Research ArticleImproved Genetic Algorithm for DistributionSystem Performance Analysis by Taking Advantage ofEssential Spanning Trees

Yanzhu Ji ,1 Zhuoqun Shi,2 and Robert M. O’Connell 1

1Department of Electrical Engineering and Computer Science, University of Missouri, Columbia, MO, USA2Black & Veatch, Kansas City, MO, USA

Correspondence should be addressed to Robert M. O’Connell; oconnellr@missouri.edu

Received 26 January 2018; Accepted 30 April 2018; Published 3 June 2018

Academic Editor: Roberto Brighenti

Copyright © 2018 Yanzhu Ji et al.This is an open access article distributed under the Creative Commons Attribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Growing interest in the smart grid, increasing use of distributed generation, and classical distribution system reconfiguration(DSR) and restoration problems have led to the search for efficient distribution automation tools. One such tool, the improvedFast Nondominated Sorting Genetic Algorithm (FNSGA), not only is effective in finding system configurations that are optimalwith respect to voltages, currents, and losses, but also considered parametric study to determine minimum values of N and Gen.In this paper, the essential spanning tree concept is expanded to improve the computational efficiency of the algorithm. Results ofthe study show that for relatively small test systems, optimum system configurations are obtained using values of N and Gen thatrequire very small CPU times. In larger systems, optimum values of N and Gen requiring reasonable CPU times can also be found,provided that certain carefully chosen branches are removed from the pool of possibilities when producing the initial populationin the algorithm. By using essential trees, the efficiency of the calculation is improved.

1. Introduction

Growing interest in the smart grid [1] and increasing useof distributed generation, along with classical problems ofDSR, have led to the need for efficient and reliable powerdistribution system simulation tools. For systemperformanceoptimization with respect to multiple objectives, such asvoltage profiles, system load balancing index, and powerlosses, various formulations of genetic algorithms (GA) [2–7] have been shown to be very effective.

One recently developed GA, the Fast NondominatedSortingGenetic Algorithm (FNSGA) [8], converges relativelyefficiently to reliable Pareto-optimal solution sets in themultiobjective DSR problem when applied to relatively smalltest systems. When applied to larger test systems, however,the original FNSGA is restricted by the use of the Newton-Raphson numerical load flow solution method [9] and theneed for large initial population sizes (N) and numbersof iterations (Gen), which leads to burdensome memoryrequirements and long computation time. Computational

efficiency is increasingly important in large (real) systems [2,3, 10]. In the improved FNSGA discussed here, the Newton-Raphson load flow program is replaced with a revised versionof the direct load flow method [11]. Also, a parametricstudy was conducted to determine minimum values of Nand Gen that lead to reasonably repeatable configurationsof a distribution system that are optimized for the multipleobjectives of voltages, currents, and power losses [12]. In thatstudy it was determined that computational efficiency couldbe improved by judiciously removing certain branches fromthe pool of possibilities when forming the initial populationfor the algorithm.

A method for further improving algorithm computa-tional efficiency, the principal subject of this paper, involvesthe use of essential spanning trees. The method involvesreplacing the so-calledMmatrix [8] and its associated processin the algorithm with the random selection of b essentialbranches, where b is the number of fundamental loops inthe system. Essential branches are those between essentialnodes, where essential nodes are buses that connect at least

HindawiJournal of EngineeringVolume 2018, Article ID 1601239, 8 pageshttps://doi.org/10.1155/2018/1601239

2 Journal of Engineering

Initialization: Define N and Gen.Generate N random switch sets.

Perform non-dominated sorting, which determines the Pareto optimal solution

set.

Perform load flow analysis. Obtain voltage profile, power loss, SLBI, etc.

Check for radiality. Removenon-radial and repeated switch sets.

Perform load flow analysis.Obtain voltage profile.

Perform guided mutation based on voltage profile.

Check for radiality. Removenon-radial and repeated switch sets.

Produce N offspring using crossover.

