a new approach to fault section estimation in power systems using ant system

Electric Power Systems Research 49 (1999) 63–70

A new approach to fault section estimation in power systemsusing Ant system

C.S. Chang *, L. Tian, F.S. WenDepartment of Electrical Engineering, National Uni6ersity of Singapore, 10 Kent Ridge Crescent, Singapore 119260, Singapore

Received 27 May 1998; accepted 8 September 1998

Abstract

In this paper, a new method to fault section estimation problem in power systems is developed using Ant System. Based on theinformation from the operated protective relays and tripped circuit breakers, the fault section estimation is first formulated as acombinatorial optimization problem. Then, a new approach known as the Ant System is applied to solve this optimizationproblem. The Ant System is suitable for solving problems of combinatorial optimization by means of agents that work in parallel,among a population and without a supervisor in a cooperative manner. Finally, a sample power system is used to demonstratethe efficiency of the Ant System based method. From these test results, it appears that the Ant System is of promise for thefault-section estimation problem. © 1999 Elsevier Science S.A. All rights reserved.

Keywords: Fault section estimation; Ant system; Genetic algorithm

1. Introduction

Communications and technologies in power net-works have been much improved. An increasing num-ber of system variables and alarms can now bemonitored and processed. System control is now han-dled by fewer operators. After the fault occurrence,operators would however have to respond to anavalanche of alarm messages and the task can be verychallenging. There are thus incentives for implementingefficient fault-section estimation methods on SCADAsystems for helping operators judge the situation beforestarting corrective procedure.

Fault-section estimation identifies the faulty device(s)in power systems by using the current status informa-tion of protective relays and circuit breakers, which isavailable from SCADA systems. Many methods havebeen adopted to solve this problem. Among these, thelogic based method [1], the expert system approach [2]and the Artificial Neural Network [3] were researchedearlier. Recently, new methods were developed usingthe optimization techniques. The principle behind these

methods is to formulate the fault section estimation asan optimization problem and use a global optimizationmethod such as Boltzmann machines, genetic al-gorithm, simulated annealing and Tabu search to solve.All of them show some advantages and disadvantages.We have done a lot of work on the application ofgenetic algorithm for solving the fault section estima-tion problem [4–7]. The results are satisfied, while insome cases the computational cost is very high. Thismotivates the application of the Ant system [8,9] forsolving this problem.

The Ant system is a kind of natural algorithm in-spired by behaviour or processes presented in nature. Itis derived from the study of ant colonies. The word‘ants’ in the approach can be regarded as agents withvery simple capabilities that, to some extent, mimic thebehaviour of ants. The way by which ants manage toestablish the shortest route paths from their colony tofeeding sources and back has inspired the interest ofecologists. They found that a moving ant lays a kind ofcommunication medium, namely pheromone, thusmaking the path by a trail of this substance. When anant moves at random, it can detect a previously laidtrail and decide with high probability to follow it.Consequently, that trail is reinforced by its own

* Corresponding author. Tel.: +65-87-42109; fax: +65-77-91103;e-mail: [email protected].

0378-7796/99/$ - see front matter © 1999 Elsevier Science S.A. All rights reserved.PII: S 0 3 7 8 -7796 (98 )00127 -8

C.S. Chang et al. / Electric Power Systems Research 41 (1999) 63–7064

pheromone. So the more the ants following a trail, themore attractive that trail becomes for being followed.During a period of time, the shorter path is visited bymore ants, thus the quantity of pheromone on it growsfaster than the longer one. A single ant thereforechooses the path with probability biased toward theshorter one. This process continues until all ants willchoose the shortest path.

