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IEEE TRANSACTIONS ON RELIABILITY, VOL. 40, NO. 3, 1991 AUGUST

A Computer Program for Generating Power-System Load-Point Minimal Paths

C. 0. A. Awosope

T. 0. Akinbulire University of Lagos, Lagos

University of Lagos, Lagos

Key Words - Minimal path, Minimal cut-set, Power-arm, Algorithm

Reader Aids - Purpose: Practical application, widen the state of art Special math needed for explanations: Network terminology Special math needed to use results: None Results useful to: Power-system reliability engineers and planners

Abstract - In reliability evaluation that uses minimal cut-set theory, the deduction of minimal cut-set orders is crucial and depends on all the identified minimal paths associated with the load- point whose reliability indices are desired. An input-reduction pro- gramming technique that automates this deduction is presented. The technique can be applied to a network of any configuration and finds its greatest application in complex networks with multi- ple inputs. The power-system structure in the form of power-arms (termination busbars, branch and protective devices) is the only initial input data needed. The results obtained, in terms of minimal paths and minimal cut-set orders for all the load-points, using the program on a sample power-system compare well with the existing results. The program is compact, modular, easy to use and can be applied to any complex-system to generate minimal cut-sets of the desired orders.

1. INTRODUCTION

The minimal cut-set method of evaluating load-point reliability indices of a system is one of the best known methods. Its general theory, assumptions, and advantages over other methods have been discussed [l] extensively. The method depends in most cases, on identification of all minimal paths from which minimal cut-sets of any desired order are deduced.

In existing algorithms [2-61, minimal paths are deduced from a list of predecessors for each element of the system. The list of elements together with their predecessors, representing the topology of the network, are fed into the algorithms as in- put data and processed for minimal paths. These algorithms first deduce all cut-sets, from which non-minimal cut-sets are then eliminated. In our Input Reduction Technique (INRET) method, only the minimal cut-sets are deduced from the minimal paths identified, without recourse to determining whether a cut-set is minimal or not. INRET involves 3 steps:

1. Obtaining a suitable reliability diagram from the 1-line diagram of the power-system in a systematic way.

2. Identification of all minimal paths between the desired load-point and the sources.

3. Deduction [5, 131 of minimal cut-sets of any desired order. This is incorporated to show how versatile the INRET is. The minimal cut-set elements can then be combined [5-81 to compute the load-point reliability indices.

INRET, written in Fortran, has been applied to an operating 16-bus power system (figure 4) and results are presented. Several existing algorithms [8-131 can benefit from INRET. Interested readers can contact the authors for further information on the algorithm.

2. ANALYSIS OF INRET DEVELOPMENT

An electric power-system is made up of generating and load buses, protective devices (circuit breakers) and branches (transmission line - overhead or underground, reactors, distributors, transformers). The assemblage of symbols of these power system components constitutes the 1-line diagram of the power-system. The reliability diagram of the system is obtain- ed by giving unique identification numbers to the system com- ponents. This forms the basis of INRET.

In between two busbars, there are two circuit breakers (one on either side of the branch) and a branch. This is the minimum building block of a power-system referred to as the power-arm. Figure 1 shows the arrangement of components in a power-arm. Therefore, a power-system is an assemblage of power-arms. The power-system structure is thus entered into INRET as a set of power-arms. The number of rows in the structural data is equal to the number of power-arms making up the system. In this way, an (NL x 5) matrix of structural data is built where NL is the number of power-arms in the system. An additional column, containing information on the direction of power flow in each branch, is added. Thus, the resulting (NL x 6 ) matrix is the reliability matrix of the system.

B B BR R E

V v A CB CB

BR = Branch

Figure 1. 5-Component Power-Arm Structure of a Power-System

BB = Busbar CB = circuit breaker

The reliability matrix has busbar identification numbers in columns 1 and 5 , protective device identification numbers in columns 2 and 4, branch identification numbers in column 3, and branch power-flow direction codes in column 6. The

0018-9529/91/0800-0302$01.00@1991 IEEE

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AWOSOPE/AKINBULIRE: A COMPUTER PROGRAM FOR GENERATING POWER-SYSTEM LOAD-POINT h " M A L PATHS 303

codes for power-flow directions are used to construct the horizontal-image matrix which is the mirror image of the reliability matrix. The reliability and horizontal-image matrices constitute the system matrix [IA]. The system matrix is pro- cessed for all minimal paths from all sources to the desired load- point (LP) .

