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E L S E V I E R Electric Power Systems Research 31 (1994) 97-102

Optimal sectionalizer allocation in electric distribution systems by genetic algorithm

Gregory Levitin, Shmuel Mazal-Tov, David Elmakis Reliability Department, Research and Development Division, Israel Electric' Corporation Ltd., Haifa, Israel

Received 1 February 1994; accepted 9 May 1994

Abstract

This paper presents an economics based model of sectionalizer allocation in single radial feeder distribution systems. The model considers both cost of energy losses and capital investment in the sectionalizer installation. The cases when sectionalizers are not fully reliable and when they may cause additional short-circuits are investigated. To solve the problem of optimal sectionalizer allocation a genetic algorithm based procedure is developed. An illustrative example is presented.

Keywords: Distribution network reliability; Sectionalizer allocation; Genetic algorithm

I. Introduction

To improve customer services, reliability has become the focus of electrical industries. Customer failure statistics show that the distribution systems make the greatest individual contribution to the unavailability of supply to a customer [1]. Therefore considerable effort has been devoted to the reduction of unsupplied energy in distribution networks [2, 3]. The single radial feeder distribution systems consist of series of lines and cables connecting any load point to the single supply source. These lines and cables are divided into sections by connection nodes and load points. Each section has its individual failure rate which depends on the type of line construction, section length and environmental condi- tions. In practice it is normally found that the failure rates are approximately proport ional to the length of the section.

Short-circuits in each system component cause the main feeder breaker to operate. I f there are no points where part of the system can be isolated then each failure must be repaired prior to closing the breaker. In this case no customers are supplied during the repair time. Sectionalizers are installed in order to reduce the damage caused by failures and to improve the system reliability. The sectionalizers isolate the failed subsys- tem while the rest of the system is supplied normally. When an interruption occurs, the outage duration for an individual customer depends on its location relative to the failure and to the sectionalizers. The allocation of

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sectionalizers effects the distribution system's reliability. Since economic factors limit the number of sectionaliz- ers, their optimal allocation will lead to the greatest possible improvement in the reliability of the system.

In this paper an economics based criterion for sec- tionalizer allocation is used which considers both the cost of unsupplied energy and capital investment in the installation of the sectionalizers. A mathematical model for the cost of losses is developed for a single radial feeder distribution system. The model considers the cases when sectionalizers are not fully reliable and when they may cause short-circuits in sections where they are installed. A genetic algorithm based procedure is used to achieve the optimal allocation of sectionalizers.

2. Mathematical formulation

The radial distribution system may be represented by a tree graph G in which nodes oi (0 ~< i ~< N) corre- spond to connection or load points. Since in such tree graphs each edge has a unique end node, the edge (oi, oj) may be denoted as j. Edge j corresponds to the distribution system section j. Let s(i) be the immediate predecessor of edge i if the graph G contains edge (a~o, oi). One can see that isolation of a section k also causes isolation of those sections i for which k = s(i).

Let us define the cost Zk of energy supply interrup- tion caused by the disconnection of arbitrary section k for one hour. For each k ~ L, where L is a set of loaded

9 8 G. Levitin et al./Electric Power Systems Research 31 (1994) 97-102

sections, that is, sections directly connected to the load transformers (note that, for k~L, sections i for which s(i) = k do not exist):

Zk =PkCe

where Pk is the average load at node k and Ce is the cost of unsupplied energy (per kwh). For a section k which does not belong to the set L,

Z k = ~ Zi (1) s(i) k

Using this definition, Zk can be determined for all 0 ~< k ~< N using the following simple procedure:

(1) For each k (0~<k ~<N) set Z~ =0 . (2) For each i ~ L set Zt=PtC~; add Zt to all Zj,

j ~ S~, where St is the set of predecessors of section j:

S~ = {j(0), j ( 1 ) , . . . ,j(n)}

j(O) = s(i), j(k) = s(j(k - 1))

for 1 ~< k ~< n, where j(n) = 0 is the root section. It should be noted that the cost of the total system

isolation is

z0= Z k e L

Suppose that sectionalizers are already allocated in the radial distribution system. For each section i we can find the closest protected predecessor (CPP), that is, the closest predecessor with sectionalizer installed, D(i) ~ {St, i}. Actually, D(i) is the number of the section containing the point of isolation during the repair process in the section i. If the fault occurs in a section j with sectionalizer installed, then D(j) =j. A short-cir- cuit in section i causes the main breaker to isolate the whole system. After some time T s (detection of the fault and switching) the sectionalizer in section D(i) is opened and the breaker reclosed. During the repair time rt section D(i) will be disconnected.