Dorigo et al. [8] defined the new computationalparadigm as Ant system since it simulates the processby which ants find the shortest path from their colonyto feeding sources and back. The algorithms they intro-duced are called as the Ant algorithms. The artificialants in the Ant system have different characters withthe real ones. First, they are not completely blind sincethey choose the next destination that constitutes theshortest path. Second, they have some memory, i.e.each ant can remember the places it has just visited.The characteristics of an Ant algorithm are as follows[9]:� it is a natural algorithm since it is based on be-

haviour of ants in establishing paths from theircolony to feeding sources and back;

� it is parallel since all the ants (agents) in the Antsystem move simultaneously, independently andwithout a supervisor;

� it is cooperative since each agent chooses a path onthe basis of the information (pheromone trails) laidby the agents which have previously selected thesame path;

� it is robust since it can be applied with only minimalchanges to other combinatorial optimization prob-lems.The Ant system has been applied successfully in a

variety of optimizaition problems [8,9] that can beexpressed as searching for optimal paths on graphs,such as the travelling salesman problem, quadratic as-signment and job-shop scheduling. The results showthat the Ant System is better than or as effective asmany other heuristic approaches (for example, geneticalgorithm, tabu search and simulated annealing) in thearea of application.

In this paper, we attempt to resolve the fault sectionestimation problem using the Ant system with theobjective for seeking a more efficient method. Themathematical model for the fault section estimation hasalready been set up in our previous work. In using theAnt system to solve this problem, two difficulties needto be overcome. The first one is to find an appropriategraph representation for the problem. Then, the secondone is to set up a relationship between finding theshortest path on the graph and searching the optimiza-tion solution for the fault section estimation problem.As we know, the travelling salesman problem selects allthe cities in the graph to compose the path. On theother hand, the fault section estimation is to select only

some sections from the whole sections. Hence, somemodification needs to be done about the Ant system.

The remainder of this paper is organized as follows.The mathematical model for the fault section estima-tion problem is described in Section 2. Then, in Section3, we present the Ant system and Ant algorithm indetails. In Section 4, the issue of applying the Antsystem for solving the fault section estimation problemis discussed. The paper ends with some test results andconclusions.

2. Mathematical model for the fault section estimationproblem

The fault-section estimation problem is to find themost probable hypothesis that explains the reportedalarm information. That is, when faults occur undersuch a hypothesis, the expected status of breakers andrelays should be consistent with the reported alarms asmuch as possible. Mathematically, it can be formulatedas a 0–1 integer-programming problem as in [5]:

min E(S)= %nr

k=1

�rk−rk*(S)�+ %nc

j=1

�cj−c j*(S, R)� (1)

where: nr is the total number of protective relays; nc isthe total number of circuit breakers; S is an n-dimen-sion vector, and n is the total number of sections in agiven power system. The ith element, si, represents theith section and its state, and si=0 or 1 corresponds toits normal or faulty state. S is a vector to be deter-mined; R is an nr –dimension vector and denotes theactual status of the nr protective relays. The kth ele-ment of R, rk, represents the kth protective relay and itsactual state, and rk=0 or 1 corresponds to its non-op-erated or operated state, respectively; R*(S) is an nr

–dimension vector and denotes the expected status ofthe nr protective relays. The kth element of R*(S),rk*(S), represents the kth protective relay and its ex-pected state. If the kth protective relay should notoperate, then rk*(S) should be 0, otherwise it should be1. R*(S) is dependent on S. C is an nc –dimensionvector and denotes the actual status of the nc circuitbreakers. The jth element of C, cj, represents the jthcircuit breaker and its actual state, and cj=0 or 1corresponds to its closed (non-tripped) or tripped state,respectively. C*(S, R) is an nc –dimension vector anddenotes the expected status of the nc circuit breakers.The jth element of C*(S, R), c j*(S, R), represents the jthcircuit breaker and its expected state. If the jth circuitbreaker should not trip, then c j*(S, R) should be 0,otherwise it should be 1. C*(S, R) is dependent on Sand R.

About how to determine the expected states of circuitbreakers and protective relays, the detailed descriptioncan be found in Appendix A.

