Distribution feeder reconf iguration with refined genetic algorithm

W.-M.Lin, F.-S.Cheng and M.-T.Tsay

Abstract: A refined gcnctic algorithm for a distribution fecdcr reconfiguration to rcducc losses is presented. The prohlcm is optiniised i n a stochastic searching manner similar to that of the conventioiial CA. The initial population is determined by opening the switches with the lowest current in every mesh derived in the optimal power llow (OPF), with all switches closed. Solutions provided by OPF are generally the optimum or ncar-optimal solutions for most cases, so prematurity co~ilcl occur. To avoid prematurity, the convcntional crossovcr and mulation scheme was refined by a competition mechanism. So thc dilcmrna oT choosing a proper probability for crossovcr and mutation can be avoided. The two processes were also combincd into one to save computation time. Tabu lists with heuristic rtilcs wcrc also employed in the searching process to enhance perrormance. The new approach provides an overall switching dccision instead of a successive pattern, which tends to converge to a local optimum. Many tests wcrc conductcd and the results have shown that RGA has advantages over many other previoiisly developed algorithms.

1 Introduction

Most electric distribution systems arc radially configured. There arc a few norinally closed and normally opened switches in a distribution feeder. Uiidcr normal operating conditions, distribution engineers could reconfigure feeders to increase network rcliability and rcducc line loss. There are nLnncroLis numbers of switches in the distribution sys- tem, and the number of possible switching operations is tre- mendous. Feeder reconfiguration thus becomes 21 complex decision-making process for dispatchers to follow.

Many algorithms have been developed to deal with the problem [ I ~ 161. In [1-4], analytical approaches were employed to minimise the power loss. The method described in [5] achieves the optimal configuration, by opening the switches with the lowest current dclcrinined by the optimal flow pattern. Heuristic methods wcrc suggested in [GO], and switching indiccs wcrc dcsigncd in [ IO] 'lo solve the problem.

Recently, new algorithms based on artificial intelligence (AI) have been developed, such as simulated annealing (SA) [ I I ] , genetic algorithms (CA) [ 12-14], artificial neural networks [I51 and expert systems [16]. The switching stratc- gies proposed by most AI algorithms need to consider a large solution space. Extensive numerical computation is often required, especially when the load flow tcchniquc has to bc used. On the other hand, conventional methods nlay be faster; they may diverge or co~ild lead to ii local mini- nium.

The reconliguration problem can be formulated as a mixed-integer nonlinear optiinisation problem. It is diflicult to design a quick and optimal procedure by using only one algorithm. This paper proposes ii refined genetic algorithm (RGA) which takes advantage ol' the optimal llow pattern [5], genetic algorithm (CA) [12-141 and tabu search method [17, 181. Crossover and mutation were combined in RGA, and a competition inechanisni was implemented to auto- matically determine the choice of either one. With the advantages of both heuristic ideals and Al, RGA super- cedes the original ideals in thrcefolds: the complicated problem is solvable, with a better pcrforinancc than AI, and the more liltelihood to get a glohal optinitiin than heu- ristic methods. This papcr Iocuscs on the minimisation of the real power loss and the number or switchings subjected to the radial network structure. Numerical examples also have been providcd to show its cffcctivcncss.

feeder1 feeder2 feeder3

w w w 13

I 5 11 L 1 2 I s i 3 s23

L a a 4 6 s i 4 7 s26 16 s25 15

Fig. 1 7lii.i.c,/~cdi~i. .s(iiii/ilc .\ysIei)i .i = RWilCI, ,,umhcr

2 Problem description

Fig. I shows a sample distribution network [7] consisting of three feeders with thirlccn normally closcd scclionalising

149

switches and three normally opencd tic switches, namely, s15, s21 and s26. The notation (X, Y) is used to denote the operation of opening switch Y and closing switch X. For instance, operation (s12, s15) transfers load 5 from fccdcr I to fccdcr 2 by opening a scctionalising switch s12 and clos- ing the tie switch s1S. Hence, closing a switch shotild always follow the opening of a switch. The load at busbar I 1 can be transferred to fceder 1 by closing tie switch I5 and opening the sectionalising switch 19. Similarly, other loads can be transferred From feeder to fccdcr by switching operations.

Thc loss-minimising problem can bc formulated [SI as

iiiiri fic).9.s(Su) ( 3 1

open switches callcd genes. So IS, 21 and 26 are the genes of this chromosome, and reprcscnt the open switches in three mcshcs, respectively.

3.2 Initialisation The initial population is {X,li = 1, ..., I J } , where p is the population size. Half population is randomly created, and another half of the population stores the near-optimal solu- tion obtained by opening the branches with the lowest cur- rent in powcr flow [5].