Gen = 1

Produce N offspring using crossover

Keep the previous optimal solution. Generate new switch sets randomly so

the total is N

Gen = 2

．４ = 32

．４ = 64

．４ = 42

．４ = 5

．４ = 40

．４ = 40

．４ = 20

N = 20

．４ = 20

· · ·

Figure 1: Illustration of the first two iterations (generations) of the genetic algorithm as applied to the 16-bus system shown in Figure 2.

three regular branches. The usual check for system radialityis then made based on the b chosen essential branches. Ifthe system is radial, one regular branch within each essentialbranch is then chosen randomly to form a possible solutionswitch set. This switch set is then checked for repetition.Finally, this process is repeated N times to determine theinitial population for application of the genetic algorithm.

In Section 2, the FNSGA is briefly described, withemphasis on the roles of the parameters N and Gen in thealgorithm. In Section 3, the use of essential systems to furtherreduce CPU time is described. In Section 4, the results ofparametric studies to optimize the values of N and Gen viafull switch pool and essential tree approaches are presented.It is shown that the essential tree approach can result insignificant computation time reductions as compared to thefull switch pool approach. Section 5 concludes the paper witha summary.

2. Brief Description of the FNSGA

The FNSGA is described in detail in [8, 13]. Here, abrief description, illustrating its application to the above-mentioned 16-bus test system and emphasizing the rolesof the two important parameters N and Gen, is providedinstead. The FNSGA is illustrated in Figure 1 for applicationto the 16-bus test system shown in Figure 2. The exampleillustrates the first two generations, i.e., iterations of the mainbody program, with an initial population of 20 randomlygenerated sets of tie or open switches in the initially meshedtest system. One such possible switch set consists of branches7, 9, and 16, as in Figure 2.

The left column shows the current total population NTof open switch sets under consideration during the process.During the first iteration, the initial population, which isequal to the size predefined by the user (20 in the example),

Journal of Engineering 3

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952

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16 15

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1286

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Branch

Node

Figure 2: 16-bus test system consisting of three fundamental loops.Open (tie) switches in branches 7, 9, and 16 determine one possibleradial system.

is generated randomly. In the next step, an equal populationof offspring is generated from the initial population usingthe genetic crossover operator. The total population in theexample now consists of 40 individuals. Due to the random-ness involved in the first two steps, there might be repeatedindividuals among the 40, and some of themmay not actuallycreate a radial topology. Thus, the next step (the third boxin the algorithm) is to detect and remove both the repeatedindividuals and those that do not produce radial topologies.At this point the population will consist of fewer than twicethe original number, e.g., 32, as shown in Figure 1.

Next, load flow analysis is conducted on all 32 radialsystems defined by the 32 valid switch sets, to generate voltageprofiles in preparation for the next step, which is guidedmutation. During guided mutation [8, 13], the branch thatis connected to the lowest voltage bus is replaced randomlywith one that is also connected to the lowest voltage bus, butnot present in the current switch set combination. If thereis not such an alternate, mutation for this bus is skipped.This mutation process is conducted on the voltage profiles ofall 32 radial systems in question. Thus, the total populationis now doubled, to 64 in the example. At this point, theremight be repeated individuals among the 64, and some ofthem may not actually create a radial topology. Thus, it isagain necessary to remove both the repeated individuals andthose that do not produce radial topologies. For the sake ofthe example, it is assumed that 42 valid individuals remain,as shown in Figure 1.

After the guided mutation step, load flow analysis isconducted again, this time to determine all the performanceobjectives defined by the user. For this study, the definedmultiple objectives were voltage profiles, total system powerlosses, and the system load balancing index (SLBI) [5]. Theseare the data needed for the nondominated sorting process[13], which produces the Pareto-optimal (nondominated)solution set. For the example, the population of the Pareto-optimal set is assumed to be 5, as shown in Figure 1.

At this point, one genetic generation is complete, i.e.,one iteration of the user-specified number of generationsGen. The entire process is repeated for each remaininggeneration, with the only change being that the Pareto-optimal population set from each generation is kept and

1

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Figure 3: Meshed 32-bus test system consisting of five fundamentalloops.

included as part of the population N at the start of thesucceeding generation. Thus, the solution “evolves” to betterand better results with each generation, and, thus, largervalues of Gen lead to better results. Similarly, larger valuesof N allow the algorithm to consider more potential optimalsolutions, which also leads to better results. Of course, largervalues of N and Gen result in increased CPU times.