C.S. Chang et al. / Electric Power Systems Research 41 (1999) 63–70 65

In genetic algorithms, the fitness function is maxi-mized and positive. The objective function in Eq. (1) isexpressed as the following:

max f(S)=w− %nr

k=1

�rk−rk*(S)�− %nc

j=1

�cj−c j*(S, R)�(2)

where w is a specified very large positive number whichis used to guarantee the fitness function to be positive.w=103 is used in this work. In order to compare thecomputational performance of genetic algorithm andAnt system, we will adopt the objective function ex-pressed in Eq. (2) in the following sections.

3. Ant system model

The Ant system is first applied to the travellingsalesman problem in [8]. In order to facilitate theexplanation, we introduce the Ant system in the contextof the travelling salesman problem.

3.1. Ant system

In Ant System, the travelling salesman problem isexpressed as a graph (N, E), where N is the set of townsand E is the set of edges between towns. The objectiveof the travelling salesman problem is to find the mini-mal length closed tour that visits each town once. Eachant is a simple agent to fulfil the task. It obeys thefollowing rules:1. It chooses the next town with a probability which is

a function of the town distance and of the amountof trail present on the connecting edge;

2. before a tour is completed, it can not choose thealready visited towns;

3. when it completes a tour, it lays a substance calledtrail on each edge (i, j ) visited;

4. it lives in an environment where time is discrete. Itmust choose the next town at time t, and be there attime t+1.

Let m, n be the total number of ants and towns. Aniteration of the Ant system is called, as the m ants allcarry their next moves during time interval (t, t+1).The n iterations constitute a cycle. In one cycle, eachant has completed a tour. Let tij(t) denote the intensityof trail on edge (i, j ). After a cycle, the trail intensity isupdated as:

tij(t+n)=rtij(t)+Dtij (3)

where r is a coefficient such that (1−r) represents theevaporation of trail between time t and t+n,

Dtij=%mk=1 Dt ij

k (4)

where Dt ijk is the quantity per unit of length of trail

substance (pheromone in real ants) laid on edge (i, j ) bythe kth ant between time t and t+n ; it is given by:

Dt ijk=Q/Lk if the kth ant uses edge (i, j )

in its tour between time t and time (t+n)

Dt ijk=0 otherwise (5)

where Q is a constant and Lk is the tour length of thekth ant.

Let dij be the length of the path between town i andtown j. Let 6isibility hij denote the quantity 1/dij. Thetransition probability from town i to town j for the kthant is:

pijk(t)=

[tij(t)]a[hij ]b

%k� allowedk

[tik(t)]a[hik ]bif j�allowedk

p ijk(t)=0 otherwise (6)

where allowedk={unchosen towns for the kth ant). a

and b are parameters that control the relative impor-tance of trail versus visibility.

From Eq. (6), it is obvious that the transition proba-bility is proportional to the visibility and the trailintensity at time t. The visibility shows that the closertowns have a higher probability of being chosen. Themechanism behind this is a greedy constructive heuris-tic. While the trail intensity shows that the more trailon edge (i, j ), the more attractive it is. The process canbe characterized by a positive feedback loop, in whichan ant chooses a path thus reinforces it.

In order to constrain the ants not to visit a previousvisited town, a data structure called the tabu list isassociated with each ant. All the visited towns are savedin it. When an ant finishes a cycle, the tabu list is thenemptied and the ant is free again to choose. Let tabuk

denote the tabu list of the kth ant.

3.2. Ant algorithm

With the definitions in the above, the Ant algorithmcan be simply stated as follows.1. Initialize:

Set NC to 0 (NC is the cycles counter).Set c to the initial trail intensity for every edge.Place the m ants on the n nodes.

2. For k :=1 to m doPlace the starting town for ant k and save it intabuk

3. For k :=1 to m doCalculate p ij

k(t) given in Eq. (6).Choose town j with the highest probability.Move kth ant to town j.Insert town j in tabuk.


Repeat this step until all ants’ tabu lists are full,thus a cycle is finished.

4. Update the trail intensity on all edges according toEqs. (3)–(5).

Update the shortest tour found.Empty all tabu lists.NC:=NC+1.