3.3 Statistics With load-flow computation, eqn. I will be used as the fit- ness function by adding constraints as

such that

wlierc P,,,,, = total line losscs of distribution lccdew,

S, Ni, X i I, I,,,;,, = upper limit of branch current magnitude, N V, V,,,, = upper limit of busbar voltage magnitude, V,,,,. = lower limit of‘ busbar vollage magnitude.

lowing rules: (i) no fccder section can be left out of servicc, (ii) radial network structure must be retained.

3 Solution algorithm

A genetic algorithm was suggcstcd in [I21 in ordcr to solve this problem, and is callcd the simple genetic tilgorithm (SGA) in this paper. C A is a search algorithm based on the mechanism of nature selection and genetics. To enhance the performance of GA, a tcfined GA (RGA) was devel- oped as l‘ollows:

3.7 Encoding and initialisation

= the status vector or switches, = total number of branches in the whole system, = resistance of blanch i, = current magnitude or branch i:

= tolal number of buses, = voltage magnitude of busbar,j,

In addition, fceder reconfiguration has to obey the Col-

The coding scheme is illustrated in Table 1, where ‘0’ indi- cates an ‘open’ switch, and ‘c’ indicates it ‘closed’ switch, hi’ denotes the tolal number of meshes, and is equal to the total number of open switches. To retain a radial network structure, only one switch is opened in each mesh. Table 1 shows a coding scheme of Fig. I . In this paper, notation X = [M’, Iv2 ... LVJ represents a chromosome, which is [I5 21 261 for Table 1, Each chroinosomc comprises a number of

j = l

where and iirc tlic penalty factors that can be adjusted in the optimisation proccdure. Ai,,, and V,,,, arc defined by

( 3 ) if I , 5 I,,:,

Imaz if 1.i > Imas; I h L = { It

yj if V,,,,,, I V, I 1fm.n.3: Kin, = V,,,,, if V, < %in (4) i V,,,.,, if V, > K,,,,,

lf onc or more variables violate their limits, the penalty fac- tors will increase, and the corresponding chromosome will bc entered into the tabu list to avoid generating thc same infcasiblc solution again.

The fitness of each chromosome, i.e. each network topol- ogy, is calculated, and the minimum litness F,,,,,, and the average fitness will be found.

3.4 Offspring Thc offsprings arc the new chromosomes obtained from crossover and mutation. Crossover is a structured recombi- nation operation by exchanging genes of two parents. Mutation is the occasional random alteration or genes. For mutation in RGA, the neighbouring genes, i.e. switches besides the one to be mutated, llavc a higher priority to be selected than other switches in the same mesh. In Fig. 1, switch 12 and switch 19 have higher priorities to replace switch 15. A refined crossovcr and mutation (RCM) scheme is used in this paper and will be explained in Section 4.

3.5 Tabu list A Pabu list will bc constructed to define forbidden moves, such as (i) the solutions just visited except the bcst solution in the current generation,

Table 1: A coding scheme of the sample feeder in Fig. 1

m l 2 3

s 11 12 15 19 18 16 16 17 21 24 22 13 14 26 25 23 24 21 17 18 19 15 12

a c c o c c c c c o c c c c o c c c c c c c c c

w 15 21 26

m = mesh number, s = switch number, a = switch status, w = coding scheme

350

(ii) tlic local optima ever visited, (iii) the chromosomes violating electric constraints, (iv) the heuristic rules incapable ol' retaining tllc radial structure, and leaving load unserved. For example, the edge of two adjacent meshes cannot contain inore tlian one open switch siniultancously.

3.6 Elitism selection The 2p chromosonies (p parents and 1) ofkpring) are tlicn ranked in ascending order according to their fitness values. '1'1' individuals with the best fitness arc kept a s the parents for the next generation. Othcr individuals in the combined population of size (211 ~ / I ) have to compete by adopting the 'roulette wheel' approach to get sclcctcd in tlie ncxt g e n e w tion.

3.7 Stopping rule Tlie proccss of generating new trials with the best fitness will bc continucd until the litncss values arc optimised or the maximum generation ntinibcr is reached.

The flowchart of the RGA process is shown in Fig. 2.

sfart

I I initialisation I I t

refined crossover and mutation (RCM) I- 1 I check if the chromosomes

are in tabu list

no

elitism

i converged

no yes

optimal switching strategy I

4

To prevent prematurity and cnliancc performance, RGA also refines the crossover and mutation scheme. The proc- ess of simple C A (SCA) is first delineated [12] and then relined as follows:

4. I Simple crossover and mutation scheme (SCM) The crossovcr process randomly (iinil'orni distribution) selects two parents to exchange genes with a crossover rate P,. Tlie location of tlie gene within the chromosome is called 'locus'. The crossover point is also randomly chosen from tlie loci. IT one (or both) offspring is infeasible, another inate will be chosen again for crossover.