Now we can define the cost of energy not supplied due to faults which occurred in section k during a year:

C k = ~ ' k ( T s E Z i + r k Z D ( k ) ) = O ~ k + f l k Z D ( k ) i c L

where 2k is the failure rate in section k, r k the repair time in section k,

~k = & T~ y, Zt = X~ T~Zo tEL

is the cost of losses caused by total isolation, and

3kZD(k) = 2krkZD~k)

is the cost of losses caused by local isolation. For the whole distribution system the annual cost of

unsupplied energy is N N N

k = l k = l k = l

If the number of sectionalizers is given, our objective is to find the distribution function D minimizing C(D):

D = a r g { C(D)= ~ ~k+ ~ ~=,

As £ ~k does not depend on D, only the functional Z fi~ZD(k) should be minimized.

For an arbitrary number of sectionalizers, M, the cost of the sectionalizer units should be considered as well. The objective function in this case is

N

( ( D , M ) = ~, flkZD(k) + Mh (2) k=l

where h is the annual cost of a sectionalizer. Let us now estimate the effect of sectionalizer addi-

tion to section k assuming the initial allocation (D, M) is given. To do this we introduce an auxiliary function d(i) which gives the CPP number for section i, assum- ing that the sectionalizer in section i is removed (if in section i a sectionalizer was not installed, d(i) = D(i)). d(0) = 0 by definition.

The fault in each section j after the new sectionalizer addition (D( j )= i) now causes losses Zt per hour, whereas before this addition it caused losses Zd( o per hour. Therefore

A C ( D , M , i ) = ( Z i - Z a u ) ) ~ f l j+h D ( j ) = i

We have now a criterion for sectionalizer addition: the addition to section i is worthwhile if AC < 0, that is

h - - + Zt < Za(o

D ( j ) = t

For all the load sections i eL ,

D( j) = i

and the previous expression may be rewritten as

h + Zt < Zdco

2.1. Effect of sectionalizer operation failures

The sectionalizers occasionally fail to operate. If the sectionalizer in section i cannot be opened it is up to the sectionalizer in section d(i) to open and isolate the relevant subsystem. Let p be the probability of section- alizer failure. Following a short-circuit in section k the sectionalizer in section D(k) isolates the subsystem with probability 1 - p , the sectionalizer in section d(D(k)) with probability p ( 1 - p ) , etc. The cost of unsupplied energy caused by the failure in section k is

ck = ak + fl~ [( 1 - p)Zo(k) q- p ( 1 - p)Za(o(k) )

+p2(1 --p)Za(a(D(km + • • .]

G. Levitin et al./Electric Power Systems Research 31 (1994) 97-102 99

Assuming that p is small enough, we can neglect all elements multiplied by pn for n > 1. Doing so, we will get

Ck = ~, + fig[(1 - -p)Zn(k) q-pZd(D(k))]

As in (2) we can estimate C(D, M):

N

( (D, M ) = Y. fl,[(1--P)ZD(o +PZd(D(O)] + M h i = l

To estimate the effect of sectionalizer addition to section i, we have to compare the losses caused by a one hour failure in section j ( D ( j ) = i) before the addition,

Col d = ( 1 - - p ) Z d ( i ) ~- pZd (d ( i ) )

and after the addition,

Cn~w = ( 1 - p ) Z i + pZd(i)