5. If (NCBNCmax) goto step 2)Else print the shortest tour.

The above algorithm is also called as the ant-cyclealgorithm, because the trail is updated only after a cycleis completed. There are two other algorithms differentin the way the trail is updated. One is the ant-densityalgorithm, the other is the ant-quantity algorithm. Inthe former one, Dt ij

k is defined as:

Dt ijk

=Q if the kth ant uses edge (i, j )

in its tour between time t and t+1


In the latter one, Dt ijk is defined as:

Dt ijk

=Q/dij if the kth ant uses edge (i, j )

in its tour between time t and t+1


Ant-cycle uses global information. That is, it lays anamount of trail proportional to how good the solutionproduced is. However, both the ant-quantity and ant-density use local information. The trail in ant-quantityis independent of the solution (tour length) and inant-density, the trail is only inversely proportional to dij

that is part of the tour.

3.3. Robustness of Ant system

As we have mentioned in Section 1, the Ant system isrobust in solving other combinatorial problems. Dorigoet al. [8] suggested that to apply the Ant algorithm to acombinatorial problem, four processes are required:1. an appropriate graph representation with search by

many simple agents for the problem;2. the autocatalytic (i.e. positive) feedback process;3. the heuristic that allows a constructive definition of

the solutions (which is called as ‘greedy force’);4. the constraint satisfaction method (that is, the tabu

list).In the next section, we will discuss how the Ant

system is applied to the fault section estimation prob-lem according to the above four processes.

4. Ant system based fault section estimation

Fault section estimation can be described as: given aset of sections, expressed as set S, which includes faultyand healthy sections, the problem is to select subset Ffrom S in such a way that if all sections in F are faulty,then the objective function in Eq. (2) takes its maxi-mum value.

To apply the Ant system to the fault section estima-tion problem, we first construct an ‘imaginary’ graph. Itis different from the real Ant system graph whoseobjective function is about the distance between nodes.The ‘imaginary’ graph is a directed graph Q= (S, A),where S is the set of nodes representing sections in apower system, and A is the set of directed arcs. Arc aij

from node i pointing to node j represents that section jis in the protection area of section i. Now, the task ishow to determine the two concepts ‘trail level’ and‘visibility’ defined in the Ant system for the fault sectionestimation problem. The first concept corresponds tothe positive feedback process while the second one tothe ‘greedy force’. Before we define them, we shouldnotice that the connection between two nodes is looserthan that in the travelling salesman problem. Besidesthe relationship set up by the arc set A, there is no closerelationship between sections. Hence, we modify thedefinition of ‘intensity of trail’ for edge (i, j ) as similaras for node (i ). This means that when an ant chooses atown, it lays the trail in the town, not on the path fromthe starting town to its destination.

The Eq. (3) to Eq. (5) are defined again for ourproblem as follows:

ti (t+n)=rti (t)+Dti (9)

Dti= %m

k=1

Dt ik (10)

Dt ik=Q*Fk if the kth ant uses node i in subset F

between time t and t+n

Dt ik=0 otherwise (11)

where Fk is the objective function value of the kth ant.We set the intensity of trail at time 0, ti(0), to a small

positive constant c %. The visibility is defined as:

hij=h1, if section j is protected by section i

hij=h2, otherwise (12)

h1 is a large positive constant, and h2 is small. Forexample, h1� [1, 5], and h2� [0, 1].

The transition probality from node i to node j for thekth ant is:

pijk(t)=

[tj(t)]a[hij ]b

%k�allowedk

[tk(t)]a[hik ]bif j�allowedk


pijk(t)=0 otherwise (13)

Since the trail intensity is a more important factor thanvisibility in deciding whether the node j is chosen, a

should be chosen larger than b.Now, let’s find the constraint ‘tabu list’ for the fault

section estimation. The choosing process for the faultsection estimation should satisfy the following two con-straints:� If a node is chosen with the maximum p in Eq. (13),

it should not be chosen again in the following stepsin one cycle;