All the minimal paths identified are used to construct [6] a minimal-path array using binary digits. The binary digits 1 and 0 signify the presence and absence of a component respec- tively in a minimal path. The size of the minimal-path array (MPA) is (MP x NC) where MP and NC are numbers of minimal paths and system components respectively. The Binary Formulation Algorithm [ 131 of minimal cut-set theory is used to deduce all desired minimal cut-set orders.

INRET consists of a main program in which several subroutines are called. The linkages and possible interactions among these subroutines within the main program are illustrated in figure 2.

1 S t a r t 1

I 1 ,/, MCS

N

7 Any more Loadpoint?

-q Any fnore'$y$tem? I MPA - Minimal Path Array C I A ] - System M a t r i x MCS - Minimal cut -set .

MPA - Minimal Path Array [IA] - System Matrix MCS - Minimal Cut-Set

Figure 2. Program Lay-Out

3. STAGES OF INRET

INRET is developed in 3 stages:

1. Provision of raw and generated data 2. Identification of all minimal paths and construction of

MPA. 3. Deduction of minimal cut-set orders.

Each stage, as it affects program development, is described in detail.

3.1 Provision of Raw and Generated Data.

The starting raw data are fed into the program through the data handling subroutine TOLU. These raw data include the system reliability matrix and the system characteristics: total number of buses (NB), total number of branches (NL), number of generating points, and desired loadpoints.

New data are also generated within this subroutine. These are the horizontal image of the reliability matrix and the total number of components (NC) making up the system. This is com- puted using (1).

NC = NB + 3(NL) (1)

The horizontal image of the reliability matrix is formed as follows. For each row of the reliability matrix, the code in column 6 is checked. If it indicates a unidirectional branch, the image of the row is not formed. If, on the other hand, the branch is bidirectional, the image of the row is formed. The image row formed occupies row R in the system matrix. R is the sum of the total number of branches in the system and the total number of bidirectional branches encountered up to the row of the reliability matrix. In this way, the element of column 1 in the object row occupies column 5 in its image row, that of column 2 occupies column 4, and so on, thus building up the horizon- tal image matrix.

3.2 Identijication of Minimal Paths and Construction of MPA

This most important stage of the program, is made up of subroutines SEGUN, XTEND, AREE.

AREE increases the minimal-path counter by one. The con- tent of the row identified as a successful minimal path and the path number are used to construct the corresponding row in the MPA [6] using the Binary Algorithmic Formulation [13].

XTEND extends the minimal path under construction by the content of a successful row of the system matrix. The con- dition under which a row is successful is discussed later in this section. The last entry of this path is tested. If it is the desired load-point, the minimal path being constructed is a success. AREE is called to work on the path.

SEGUN determines all minimal paths to the desired load- point (LP) from specified generating buses (LG) in the system. To do this, the system matrix and the system characteristics are used. These paths are determined in the following systematic way.

Loop 1 checks the entries in column 1 of the system matrix from the first row to the last row, NL2 (NL2 = twice the number of links in the system less the total number of unidirec- tional ones). Each time it finds a source identification number, it picks up the row of the system matrix as a possible minimal path. The content of the row is transferred into a dummy vec- tor IB which accommodates the elements of the minimal path under construction. If the last entry in IB is the desired load- point, the content of IB is taken as a successful minimal path. This is passed on to AREE for necessary action. If it is not the desired load-point, the image or the 'object' row numbers of the row in the loop is computed from the total number of system

304 IEEE TRANSACTIONS ON RELIABILITY, VOL. 40, NO. 3, 1991 AUGUST

branches and the total number of unidirectional branches en- countered up to that row of the system matrix.

For loop 2 upwards, the continual generation process of the minimal path is similar.

Each counter in the general loop runs from 1 to NL2 of the system matrix. If the counter value is less or equal to NL, then the image row number is computed, otherwise the object row number is computed. The position in IB, of the last entry from each loop (busbar number) is determined from the loop number N. N increases by 1 each time XTEND is called within the cycle of a minimal-path generation process. Lp denotes the elements in columns P of IB:

P = 4 x + l x = 1,2, ...,( N-1) (2)

Lp is the element in column P of IB when x takes its highest value.