The mutation process randonily (Liniforni distribution) selects one parent with a mutation rate P,,,. The authors could randomly select a loc~is to mutate. IT tlie offspring is infeasible, another parent will be chosen until 21 feasible solution can be obtained.

Refined crossover and mutation scheme (RCM)

/ K K /+oc.-(hvr, 7 ' r ~ t ~ u n ~ . / h I r i / 3 , , Vol. 147. No. 6 , A'osi~iilhrr. 20110

4.2 Refined crossover and mutation scheme (RCM) Crossover generally executes before mutation, throughout the SGA searching process. In SGA, 21 higher crossovcr rate allows the exploration of solution spacc around tlie parent solution. The mutation rate controls tlie rate new genes are introduced, and explores new solution territory. If it is too low, the solution might settle at a local optimum. On the contrary, 21 high rate could gcnenitc too miniy pos- sibilities. The offspring lose their rcsctnblancc to the par- ents; tlie algorithm will not learn Trom the past and could become Linstablc. It is a dilcmma to choose suitable crosso- ver and mutation ratc for SGA. A refined crossovcr and mutation sclienic (RCM) is thus proposed in the following to iivoid sLicli a difficulty.

(i) Randomly sclcct two parents, and generate oflipring by introducing C(g) with

(U ) I f rtu7tl < C(g) : use mutation; (h) I f ~ m t l > C(x) : use crossover.

r tod = tlie uniform random number in (0, I), C

g

Where

= the control paraiiictcr with initial valuc set to

= tlie current generation nunibcr. 0 . 5 , 0 s C 5 1 ,

The offsprings will be generated until all parents arc proc- csscd. Fig, 3 shows the initial relationship of crossover and mutation in RCM. Mutation operation will play a more important role t h a n that i n SGA, because mutation is more capable of exploring new regions. If the scarch is very close to tlic local or global optimum, nititation may need to become dominant, especially in tlic absence of the critical good genes in a generation. As crossover and mutation arc both random operators, thcrc is no telling which one is bct- tcr of tlie two. A competition mcclianisni is thus iniplc- mcntcd in the searching process, according to tlie fitness score. If the best current solution conies from crossover, there is a more liltcliliood for crossover to generate better orkpring for the next population. On tlie contrary, there is a more likelihood for mutation to generate better offipring. If the best solution remains the same, the operation of crossovcr or mutation needs to hold hack. Tlie probability of crossovcr and mukation sums to one

0 0.5

J Lo

following holds: Ii

g7na I

C(g + 1) = C ( g ) ~ -

where K is the rcgulating factor and g,,,,, is the iiiaximum generation number. Fig. 4 shows the variation of probabil- ity or crossover and mutation.

1

(iii) If Fn,,,,(g) derives from ni utation, the control parame- ter Ck + 1) will incrcase. For F, ,,,,, (g ~ 1) > F, ,,,, (g):

(iv) It" F,,,,,,(g ~ I) = F,,,,,,(g), the control parameter needs to hold back. If C(g) > C(g ~ I) , thc Ibllowing is givcn:

(7) IC

!/ma? @(.9 + 1) = C(.9) - __

othcrwise

5 Test results

Many tests had been run and the distribution systciii pre- sented in [7] is used for example. Tlic network contains 32 busbars and 5 tie switches (s33, s35, s34, s37, s36}, as shown in Fig. 5. The total loads are 5048.261tW and 2547.32kvar. Thc real power loss is 0.0203p.u.. The pro- posed method has been iniplcmentcd in C language, and ran on an IBM compatible Pciitium 133 computer.

5. I Convergence test Fig. 6 shows the optiinised losses with 100 difl'crent initial population trials. Each trial reaches the same point. Fig. 7 shows the number or gencrations required to reach the optimum. It can bc shown that all the trials reach the opti- mum in less than 10 gencrations. The iwcrage number of gcnerations to reach the optimum is 6.1, with a population size of 10.