� Even if a node pushes itself forward with the maxi-mum p, whether it is selected by subset F or notdepends on whether it can bring a better solution toEq. (2) or not. Let F0 be the set of the current chosennodes. Let F0 %be the set of the chosen nodes includingnew node j. Only when f(F0 %)\= f(F0 ), then node jcan be chosen in F.Compared with the Ant system algorithm for the

travelling salesman problem, the Ant system algorithmfor the fault section estimation consists of two selectionprocesses introduced above. So two data structures areneeded to save the results in these two steps. The firstone is also called as the tabu list, which saves the nodesalready visited according to the transition probalititydefined in Eq. (13). The second one is called as the pathlist, which saves the nodes selected by F.

From Eq. (2), we can deduce the following rules:� If si is a faulty section, then f(S) � si=1\ f(S) � si=

0;� If si,sj are faulty sections, then f(S) � si=1, sj=1\

f(S) � si=1, sj=0With these two rules, we can testify that the two

selection processes can bring out the desired F, that isthe faulty section set.

Thus, the four requirements for Ant System applica-tions mentioned in Section 3 are resolved. There aretwo other things needed to mention: (a) In each cycle,we record the best solution produced by the m ants. Inthe next cycle, if the current best solution is better thanthe former one, then replace the record. If two or moredifferent best solutions exist at the same time, all ofthem should be stored. (b) Basically, the ants number mcan be set to be equal to the nodes number n. At thefirst cycle, each ant is located in a node. To find theoptimal solutions quickly, we can adopt other strategiesin distributing the m ants. For example, it is a goodidea to scatter more ants on the sections whose corre-sponding protective relays have operated.

5. Numerical results

We have used a sample power system [5] as shown inFig. 1 to test the developed method. It consists of 28

sections, 84 protective relays and 40 circuit breakers.The operating logic of protective relays and circuitbreakers are described in the Appendix of Ref. [7], andwill not be presented here to save space.

More than 40 fault scenarios have been tested.Among them, some fault scenarios belong to a single-fault category and the corresponding main protectiverelay(s) have correctly operated to successfully trip theassociated circuit breakers. The others are complicatedfault scenarios consisting of the cases of malfunctionsof protective relays and/or circuit breakers for singlefault cases and multiple fault cases. In accordance withthe operating logic of the protective relays, all theresults are correct. Due to space limitation, only thedetailed results for 10 test cases are shown in Table 1.

Simulation results show that the Ant method is appli-cable in cases with incomplete information of relays, incases with malfunction of relays and/or breakers, andin cases with multiple faults. For example, the informa-tion of protective relays in tests 8, 9 and 10 is incom-plete and correct results are obtained. In tests 1 through7, some malfunction of relays and/or breakers occurs,and correct results are obtained also. The results are assame as that obtained by genetic algorithms, but thecomputational speed has been improved a lot. The bestsolution can be obtained only in a few cycles. Forsimple cases, one cycle is often enough for obtainingthe optimal solution. While for complicated ones, sev-eral cycles (for example 10 cycles) are usually sufficient.This shows that Ant system can solve the fault sectionestimation problem more efficiently than genetic al-gorithm. In the test, the parameters in the Ant systemare chosen as: Q=1, a=5, b=1, r=0.5, C%=0.5,h1=2, h2=0.2, NCMax=10

The number of ants is set to be equal to that ofsections. At the beginning of the algorithm, one ant ispositioned on one section. The stopping criterion is thatthe maximum permitted cycle number has beenreached.

6. Conclusions

In this paper, a new optimization method named Antsystem for solving the fault section estimation problemis proposed. To the best of our knowledge, this workexplores the first application of Ant system for solvinga power system problem. The three characteristics ofthe Ant system, i.e. positive feedback process, greedyconstructive heuristic and distributed computation,work together to find the solutions to optimizationproblems fast and efficiently. The positive feedbackprocess is similar with the reinforcing learning processin Neural Networks. This process leads to very rapidconvergence, and if no limitation mechanism exists, theexplosion may occur. To prevent this, the distributed


Fig. 1. Sample power system.

computation is adopted by starting the search processat different points. The greedy heuristic helps findacceptable solutions in the early stages of search pro-cess.