In any loop N , row-counter k is increased by 1 (goes to the next row) if any of the following conditions is satisfied:

1. Row k is equal to any of the selected rows, their ‘im- age’ rows or ‘object’ rows as the case may be in the previous

2. Element in column 5 of row k is the specified generating

3. Element in column 5 of row k is not equal to Lp.

If none of these first terminating conditions is satisfied, then XTEND is invoked. Row k goes to the next one if any of the following second conditions is satisfied:

1 . Element Lp is equal to the desired load-point or specified generating bus

2. Element Lp equals any of the elements Lp.

If none of the second terminating conditions is satisfied, the process goes to the next loop. The process returns to the previous loop whenever k > NL2.

The number of loops to use depends on the complexity of the network. INRET goes up to 8 loops, making the maximum number of elements per minimal path to be 33 from (2). If throughout the last loop none of the terminating conditions is satisfied, the situation diverges. The content of IB, the loop counter Nand row k may be printed. The number of loops can be increased with ease.

loops.

bus or the desired load-point.

3.3 Deduction of Minimal Cut-Set Orders

The minimal-path array constructed in section 3.2 is pro- cessed for minimal cut-set orders. There are M components in each minimal cut-set of order M . The highest order considered in this paper is M=3. The deduction of cut-set orders higher than 3 can be deduced by CUT. The general procedure is il- lustrated below. However, the highest order to consider depends on the complexity of the network, capacity of the available com- puter, and the desired accuracy. The elements of order 1, order 2, etc up to the desired order are assembled in the CUT subroutine.

The order 1 minimal cut-sets are first deduced [6] in CUT. Each set of this order contains one component. To obtain the minimal cut-set of this order, the algorithm searches for col- umns of the minimal path array in which every entry is logic digit 1. The components associated with these columns are order 1 minimal cut-sets. In order to prevent the deduction of non- minimal cut-sets in second and higher orders, all columns of the MPA that contain only logic digit 0’s are replaced with col- umns of logic digit 1’s. The deduction of second and higher order minimal cut-sets is similar. The general algorithm for deducing minimal cut-sets of orders up to M is as follows.

For a system of NC components, there are NC columns in MPA, and the number of rows is the number of minimal paths. To deduce all minimal cut sets up to order M , counter L takes values from 2 to M . For each value of L, columns n ( 1 ) , n(2 ) , n (3 ) , ..., n ( L ) are logically combined for n(1) = 1 to [NC-(L-l)] ; n(2) = [n(1)+1] to [NC-(L-2)]; n(3) = [n(2)+1]to[NC-(L-3)]; ...; n ( L ) = [ n ( L - l ) + l ] t o NC. The logic addition is done for all possible combinations of L. If the columns resulting from the logic addition of all possi- ble L columns contain, as elements, only logic digit 1, the com- ponents corresponding to the L columns constitute the elements of order L minimal cut-sets. The minimal cut-set is written as [ n ( l ) , n (2 ) , n (3 ) , ..., n(L)]. This is illustrated for M=3.

Given the highest order of interest as 3, order-1 minimal cut-sets are deduced as already explained. For order-2 ( L = 2) , two columns n ( 1 ) and n (2 ) are logically added for n ( 1 ) = 1 to (NC-1) and n ( 2 ) = n(1) to NC. If the resulting col- umn contains only logic digit 1 ’s, components corresponding to columns n ( 1 ) and n (2) make up an order-2 minimal cut-set [n( l ) , n(2)]. For order-3 (L=3), columns n( l ) , n ( 2 ) , n(3) are logically combined for n ( 1 ) = 1 to (NC - 2) ; n (2) = [n(1)+1] to (NC-l) , and n(3) = [n(2)+1] to NC. Suc- cessful resulting columns of logic digit 1’s confirm that com- ponents corresponding to n ( 1 ) , n (2 ) , n (3) are the elements of a minimal cut-set [ n ( l ) , n (2 ) , n(3)I.

The sequential steps involved in implementing the entire program are listed in section 4. It includes CUT in its generalised form for deducing minimal cut-sets up to order M .

4. ALGORITHM FOR INRET

A. Identification of Minimal Paths.

1. Provide data in form of system reliability matrix and characteristics

2. Using 1, construct the ‘image’ matrix. Augment the system reliability matrix row-wise with its ‘image’ matrix to form the system matrix

3. Check element 1 of row 1. If it is a generating bus, the row is a probable minimal path but if not, go to the next row. This is the first loop. Each loop goes from 1 to NL2.