5.2 Performance test To check the perrormance of RGA, 8 cases were run with different rules enabled. Thc convcntional newton-Raphson load flow program was used for each method to find the titness scores. Thc first case is SGA and the eighth casc is

152

RGA. 100 differcnt trials were tested for each case. The results are shown in Tablc 2. With the stochastic nature of CA, test runs may not always converge at the samc point Tor various cases. It can bc shown that a good initial condi- tion providcs a bctter perrorinancc. Using KCM will improvc the performimce significantly, and enforcing the tabu list will slightly degradc the pcrforniancc. As tlic load flow program w ~ s uscd extensively, i.e. 10 rims per geiiera- tion, the usc of fist load flow technique could also enhance tlic performance substantially [ 191.

0

0 .

5.3 Robustness test A robustness test was also conducted for the 8 cases, and the results are shown in Table 3 . It c m be seen that a good initial condition provides a bctter solution or has a bcttcr chance to rcach the global optimum. RCM cnhanccs the performance and may not guarantee a global optimum.

IEE Ploc.-Gerter. T,wrz.v,it. KWd>., I 'd 147. N u . 6. N O W W ~ J < N 2000

Table 2: Performance test

Rules Results

Average

time, s

Average Case

Initialisation RCM Tabu number Of execution list generations

to converge

1 (SGA) no no no 38.5 142.2

2 no no yes 41.6 148.6

3 no yes no 30.4 109.4

4 no yes yes 32.6 113.8

5 yes no no 9.3 32.1

6 yes no yes 11.4 36.8

7 yes yes no 5.6 21.6

8(RGA) yes yes yes 6.1 22.1

Rule 1 (Initialisation): no = all initial population created randomly yes = half initial population obtained from OPF Rule 2 (RCM): no = SCM were used with P,= 0.6 and P,,, = 0.1 yes = RCM were used

Table 3: Robustness test

Results

Case Average loss Number of trials to reduction, % reach global optimum

1 (SGA)

2

3

4 5

6 7

8 (RGA)

28.148

30.532

29.714

30.987 31.126

31.148

31.138 31.148

79

87

85 89 92

100

96 100

The use of tabu list enhances the solution quality, and pro- vides a better opportunity to reach the global optimum.

With the ciikminciit of all rulcs, RGA enstires the pcr- formance, solution quality and the robustncss. From Table 3, it is clear that RGA providcd "the highest proba- bility to reach the global optimum.

5.4 Effectiveness,comparison The el'fectiveness or the propoqd incthotls was compared with many other methods. Among stfch niethods, the thrcc niethods dcvclopcd by Goswami [9] and the three methods dcvclopcd by Baran and WLI [7] had used the same test sys- tem as shown in Fig. 5, and a r ~ thus shown here for com- parison in Table 4. Tlie number. of steps rcquir@ by each mcthod is shown in column I . The number of the opened switch is shown in column 2, where 'old' indicates the orig- inal open switch and 'new' is the chosen switch to open from the algorithm. Nolc that B a r k mr,thod 2 and 3 yield t h e same result. From Tahle 4, it can bc seen that the pro- posed algorithm is effective with minimiini loss and switch- ing operations. Compared with Baran method 1, the proposed mcthod can avoid complicated numerical compu- tation, and have thirty pcrccnt morc loss reduction.

6 Conclusions?

In this papcr, RGA has bcen proposed to rccducc distribu- tion system losses. The en'ectiveness of RGA has bccn dcm- onstrated by nuniei:ical examplcs. The proposed mcthod can guarantee a good solution quality, less number of switchings, robustness and a better p c r h " m . RGA is superior to SGA in two ways: onc is the use of tabu lists to avoid invalid searches, especially in applications with many constraints and local optima; another one is the automatic regulation of the frequency of crossover. and mutation operations, particularly in applications sensitive to the probabilistic rates. R G A has great potential to be furlhcr applied to many ill-conditioned problems in power system planning and operations.

Table 4: Comparison chart of selected open switches for various methods

new 8 28

2 old 37 33

new 28 7

3 old 36 35

new 32 11

4 old 34 34

new 14 14

5 old 8 36

new 9 32

6 old 33 28

new 7 37

7 old 28 11

new 37 9

Optimum solution 7,9, 14, 7,9, 14, 32,37 32.37

Switching times 7 7

Loss reduction,% 31.148 31.148

G3 61 82 83 SGA or RGA

33 33 33 33 33, 34, 35, 36

7 6 6 6 7,9,14,32

34 35 35 35

9 11 11 11

35 36 36 36

14 31 31 31

36 37

32 28

6

33

7.9.14, 6.11.28. 6,11,31, 6,11,31, 7,9,14,32,37 32,37 31.33 34.37 34,37 4 5 3 3 N/A

31.148 27.830 23.826 23.826 31.148

Note: G1-G3 = Goswami method 1,2,3, respectively; 81-63 = Baran method 1.2.3, respectively.

353

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