3. Applying Essential Spanning Tree to ReduceSwitch Pool Size

As described previously, the FNSGA randomly selects onebranch from each fundamental loop in the system to bemade radial; it checks to see whether the selected set ofopen branches (a switch set) defines a radial system, andif so, whether it has been selected already; it then repeatsthe process N times. The N sets of open branches thenevolve to better solutions as the algorithm proceeds. If N isvery small in comparison with the number of possibilities, avery large number of generations will be required in orderfor the algorithm to consider a sufficient number of switchsets to produce repeatable results. This will result in theconsumption of excessive CPU time. Thus, it should bepossible to decrease CPU time by reducing the number ofswitches to be considered in forming the initial populations,i.e., the size of the eligible switch pool.

Here, we use the essential spanning tree concept [14] tosimplify the original graph describing the system in questionto an essential graph. For example, the complete 32-bus testsystem and its corresponding essential graph or mesh areshown in Figures 3 and 4, respectively. To make use of theessential mesh, the random switch sets are created usingit, rather than the complete system. The FNSGA shown inFigure 1 is then accordingly modified, as shown in Figure 5.Thus, following initialization, radiality of the essential tree in

4 Journal of Engineering

Table 1: Results summary from 25 runs of the FNSGA on the 16-bus test system with various values for N with Gen = 5.

Gen N Average CPU time(seconds) Total no. of Solutions Number of unique

solutionsMaximum occurrence

rate (percent)5 2 0.048 49 17 245 4 0.073 66 18 485 8 0.146 97 16 445 12 0.203 110 12 805 16 0.265 118 11 925 20 0.330 118 9 965 24 0.369 118 10 965 28 0.436 118 7 1005 30 0.448 122 8 100

1

10

4

3

5

6

8

7

9

2

1

12

43

147

6

1311

8 2

5

10

9

Figure 4: Essential graph of the 32-bus test system shown inFigure 3.

question is determined, followed by random selection of oneswitch from each branch in the radial essential tree. Load flowanalysis is then performed on the associated system to calcu-late voltage profile, power loss, and SLBI information. Next,guided mutation based on the voltage profile is performedand repeated branches are deleted. Because mutation maycause the radiality situation to change, new essential branchsets are found, and the above steps are repeated. At this pointnondominated sorting to obtain the Pareto-optimal solutionsets is performed, and one iteration of the entire algorithm iscomplete.

After the algorithm has been repeated the specified num-ber of times (Gen), a reduced switch pool will have been pro-duced, which consists of only the branches in selected essen-tial tree branches.The FNSGA algorithm can then be appliedto this reduced size switch pool, rather than either the full 37branches in the system or the 22 branches described in [12].

4. Parametric Study to Reduce CPU Time

In order to optimize N and Gen by finding minimumvalues of the two (for CPU time purposes) that lead to

highly repeatable Pareto-optimal configurations, the above-described FNSGA was used to determine Pareto-optimalradial configurations of the initiallymeshed (no initially openswitches) 16- and 32-bus test systems.N andGenwere definedas variable parameters. The algorithm was programmed inMATLAB and implemented on a 2.7 GHz, 8 GB RAM laptoppersonal computer.

4.1. 16-Bus Test System. To optimize the initial populationsize N, Table 1 shows a summary of the results of applying theFNSGA to the 16-bus system shown in Figure 2. All switches,including numbers 7, 9, and 16, were initially closed, so as notto bias any one possible solution. Bus and branch input datadescribing the system were taken from [15]. The number ofgenerations was fixed at 5 and the initial population size wasvaried from 2 to 30. For each (Gen, N) pair the algorithmwasrun 25 times. Data in the table include, for each (Gen, N) pair,the average CPU time per run, the total number of Pareto-optimal solutions in the 25 runs, the number of uniquesolutions, and the occurrence rate of the most frequentlyoccurring solution in the 25 runs (the maximum occurrencerate).Thus, for example, in the case ofN= 16, the averageCPUtime was 0.265 seconds, eleven unique three-open-switchPareto-optimal solutions appeared 118 times, and at least onesuch solution appeared 92 percent of the time or 23 times inthe 25 runs of the algorithm.