4. The last element of the probable minimal path is tested. If it is the specified load-point, then the row is a minimal path and the corresponding row in the MPA is constructed [5, 131. Otherwise start the next loop.

I

AWOSOPE/AKINBULIRE: A COMPUTER PROGRAM FOR GENERATING POWER-SYSTEM LOADPOINT MINIMAL PATHS

1 4

0 3

1

7

8 2 Lll Figure 3. One-Line Diagram of NEPA 330kV Grid for 1980/81 Stage

15

2 Jll

SAPELE - G Q Figure 4. NEPA Reliability Network

305

1 6

L

U

Figure 5. Reliability Block Diagram for Bus 5 of Figure 4

5. Compute the ‘image’ row if row of interest I NL, or the object row otherwise.

6. If any of the first terminating conditions is satisfied, go to the next row of the loop. Otherwise, add the elements of this row to the probable minimal path.

7. If any of the second terminating conditions is satisfied, go the the next row of the loop. Otherwise start the next loop.

8. Repeat steps 5 to 7 until a minimal path is formed and the corresponding row in MPA constructed. Each loop is ter- minated when the last row (NL2) has been considered or when a minimal path has been identified. The process then reverses to the immediate previous loop when one is terminated.

B. Deduction of minimal cut-sets.

1. Identify column j of the MPA that contains only 1’s from column 1 to the last, NC. The element corresponding to column j is an order-1 minimal cut-set.

2 . Set columns of only 0’s in the MPA to columns of only 1’s

3. Combine the columns of the MPA logically from 2 to M where M is the numerical value of the highest order of in- terest. If the resulting vector contains only l’s, then the elements corresponding to the columns combined form the elements of the minimal cut-set of order equal to the number of columns combined.

306 IEEE TRANSACTIONS ON RELIABILITY, VOL. 40, NO. 3, 1991 AUGUST

5. CASE STUDY APPENDIX

The INRET algorithm described in section 4 has been coded in Fortran and run on a PC-AT microcomputer MFC-6000 using the WatFor 77 compiler. INRET has been ap- plied to many networks of various complexities. One of the net- works [14], shown in figure 3, is analyzed.

The reliability network of figure 3 is shown in figure 4. It is made up of 2 generating busbars, 14 load-points, and 23 branches which are assumed bidirectional. Each branch is pro- tected with two circuit breakers, one at each end. Figure 5 shows the block diagram of load-point 5 with typical first, second, and third order minimal cut-sets. The load-point reliability indices can be obtained using existing expressions [5-71 if desired. Table 1 shows the results obtained in terms of numbers of minimal paths, minimal cut-sets up to the third, at each load-point while table 2 contains the time taken, in seconds, to accomplish each process. A typical result for load-point 5 is shown in the appendix.

BUSBAR - 1 BUSBAR - 2 BUSBAR - 3 BUSBAR - 4 BUSBAR - 5 BUSBAR - 6 BUSBAR - 7 BUSBAR - 8 BUSBAR - 9 BUSBAR - 10 BUSBAR - 11 BUSBAR - 12 BUSBAR - 13 BUSBAR - 14 BUSBAR - 15 BUSBAR - 16

KAINJI SAPELE

JEBBA OSHOGBO IBADAN

AKANGBA BENIN ONITSHA ABA ENUGU GOMBE JOS KADUNA KANO

BIRNIN-KEBBI

IKEJA-WEST

NO. OF TRANSM. LINKS = 23

NO. OF COMPONENTS = 85 TABLE 1

Number of Minimal Paths and Cut-Sets for NEPA Network TOTAL NO. OF BUSES = 16

MATR,X oIp CODE =

Load Points NO. OF GENERATING BUSES = 2

TOTAL NO. OF NON-GENERATING BUSES = 14 Number of 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Minimal Paths 41 22 14 28 24 48 22 22 22 22 44 44 44 44 Sets in Order-1 5 1 1 1 1 2 1 5 9 9 10 6 2 6 sets inorder-2 0 3 4 11 2 11 3 3 3 3 12 12 12 12 Sets in Order-3 0 48 60 36 36 36 48 48 48 48 48 48 48 48 = = = = = = = = = = = = = = = = = = = = = = = = = = = = SYSTEM RELIABILITY MATRIX