The total cost difference after the sectionalizer addition is

A ( ( D , M , i ) = ( e ~ w - - C o l d ) ~ f l j + h D(j) -- i

The condition for sectionalizer addition derived from this expression is

h V - - ~ _ + ( 1 - p)Z~ < ( 1 - 2p)Zd(i) q- pZd(d(i) ) (3) A_a r-J

D( j ) = i

2.2. E f fec t o f shor t -c ircui ts caused by sect ional izer fa i lures

The sectionalizer itself may cause the short-circuit in its section. To estimate this effect we introduce the failure rate /~ and repair time z for sectionalizers. The short-circuits caused by sectionalizers lead to an in- crease in the total isolation time as well as to an increase in the failure rate of the protected sections. Note that in the case of a fault caused by a sectionalizer in section i, it can be isolated only by the sectionalizer in section d(i ) (not D(i ) because the sectionalizer in section i has failed). The annual cost of losses is now

N

C(D, M ) = ~ fli [( 1 - p)ZD(i) + pZd(D(i))] i = 1

N

4- ~ 7[(1 --P)Zd~i) +pZd(d(i))]6i i = l

+ M ( h + #TsZo) N

= Z {(~ifli[( 1 - -P)Zi -[ -PZd( i )] i = 1

+ Ei [( 1 - p)Za( i ) + pZ,~( , . , )] }

+ M ( h + #T~Zo) (4)

where

7 ----i tz

j ' l if there is a sectionalizer in section i 6i 10 otherwise

E i ~- (~i~) ~ - ( 1 - - ~i ) f l i

It can be shown that the addition of a sectionalizer to section i causes the following differences in C:

AC(D, M, i) = [( 1 - p ) Z i - ( 1 - 2p)Zdt o -- pZdtd(O) ]

X Z flJ AV •( 1 - - p ) Z d ( i ) -'~ •pZd(d(i) ) D(j) = i

+ h + u T s Z o

The criterion for sectionalizer addition is therefore

h + p T s Z o + ( 1 - - p ) Z i O i

< [B~ - ~ - - p ( 2 B , - - ~,)]Z,~(i~ + (B, - - ~ ' )pZd(~ . . ( 5 )

where

B i = Z ~ j D(j) = i

For the loaded sections B i = fli. The expression then takes the form

• i < ~) iZd( i ) ~- LlAiZd(d(i)) (6)

where

Oi = h + #T~Zo + (1 -- p ) Z , fl,

6), = fl, - 7 - p ( 2fl, - 7)

q'i = (fie -- ~')P

do not depend on the sectionalizer allocation D. Assuming/~ = 0 we get expression (4), and assuming

= p = 0 we get expression (3), as special cases of the general expression (5).

3. Solution method

In this paper a genetic algorithm (GA) based method for optimal sectionalizer allocation is suggested. Origin- ally inspired by biological genetics, this search technique has been applied in various areas [4-6] such as optimal reconfiguration of distribution networks, optimal capac- itor placement in distribution systems, load flow prob- lem solving, etc.

Unlike various constructive optimization procedures which use sophisticated methods to obtain a good single solution, the GA deals with a set of solutions and tends to manipulate each solution in the simplest way. There is a fairly rich literature on genetic algorithms [7]. However, in order to follow the algorithm developed in this paper, the following basic steps of the G EN ITOR version of the GA [8] (which, as reported in Ref. [9], outperforms its basic version), are given below.

100 G. Levitin et al . / Electric Power Systems Research 31 (1994) 97 102

Step 1. Generate an initial population of randomly constructed solutions (structures). In this work the number of solutions (population size), Ns=50, is specified.

Step 2. Select two solutions randomly and generate a new solution (offspring) using a crossover procedure providing inheritance of some basic properties of the parent structures in the offsprings.

Step 3. Allow the offsprings to mutate with pro- bability Pm, which results in slight changes in the offspring structure and maintains diversity of solutions.

Step 4. Decode offsprings to obtain the objective function (fitness) values.

Step 5. Use a selection procedure which includes comparison of offsprings with the worst solution in the population, and their replacement if the new ones are better. If, as a result of selection, the population contains equal solutions, the redundant one is removed and the population size is decreased.

Step 6. Generate new randomly constructed solu- tions to fill up the population after repeating Steps 2-5 Nre p times (or until the population contains a single solution or solutions with equal quality). Run the new genetic cycle (returning to Step 2).

Step 7. Terminate the GA after Arc genetic cycles. The final population contains the best solution

achieved. It also contains different near-optimal solutions which may be of interest. It is specially important when additional criteria are considered in the decision-making process.