The data in Table 1 show that CPU time is acceptablysmall, and it increases linearly with N, at least over the rangeof N considered in the study. Also, the maximum occurrencerate saturates at well over 90 percent as N increases beyondapproximately 20, shown by the plotted data in Figure 6.Using the criterion that N is sufficiently large when at leastone solution appears at least 90 percent of the time, the fourthorder curve fitted to the data in Figure 6 shows that an initialpopulation size of approximately 20 is sufficient to producerepeatable solution sets (with Gen = 5) in the 16-bus testsystem.

To optimize the number of generations Gen, Table 2shows results (from 25 runs of the algorithm) of varyingGen with N fixed at 20, the approximately optimum valuefound via Figure 6. The data show that CPU time increaseslinearly with Gen, at least over the range of Gen consideredin the study. It also shows that the maximum occurrence rate

Journal of Engineering 5

Initialization: Define N and Gen.Generate N random essential switch sets.

Create essential mesh

Perform non-dominated sorting, which determines the Pareto optimal solution set.

Perform load flow analysis. Obtain voltage profile, power loss, SLBI, etc.

Check for radiality on the resulting essential system. Remove

non-radial essential sets.

Perform load flow analysis.Obtain voltage profile, power loss, SLBI,

etc.

Perform guided mutation based on voltage profile and delete repeated branches.

Check for radiality. Removenon-radial essential sets.

Produce N offspring using crossover.

Pick random tie branch in each essential branch.

Gen = 1

Produce N offspring using crossover

Keep the previous optimal solution. Generate new switch sets randomly so the

total is N

Gen = 2

Find essential branch sets according to existing branches

Select a random tie branch in each essential branch.

�e FNSGA applied on a reduced pool of essential branches

．４ = 32

．４ = 32

．４ = 24

．４ = 30

．４ = 30

．４ = 24

．４ = 5

．４ = 40

．４ = 40

．４ = 20

N = 20

．４ = 20

· · ·

Figure 5: Illustration of the first two iterations (generations) of the application of the FNSGA using the essential spanning tree concept.

6 Journal of Engineering

Table 2: Results summary from 25 runs of the FNSGA on the 16-bus test system with various values for Gen with N = 20.

N Gen Average CPU time(seconds) Total no. of Solutions Number of unique

solutionsMaximum occurrence

rate (percent)20 1 0.079 81 22 3620 2 0.146 96 14 6820 3 0.203 106 18 7620 4 0.263 111 9 8020 5 0.318 118 9 9220 6 0.359 119 9 9620 7 0.400 118 11 9220 8 0.435 122 8 10020 9 0.442 121 9 10020 10 0.500 122 5 100

0

10

20

30

40

50

60

70

80

90

100

Perc

ent

5 10 15 20 25 30 350N

Figure 6: Maximum occurrence rate versus N, with Gen = 5, fromTable 1.

saturates at well over 90 percent as Gen increases beyondapproximately five; i.e., five generations are sufficient toproduce repeatable solution sets (with N = 20) in the 16-bustest system.

4.2. 32-Bus Test System. The 16-bus parametric study showedthat the maximum occurrence rate, which indicates theachievement of repeatable Pareto-optimal solutions, saturatesabove the 90 percent level for reasonable values of N andGen, approximately 20 and 5, respectively. Also, the averageCPU time for these two parameters is only approximately 0.32seconds, as shown in Tables 1 and 2.Thus, there is no consid-erable computational burden with the 16-bus test system.

A similar parametric study to that done with the 16-bustest system was conducted on the 32-bus system shown inFigure 3. Bus and branch input data describing this systemwere taken from [16]. Note that since the system consists offive fundamental loops, Pareto-optimal solutions consist offive open switches, rather than three, as in the 16-bus case.

Tables 3 and 4 summarize the results of 25 runs of theFNSGA on the 32-bus system with N varied and Gen fixedat 16, and with Gen varied and N fixed at 160, respectively.

The data in the tables show that CPU times are relatively longand increase approximately linearly over the range of valuescovered by N and Gen and that the maximum occurrencerates are no higher than approximately 60 percent in anycase considered. Thus, the values of N and Gen requiredfor saturation of the maximum occurrence rate (and thusrepeatable Pareto-optimal solution sets) are much larger thanany of the values considered, and the corresponding CPUtime will be impractically large. A way of dealing with thisis discussed in the next section.

