/l",)WIo s the weight for IoV , is the sum of the ratios of voltage deviations from limitsfor each node (Cz,d x l / A V",)

WvD is the weight for V Dns w is the number of switches that can be used for restoringC, s the cost of operating switch i from the existing state tothe state required by the evaluated configurationWsws the weight for the switch operation cost term

supply

The first term in the objective function is defined as lostload: W L L PLL where PU =:E,,,P u ,/E,WL, L, Thepriority of each load is reflected by the weighting factor WL,The lost load term is meaningfill when some load cannot beconnected to any source, or load has to be shed for thesystem to meet other constraints. Heavy penalties areapplied for lost load, but where necessary, the GA candecide the optimal load shedding points to reach the bestsolution for the supply restoration problem.The second term accounts for the network losses. It hasthe minimum weighted value since compared to other itemsit is the least crucial in the supply restoration scheme. Thethrd and fourth term incorporate the traditional steadystate security constraints into the objective function. Sincebreahng these constraints mal bring about serious damageto the system, heavy penalties are applied to these twoitems.The fifth term accounts for the cost of switch operationsrequired during network reconfiguration. The parameter C,denotes the cost of changing switch i from the existing stateto the state required by the desired configuration, tahnginto account factors such as whether the switch is remotecontrolled or manual operatt:d, the access time for theswitch, etc.In the current objective function, most terms are definedas ratios. It is expected that u,sers will adjust the weightingcoefficients to reflect the relative importance of the variousobjective terms and according to experience with thesolutions obtained. Typical weight values, used in theexperiments reported here, an: 10.0 for lost load, 10.0 forpower losses, 500.0 for current overloads, 50.0 for voltagedeviations and 1O for switch operation. Varying the weightscan lead to alternative solutions being produced, but doesnot seem to have much influence on the speed ofconvergence.5 Test resultsThe GA-based supply restoration and optimal loadshedding algorithm has been tested on an example systemthat is part of a practical system in the UK. The system isshown in Fig. 1.For ths system, suppose the fault occurs on the linewhere switch S102 is located. Switch S102 is tripped off toisolate the fault. Since normally those switches located atthe power sources will not be operated during thereconfiguration, restoring the supply to the loads connectedto the network via S63 and S85, requires only the states of63 switches (from a total of 114) to be considered in thestudy. The chromosome is of length 64, composed of 63genes denoting switches plus the '0'gene. The test systemshown is a typical sized problem for postfault restoration inan urban distribution network. For online implementationa preprocessing stage is required to limit the dimensionalityof the network model. The model should include sufficientlines and substations adjacent to the fault, so that allreasonable switching options are possible. On the otherhand, it is desirable to limit the dimensionality so that thecomputer time required for solution is not excessive.

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I d

I *s44

0

- oadClosed SwitctOpenSwltch

Fig. 1 Initial state of test system

5. GA for full supply restorationSuppose the fault occurs at the base load level. The GA hasa pool size of 50. By tuning the GA parameters, the optimalperformance was reached for a crossover rate of 0.6, andmutation rate of 0.08125.After 337 iterations of evolution,the GA reaches its minimum objective function value2.0429, of whch 1.0429 is contributed by network loss whilethe remaining 1.0 is contributed by the single switchoperation: switch S1 is changed from open to closed. Thisis obviously the global optimal solution under thscondition. The resulting network topology is shown inFig. 2.The GA evolution process is shown in Fig. 3, where OBFshows the minimum objective function value achieved bychromosomes in the pool. From Fig. 3, it can been seen thatat the beginning of evolution, the objective function valuesare h gh indicating the existence of constraint violations orload shedding. However, in the 76th iteration, the objectivefunction value drops sharply from 14.5095 to 8.04167. Afterthat, the GA objective function values consist of only twoparts: switch operation costs and network losses. Theoptimal solution is finally achieved after three further stepimprovements occurring at the 97th, 120th and 337thiterations, each reducing the switch operation cost by 2.0. Itis obvious that the 0 gene has moved far enough back inthe chromosome in the 76th iteration not to prevent anyload from being connected to the network.Although the optimal solution for ths example is quiteobvious, a general disadvantage of using a GA is that thereis no simple algorithmic indication of when an optimal