TABLE 2 Time (sec) to Generate Minimal Paths, Construct MPA, and to

Deduce Minimal-Cut Sets for NEPA Network

Load Point M.P. & MPA Orders 1,2,3 MCS Total

3 4 5 6 7 8 9

10 11 12 13 14 15 16

1.20 0.33 0.27 0.82 0.38 1.09 0.32 1.10 1.15 1.16 1.10 1.09 0.88 1.15

8.79 8.90 9.12 7.96 7.63

10.22 7.97 7.91 7.97 7.96

12.20 11.81 11.87 11.87

9.99 9.23 9.39 8.78 8.01

11.31 8.29 9.01 9.12 9.12

13.30 12.90 12.75 13.02

ROW 1: ROW 2: ROW 3: ROW 4: ROW 5: ROW 6: ROW 7: ROW 8: ROW 9: ROW 10: ROW 11: ROW 12: ROW 13: ROW 14: ROW 15: ROW 16: ROW 17: ROW 18: ROW 19: ROW 20: ROW 21: ROW 22: ROW 23:

1 1 1 2 2 4 4 4 4 5 5 5 6 7 7 7 7 9

10 10 13 14 15

17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61

63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85

18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62

3 4 4 9 9 5 5

15 15 6 7 9 7 8 8 9 9

10 11 12 14 15 16

AWOSOPE/AKWBULIRE: A COMPUTER PROGRAM FOR GENERATING POWER-SYSTEM LOADPOINT MINIMAL PATHS 307

HORIZONTAL IMAGE OF SYSTEM RELIABILITY MATRIX

ROW 24: 3 18 63 17 1 ROW 25: 4 20 64 19 1 ROW 26: 4 22 65 21 1 ROW 27: 9 24 66 23 2 ROW 28: 9 26 67 25 2 ROW 29: 5 28 68 27 4 ROW 30: 5 30 69 29 4 ROW 31: 15 32 70 31 4

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ROW 32: ROW 33: ROW 34: ROW 35: ROW 36: ROW 37: ROW 38: ROW 39: ROW 40: ROW 41: ROW 42: ROW 43: ROW 44: ROW 45: ROW 46:

15 6 7 9 7 8 8 9 9

10 11 12 14 15 16

34 36 38 40 42 44 46 48 50 52 54 56 58 60 62

71 72 73 74 75 76 77 78 79 80 81 82 83 84 85

33 35 37 39 41 43 45 47 49 51 53 55 57 59 61

4 5 5 5 6 7 7 7 7 9

10 10 13 14 15

MINIMAL PATHS FOR LOADPOINT 5 - OSHBO GENERATING BUSBAR 1 KAlNJl

GENERATING BUSBAR 2 SAPELE _ _ _ _ _ _ _ _ _ _ - - - - _ _ _ _ _ - - - - - - - - - _ _ _ _ - - - - - - - - - - - - - - - - - - - - - - - - #1: 1 19 64 20 4 27 68 28 5 #2: 1 19 64 20 4 29 69 30 5

#3: 1 21 65 22 4 27 68 28 5 #4: 1 21 65 22 4 29 69 30 5 #5: 2 23 66 24 9 40 74 39 5

#6: 2 23 66 24 9 48 78 47 7 38,

#7: 2 23 66 24 9 48 78 47 7 42,

#8: 2 23 66 24 9 50 79 49 7 38,

#9: 2 23 66 24 9 50 79 49 7 42,

73 37 5

75 41 6 36 72 35 5

73 37 5

75 41 6 36 72 35 5 #lo: 2 25 67 26 9 40 74 39 5

#11: 2 25 67 26 9 48 78 47 7 38, 73 37 5

ORDER-1 MINIMAL CUT-SET: K1 = 1

(5).

ORDER-2 MINIMAL CUT-SET: K2 = 4

(1, 2) (1, 9) (2, 4) (4, 9)

ORDER-3 MINIMAL CUT-SETS: K3 = 60

REFERENCES

[l] C. 0. A. Awe*, “Critical review of the methods of evaluating power- system load-point reliability indices”, 7he Nigerian Engineer, ~0118,1983,

[2] P. A. Jensen, M. Bellmore, “An algorithm to determine the reliability of complex system”, ZEEE Trans. Reliability, vol R-18,1969 Nov, pp 169-174.