We have successfully applied the method to thefault section estimation problem. The good perfor-mance of the Ant system in solving this problemdemonstrates its strong ability for those combinatorialproblems that can be defined in terms of searching foroptimal paths on graphs. From this work, we havegood reason to believe that it is possible to applyAnt system to efficiently solve many other comb-inatorial optimization problems in power systems,such as transmission planning and reactive powerplanning.

Appendix A. Determination of the expected states ofprotective relays and circuit breakers

Now, we use a simple example as shown in Fig. 2to illustrate how to determine rk*(S) and c j*(S, R) [6].This system consists of five sections, 15 protective re-lays and six circuit breakers. The five sections (s1–s5)

are A, B, C, L1 and L2, respectively. The 15 protec-tive relays (r1–r15) are Am, Bm, Cm, L1Am, L1Bm,L2Bm, L2Cm, L1Ap, L1Bp, L2Bp, L2Cp, L1As,L1Bs, L2Bs and L2Cs. Here, m, p, s identify mainprotective relays, primary backup protective relaysand secondary backup protective relays, respectively.Am, Bm and Cm are main protective relays of bus-bars A, B and C, respectively, and the other 12 pro-tective relays are all line protective relays. Forexample, L2Bp represents the primary backup protec-tive relay of line L2 at the terminal of busbar B. Theoperating logic of main protective relays and backupprotective relays are listed in Tables 2 and 3, respec-tively. The six circuit breakers (c1–c6) are CB1, CB2,CB3, CB4, CB5 and CB6, respectively.

According to the operating logic of main protectiverelays, it is easy to determine their expected status.For example, according to the operating logic of r1

(i.e. Am), it is known that if a fault occurs on s1 (i.e.A), r1 should operate. Thus, we have:

r*1 (S)=s1 (A.1)

Similarly, we can get:


Table 1Test results of the sample power system

Test number Alarm signals (operated relays and tripped circuit breakers) Estimated faulty sec-tions

B1m, L2Rs and L4Rs; CB4, CB5, CB7, CB9, CB12 and CB27 B11B1, L1B1m, L1Sp and L1Rm; CB4, CB5, CB6, CB7, CB9 and CB112B1, B2, L1 and L23 B1m, B2m, L1Sm, L1Rp, L2Sp and L2Rm; CB4, CB5, CB6, CB7, CB8, CB9, CB10, CB11 and

CB12T3p, L7Sp and L7Rp; CB14, CB16, CB29 and CB39 T3 and L74T5s and T6s; CB22 and CB23 A35

L5, L7, B7, B8, T7,T7m, T8p, B7m, B8m,L5Sm,L5Rp, L6Ss, L7Sp, L7Rm and L8Ss; CB19, CB20, CB29, CB30,6T8CB32, CB33, CB34, CB35,CB36, CB37 and CB39

L1Sm, L1Rp, L2Sp, L2Rp, L7Sp, L7Rm, L8Sm and L8Rm; CB7, CB8, CB11, CB12, CB29, L1, L2, L7 and L87CB30, CB39 and CB40

8 L2Rm; CB8 and CB12 L2B1B1m; CB4, CB5, CB7, CB9,CB12 and CB279

10 L2Rs and L4Rs CB4, CB5, CB7, CB9, CB12 and CB27 1.B1 2.B2

r*2 (S)=s2 (A.2)

r*3 (S)=s3 (A.3)

r*4 (S)=s4 (A.4)

r*5 (S)=s4 (A.5)

r*6 (S)=s5 (A.6)

r*7 (S)=s5 (A.7)