4.3. Use of Essential Spanning Trees. To examine the com-putation efficiency when using the spanning tree approachdescribed previously, CPU times were compared for applica-tion of the algorithm to the 32-bus test system using both thefull switch pool and that determined with the spanning treeapproach. As shown in Table 5, essential branch set (4, 6, 7,9, and 10) clearly has the maximum occurrence rate, so thisset was used in the comparison study. (If multiple essentialbranch sets have similar occurrence rates, a combination ofthem can also be used.)

Having determined a reduced switch pool via the above-identified essential branch set, the FNSGA was simulated 25times using a Mac Pro with a dual-core intel Core i5 2.3 GHzCPU. The results are shown in Table 6 and compared withcorresponding results using the full 37-switch pool. The datashow that, as the (N, Gen) values increase from (30, 5) to (40,16) to (80, 16), the maximum occurrence rates improve from12 percent to 64 percent, from 36 percent to 96 percent, andfrom 52 percent to 100 percent, respectively.

In the full 37-switch pool case, a maximum occurrencerate of only 52 percent is obtained with 1.96 seconds of CPUtime, but in the essential branch case, the desired 100 percentmaximum occurrence rate is achieved with a very reasonableCPU time of 1.266 sec.

5. Conclusion

The paper shows that parametric studies of the initial popu-lation size and number of generations in the FNSGA can beused to find minimum values of the two that lead to highlyrepeatable Pareto-optimal configurations with reasonable

Journal of Engineering 7

Table 3: Results summary from 25 runs of the FNSGA on the 32-bus test system with various values for N with Gen = 16.

Gen N Average CPU time(seconds) Total no. of Solutions Number of unique

solutionsMaximum occurrence

rate (percent)16 30 2.75 113 58 2416 50 3.94 117 52 3216 70 6.07 123 51 3616 90 8.33 133 49 4016 110 10.6 147 44 4016 120 12.8 171 60 4416 140 13.3 155 46 4816 160 14.4 162 48 4416 170 16.4 154 36 5216 180 15.1 164 38 5616 190 18.3 161 33 60

Table 4: Results summary from 25 runs of the FNSGA on the 32-bus test system with various values for Gen with N = 160.

N Gen Average CPU time(seconds) Total no. of Solutions Number of unique

solutionsMaximum occurrence

rate (percent)160 5 6.64 137 78 28160 10 12.6 165 54 40160 15 14.9 152 43 44160 20 15.6 145 34 52160 25 18.3 170 43 52160 30 19.1 172 38 56

Table 5: Occurrence rate of the essential switch sets.

Essential switch sets Occurrence in 15 runs4 5 9 7 10 24 5 9 14 10 94 6 9 7 10 264 6 9 14 10 74 6 12 7 10 34 6 12 14 10 74 11 9 7 10 213 5 9 14 10 513 6 9 7 10 313 6 9 14 10 6

Table 6: Comparing average CPU time and maximum occurrence rates in 25 runs obtained with the full 37-switch pool and the essentialswitches for the 32-bus test system.

Gen N

Full (37) switch pool Reduced switch pool from selected essentialswitches

Average CPU time(seconds)

MaximumOccurrence rate

(percent)

Average CPU time(seconds)

Max Occurrence rate(percent)

5 30 0.314 12 0.192 6416 40 0.958 36 0.652 9616 80 1.96 52 1.27 100

8 Journal of Engineering

CPU times. As the size of the system increases, the number ofeligible switches to be used in the algorithm may have to belimited. This can be done with some preliminary simulationsto determine and remove from the eligible-switch pool thoseswitches that never appear in preliminary simulations. Theidea could be extended to larger systems by reducing the sizeof the eligible switch pool even further by also removing thoseswitches that appear rarely, e.g., only one to four percent of thetime in the preliminary simulations.

A more radical way to improve the computational effi-ciency of the algorithm is to take advantage of essentialspanning trees [16]. This is accomplished by replacing theso-called M matrix and its associated process in the FNSGA[8] with the random selection of b essential branches, whereb is the number of fundamental loops in the system. Theusual check for radiality is then made based on the bchosen essential branches and the process continued as usual.The results of parametric studies show the improvement incomputational efficiency due to the use of the spanning treeapproach. Continuation of this work should focus on waysto improve the algorithm and further decrease CPU times inlarger, i.e., realistic, test systems.

Data Availability

Underlying data is contained in published articles, which arecited in the manuscript.

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper.

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