I

f

0

@ lnfeedTransformer

+ oadClosed Switcl

0 Open Switch

Fig. 2levelchromosome length = 64 0 gene, pool size = 50)crossover rate=0.6mutation rate = 0.08125

Network conjgurution u@er supply restorution ut base load

iteration no

Fig.3base load levelGA evolution pr ocessfor supply restoration with 0 gene at~ OBF

solution has been reached. The usual approach is tocontinue the evolutionary process until a threshold numberof generations have been passed with no further improve-ment in objective function value. Th s inherent difficulty isillustrated in the present example, where a long period of149EE Psoc -Gener T s m Dumb , Vol 149, No 2, Munh 2002


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6 ConclusionsA GA-based method is proposed to decide the supplyrestoration and optimal load shedding strategy for distribu-tion networks. The integer permutation encoding schemeis adopted in which each chromosome represents the set ofcontrollable switches the final states of which the algorithmis to determine. Graph theory guarantees that the chromo-some will be mapped to a unique and valid radialdistribution network. A 0 gene can be introduced in thechromosome to indicate the optimal load shedding pointswhen demand is beyond the ability to supply. The algorithmhas been successfully tested on a practical system.7 AcknowledgmentsThe authors would like to acknowledge the financialsupport of Northern Electric Distribution, YorkshireElectricity, and NORWEB Distribution. They also wishto thank Mr. Bob Eunson, M r. Mark Marshall, Mr. SteveCox and Mr. John Westwood of the above companies, fortheir technical contributions to this research.8 References

1 CHA NG, G. , ZRIDA, J., and BIRDWELL, J.D.: Knowledge-baseddistribution system analysis and reconfiguration,IEEE Trum. PowerSysr., 1990, 5, (3), pp. 744749

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LIU, C.C., LEE, S.J., and VENKATA, S.S.: An expert systemoperational aid for restoration and loss reduction of distributionsystems, IEEE Truns. Power Syst., 1988, 3, (2),pp. 619-626SAKAGUCHI, T. , and MATSUMOTO, K.: Development of aknowledge based system for pow er system restoration, ZEEE Trans.,Power Appur. and Syst., 1983, PAS-102,(2), pp. 32&329SARFI, R.J. , SALAMA, M.M .A., and CH IKH AN I, A.Y.: A surveyof the state of the art in distribution system reconfiguration for systemloss reduction, Electr. Power Syst. Res., 1994, 31, pp. 61-70NAGATA, T. , SASAKI, H. , and KITAGAWA, M. : A method ofdetermining target configurations for power system restoration using amixed integer programming approach, Electr. Eng. Jpn, 1995,STANKOVIC, A.M., and CALOVIC, M.S.: Graph orientedalgorithm for the steady state security enhancement in distributionsystems, ZEEE Truns. Power Deliu., 1989, 4, (I),p. 53%544TEO, C.Y.: A computer aided system to autom ate the restoration ofelectrical power supply, Electr. Power Syst. Res., 1992, 24, pp. 119-125TEO, C.Y., and GOOI, H.B.: Restoration of electrical power supplythrough an algorithm and knowledge based system, Electr. PowerSyst. Res., 1994, 29, pp. 171-180GO LDB ERG , D.E.: Genetic algorithms in search, optimisation andmachme learning, (Addison-Wesley, 1989)NARA, K. , SHIOSE, A, , KITAGAWA, M., and ISHIHARA, T.:Implementation of genetic algorithm for distribution systems lossminimum re-configuration, IEEE Trans. Power Syst., 1992, 7, (3),pp . 10441051FUKUYAMA, Y., CHIANG, H.D., and MIU, K.N.: Parallelgenetic algorithm for service restoration in electric power distributionsystems, Electr. Power Energy Syst., 1996, 18, (2), pp . I 1 1-1 19IRVING, M.R., and MACQUEEN, C.N.: Robust algorithm forload estimation in distribution networks, IEE Proc.-Gener. Trunsm.Distrib., 1998, 145, (5), pp. 49%504M IC HALEW IC Z, Z.: Genetic algorithms+data structures= evolu-tion p rograms, (Springer, 1996)

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Genetic Algorithm for Supply Restoration and Optimal Daniel