The expected status of backup protective relays ismore difficult to determine than that of the main pro-tective relays. For example, according to the operatinglogic of r8 (L1Ap), it is known that if a fault occurs ons4 (L1) and r4 (L1Am) failed to operate, then r8 shouldoperate. Thus, we have:

r*8 (S)=s4 (1−r4) (A.8)


r*9 (S)=s4 (1−r5) (A.9)

r*10(S)=s5 (1−r6) (A.10)

r*11(S)=s5 (1−r7) (A.11)

r*12(S)=1− [1−s2 (1−c3)] [1−s5(1−c3)(1−c4)](A.12)

r*13(S)=s1 (1−c2) (A.13)

r*14(S)=s3 (1−c5) (A.14)

r*15(S)=1− [1−s2 (1−c4)] [1−s4(1−c3)(1−c4)](A.15)

The expected status of circuit breakers is moredifficult to determine than that of protective relays, andit depends on not only the associated sections’ statusbut also the associated protective relays’ status. More-over, the expected status of most circuit breakers isrelated to two or more protective relays. For example,CB2 is related to Am (r1), L1Am (r4), L1Ap (r8) andL1As (r12).

According to the operating logic of r1 (Am), it isknown that if a fault occurs on s1 (A), then r1 shouldoperate to trip c1 (CB1). Thus, we have:

c*1 (S, R)=s1r1 (A.16)

According to the operating logic of r1 (Am), r4

(L1Am), r8 (L1Ap) and r12 (L1As), we have:

Table 2Operating logic of main protective relays

Operating logicRelay name

Am If a fault occurred on A, then Am shouldoperate to trip CB1 and CB2If a fault occurred on B, then Bm shouldBmoperate to trip CB3 and CB4If a fault occurred on C, then Cm shouldCmoperate to trip CB5 and CB6If a fault occurred on L1, then L1Am shouldL1Amoperate to trip CB2

L1Bm If a fault occurred on L1, then L1Bm shouldoperate to trip CB3

L2Bm If a fault occurred on L2, then L2Bm shouldoperate to trip CB4

L2Cm If a fault occurred on L2, then L2Cm shouldoperate to trip CB5Fig. 2. Simple example used to explain the determination rk*(S) and

cj*(S, R).


Table 3Operating logic of backup protective relays

Relay name Operating logic

L1Ap If a fault occurred on L1 and L1Am did notoperate, then L1Ap should operate to trip CB2If a fault occurred on L1 and L1Bm did notL1Bpoperate, then L1Bp should operate to trip CB3

L2Bp If a fault occurred on L2 and L2Bm did notoperate, then L2Bp should operate to trip CB4

L2Cp If a fault occurred on L2 and L2Cm did notoperate, then L2Cp should operate to trip CB5If a fault occurred on B, and CB3 did notL1Astrip,ORIf a fault occurred on L2, and both CB3and CB4 did not trip, THENL1As should oper-ate to trip CB2

L1Bs If a fault occurred on A, and CB2 did not trip,then L1Bs should operate to trip CB3

L2Bs If a fault occurred on C, and CB5 did not trip,then L2Bs should operate to trip CB4If a fault occurred on B, and CB4 did not trip,L2CsORIf a fault occurred on L1, and both CB3 andCB4 did not trip, THENL2Cs should operate totrip CB5

c*5 (S, R)

=MAX �s3r3, s5r7, s5(1−r7)

r11, {1− [1−s2(1−c4)][1−s4(1−c3)(1−c4)]}r15�(A.20)

c*6 (S, R)=s3r3 (A.21)

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c*2 (S, R)

=MAX�s1r1, s4r4, s4(1−r4)

r8, {1− [1−s2(1−c3)] [1−s5(1−c3) (1−c4)]}r12�(A.17)

where the symbol MAX means to extract the greatestvalue from the set.


c*3 (S, R)=MAX �s2r2, s4r5, s4(1−r5)r9, s1(1−c2)r13�(A.18)

c*4 (S, R)=MAX �s2r2, s5r6, s5(1−r6)r10, s3(1−c5)r14�(A.19)

a new approach to fault section estimation in power systems using ant system

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