To apply the genetic algorithm to a specific problem one has to define the solution representation as well as the crossover operation and the mutation and decoding procedures. Usually a GA deals with binary strings. This representation is very suitable for the sectionalizer allocation problem. Indeed, if each bit bi (1 ~< i ~< N) corresponds to 6~ in expression (4), then bi = 1 if section i contains a sectionalizer and bi = 0 otherwise. The arbitrary binary string thus defines the sectionalizer allocation.

The crossover operation for given strings I~, I2 and the offspring string I3 may be defined as follows: first I~ is copied to I3, then all bits bj belonging to the fragment l~<k~<m (l and m are random values, l < m ) are copied to I3 from I2.

The mutation procedure just inverts a random bit in the string.

The simplest decoding procedure, using 6i defined for each section, determines D(i) for 1 ~<i ~<N and computes C(D, M ) = C(6) following expression (4). The GA seeks for the best binary string which minimizes C. To achieve a better GA performance an advanced decoding procedure may be applied. Note that for each given sectionalizer allocation (D, M) one can check whether addition of a sectionalizer to the section i is worthwhile or not using criterion (5). For all

elements i~L (sections connected to load nodes) this criterion depends only on the sectionalizer allocation in sections not belonging to the set L. Therefore the GA may generate binary strings representing N - - N L unloaded sections (NL is the number of load nodes), while for each string (solution) sectionalizers may be allocated optimally in all directly loaded sections. It should be mentioned again that f2i, Oi, and ~i in (6) do not depend on allocation D and therefore may be determined before the GA starts. The reduction of the solution representation vector by NL bits is also very important because it allows the search space to be truncated considerably.

The initial input data for the algorithm suggested are: the structure of the radial distribution system which is represented by pairs i, s(i) for 1 ~< i ~< N; the average load cost Zj for each of the NL loads (in $ per hour); the failure rates for each section, 2~, and for sectionalizers, /~; the estimated repair time for each section, r~, and for sectionalizers, r; the switching time T s, the probability of sectionalizer failure, p, and the annual cost of the sectionalizer unit, h.

In its initial stage the algorithm calculates 7 and flk for section k, Zj for unloaded sections and f2 i, Oi and ~u i for all loaded sections (/eL). It also defines the

General purpose GA • ............................................................................

INITIAL

GENERATION

1 population of ]

solutions \ CROSSOVER

MUT!TION

m ~

new solution I

t ~ SELECTION '~

Problem specif ic

software ,. ................................ ,

c~,fl,Z,fI,O,~

DETERMINATION

[ OPTIMAL

DISCONNECTS

ALLOCATION

FOR ALL

LOADED

SECTIONS

i i

- - i ESTIMATION

Fig. 1. The basic structure of the genetic algorithm based procedure.

G. Levitin et al./Electric Power Systems Research 31 (1994) 97 102 101

length o f the binary string representing solutions in the G A as N - NL. After the G A is started, the opt imal allocation o f sectionalizers in the loaded sections is defined for each solution, and the objective funct ion C ( D , M ) is estimated. The G A provides the minimal (or near-minimal) C to be found. The basic structure o f the a lgor i thm suggested is displayed in Fig. 1.

4. Illustrative example

A simple distribution system with N = 2 5 and NL = 15 is presented in Fig. 2. The tree graph representation o f this system is shown in Fig. 3. For the given system, the set o f load sections is L = {5, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 25}. Table 1 contains (i, s ( i ) ) pairs, fli and Zi for each section. Note that, for i not belonging to L, the Z/ are calculated using (1). Three solutions (D, M) are obtained for

~ ) ( ~ 18 19 (~) \ /

1,II,III ( ~ 17 y 20 1I'lI'Ill

I 24~.~II,1II

(~) 11

(1I) 5

( D lO

l,II

Fig. 2. A simple distribution system.

2 0

3 8

5 11 12 13 23 25 21 18 19 14 15 16

Fig. 3. Tree graph representation of the distribution system.