[3] J. Endrenyi, P. C. Wnhaut, L. E. Payne, “Reliability evaldonoftransmis- sion systems with switching after faults: Approximations and a computer pro- gram”, ZEEE Trans. Power Appralus & Systems, vol PAS-92,1973 Nov/Dec,

[4] Young H. Kim, Kenneth E. Case, P. M. Ghare, “A method of computing complex system reliability”, ZEEE Trans. Reliability, vol R-21, 1972 Nov,

[5] R. N. Man, R. Billinton, M. F. DeOliverira, “An efficient algorithm for deducing the minimal cuts and reliability indices of a general network”, ZEEE Trans. Reliability, vol R-25, 1976 Oct, pp 226-233.

[6] R. Billinton, M. S. Grover, “Reliability assessment of transmission and distribution schen”’, ZEEE Trans. Power A p p a w & System, vol PAS-94, 1975 May/Jun, pp 724-732.

[7] R. Billinton, M. S. Grover, “Quantitative evaluation of permanent outages indistribution systems”, ZEEE Trans. PowerApparahu & System, vol PAS-94 May/Jun, pp 733-741.

[8] C. Singh, R. Billinton, “A new method to determine the failure frequency of a complex system”, IEEE Trans. Reliability, vol R-23, 1974 Oct, pp

pp 68-79.

pp 1863-1875.

pp 215-219.

231-234.-

evaluation”, ZEEE Trans. Reliability, vol R-24, 1975 Apr, pp 83-85. #’ 2: 25 67 26 48 78 47 42’ [9] K. K. Aggarwal, K. B. Misra, J. S. Gupta, “A fast algorithm for reliability 75 41 6 36 72 35 5 #13: 2 25 67 26 9 50 79 49 7 38, [lo] Suresh Rai, K. K. Aggarwal, “Oncomplementationofpath-setsandcut-

73 37 5 sets”, ZEEE Trans. Reliability, vol R-29, 1980 Jun, pp 139-140. [ l l ] C. Singh, “A cut-set method for reliability evaluation of systems having

sdependent components”, ZEEE Trans. Reliability, vol R-29,1980 Dec, pp #14: 2 25 67 26 9 50 79 49 7 42, 75 41 6 36 72 35 5 372-375.

IEEE TRANSACTIONS ON RELIABILITY, VOL. 40, NO. 3, 1991 AUGUST

R. N. Allan, I. I. Rondiris, D. M. Fryer, “An efficient computational technique for evaluation of the cutltie sets and common-cause failures of complex systems”, ZEEE Trans. Reliabiliry, vol R-30, 1981 Jun, pp 101-109. C. 0. A. Awe%, “A binary algorithmic method of computing load- point reliability indices of electrical power-systems”, The Nigerian Engineers, vol 19, 1984, pp 95-104. “Power-system development plan”, National Electric Power Authority

(NEPA). Fed. Rep. Nigeria, 1982.

AUTHORS

is a corporate member of the Nigerian Society of Engineers and a Registered Electrical Engineer.

T. 0. Akinbulire; Department of Electrical Engineering; University of Lagos; Akoka Lagos, NIGERIA.

T. 0. Akinbulir! is a lecturer in the Department of Electrical Engineer- ing, University of Lagos. He received a BSc (EE) and an MPhil (EE) from University of Lagos in ’84 and ’87 respectively. He is on a PhD programme at the University of Lagos. His research interests are application of computer to, and development of, computational techniques for electrical power systems and reliability studies. He has Dublished Daoers in urofessional ioumals. He

- C. 0. A. A w o m [M ’741 is a Senior Lecturer of Electrical Power Systems and Reliability studies at the University of Lagos. He received a BSc (EE) 1st Class in 1972 and PhD (EE: Power System Reliability) in 1982 from University of Lagos and a MSc from UMIST in 1974. His research and teaching interests are in software reliability modeling, system reliability analysis, Power Systems and Control. He has published several papers in various journals. He

~ ~ ~ ~ ~ ~ r i ~ ~ TR87-725 received 1987 November 28; revised 1989 March 30; revised 1990 M~~ 30; revised 1991 M~~ 11,

IEEE Log Number 42715 4 T R b

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