Table 1 Distribution system data

i s(i) Z i fli

1 0 9100 0.040 2 1 6700 0.060 3 2 5000 0.160 4 3 1000 0.240 5 4 300 0.140 6 1 2400 0.195 7 8 2300 0.155 8 0 2900 0.015 9 3 3900 0.060

10 3 100 0.235 11 4 700 0.140 12 9 2300 0.015 13 9 1600 0.120 14 8 600 0.060 15 7 2000 0.080 16 7 300 0.100 17 6 800 0.220 18 6 300 0.100 19 6 600 0.175 20 6 700 0.100 21 24 400 0.135 22 22 1300 0.195 23 24 500 0.195 24 2 1700 0.045 25 22 800 0.055

Table 2 Sectionalizer allocation solutions

Solution h 7 p M no. ($) (hour) ($)

I 1000 0.02 0.000 6 14869 II 1000 0.10 0.010 5 19544

III 1000 0.25 0.005 3 27007

different system parameters p and ~ (T s = 0.03, z =1 .0) . Parameters values as well as the resulting M and C are presented in Table 2. The optimal sectionalizer alloca- tion is displayed in Fig 2. The sections where isolation units should be installed are marked by an a r row and the number o f the corresponding solution.

5. Computational results

The following G A parameters were chosen to solve the sectionalizer allocation problem: muta t ion probabil i ty pm = l; number o f crossovers per cycle Nrep= 1000; number o f genetic cycles Arc = 5. The algori thm was applied to a real problem with N = 96 and NL = 52 (a rural distribution network). Fig. 4 displays the behavior o f the G A during the optimal solutions search for this problem. The C criterion for the best solution in the popula t ion (BST) decreased constant ly during the genetic search. The average o f the criterion values over all the

102 G. Levitin et al./Electric Power Systems Research 31 (1994) 97 102

95000

90000

85000

:'~ 80000

I~ v5ooo

70000

65000

80000 5;0 ,ooo 15;o 20'00 N o o f c r o s s o v e r s

Fig. 4. Solution behavior in the genetic algorithm: - - AVG.

i I t

"\.\

25'00 3000

, BST; . . . . . ,

solutions in the population (AVG) has peaks caused by injection of new randomly generated solutions at the beginning of each genetic cycle.

The running times for the C language realization of the suggested algorithm on a DEC station 5000/240 for problems with N = 25 and N = 96 are 4s and 7 s of CPU time, respectively.

6. Conclusions

A mathematical model of the energy supply interruption cost for a single radial feeder distribution system is presented. The model allows the effect of different sectionalizer allocations on the total cost of

losses to be evaluated. The cases where sectionalizers are not completely reliable and where they may cause additional short-circuits are considered. The genetic algorithm is used to find the economically optimal allocations. The nature of genetic algorithms provides the possibility to operate with multiple solutions and allows additional criteria to be considered. Future research will aim to aggregate the additional reliability criteria for complex optimization at the stage of distribution system planning.

References

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[2] S. Civanlar, J.J. Grainger, H. Yin and S.S.H. Lee, Distribution feeder reconfiguration for loss reduction, IEEE Trans. Power Delivery, 3 (1988) 1217 1223.

[3] L.H. Tsai, Network configuration to enhance reliability of electric distribution systems, Electr. Power Syst. Res., 27 (1993) 135-140.

[4] K. Nara, A. Shiose, M. Kitagawa and T. Ishihara, Implementa- tion of genetic algorithm for distribution systems loss minimum re-configuration, IEEE Trans. Power Syst., 7 (1992) 1044-1051.

[5] G. Boone and H.D. Chiang, Optimal capacitor placement in distribution systems by genetic algorithm, Electr. Power Energy Syst., 15(1993) 155 162.

[6] X. Yin and N. Germay, Investigations on solving the load flow problem by genetic algorithms, Electr. Power Syst. Res., 22 ( 1991 ) 151 163.

[7] D. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning, Addison-Wesley, Reading, MA, 1989.

[8] D. Whitley and J. Kauth, GENITOR: a different genetic al- gorithm, Tech. Rep. CS-88-I01, Colorado State University, Fort Collins, CO, 1988.

[9] G. Syswerda, A study of reprodution in generational and steady- state genetic algorithms, Foundations of Genetic Algorithms, Mor- gan Kauffman, San Mateo, CA, 1991.