Reliability modeling and evaluation of sulaimani erbil electrical power system

Abstract-The work presented in this paper deals with the reliability evaluation of Sulaimani-Erbil electrical Power System by two different techniques, minimal cut set and disjoint tec

Abstract-The work presented in this paper deals with the reliability

evaluation of Sulaimani-Erbil electrical Power System by two

different techniques, minimal cut set and disjoint technique

Computer program is written in Basic language in order to

implement tie set and cut set algorithms for the evaluation of the

unreliability index of the power network

Keywords-Reliability Modeling, Reliability indices, Graph Theory,

Disjoint Technique

I INTRODUCTION ower system is always consists of a large number of

components which are interconnected in some

purposeful way The reliability of a power system

depends on the reliability of its components as well as its

configuration In system reliability studies, the goal is to

predict suitable reliability indices for the system based on the

component failure data and system design [1] A complete and

accurate reliability model should be able to represent the

variation characteristics of the system interested for all aspects

of performance Selection of the actual form and the type of the

reliability model depends upon a large number of factors which

should be carefully examined during the formulation process of

the reliability analysts The first factor which influences the

selection of the reliability model is the system functional

arrangement and the second factor arises from the types of

variation which may take place in the performance aspects of

the various elements of the system [2].In general, the system is

of two types (depending on the structure of the system), a

simple structure system and a complex structure system

The purpose of investigating the reliability for the area of

Sulaimani-Erbil electrical power system is that: i) This system

has been operated for more than 10 years as a split network due

to the political and economical sanction against the last regime

of Iraq in the Kurdistan region ii) UN agencies were

responsible to develop the network due to 986 UN resolution of

oil for food program, and the agency UNDP was in charge of a

large number of rehabilitation program and iii) It was necessary

to identify the system maintenance requirements and to specify

the weak points of the network and to list their priorities

II DESCRIPTION OF SULAIMANI-ERBIL ELECTRICAL

POWER SYSTEM Prior to about 1990, there were twelve 132 kV tie lines to the

region from the other Governorates of Iraq Now, there are

only two 132 kV circuits which connect Dohuk Governorate to

Mosul region At present, there is no any connection to Erbil

and Sulaimany Governorates from the national grid [3] The energy supply to all three Governorates is restricted due to the shortages of power supply and even the available supply is not reliable due to the present network situation The power network of Governorates Erbil and Sulaimani had been cut off from the national grid The power supply of the two Governorates had to rely on the two hydropower stations at Derbandikhan and Dukan, located in the Sulaimani region with the only one 132 kV line connecting Dohuk with the original national grid However, the electrical power supply of this line was also limited, infrequent and unreliable The capacities of the power stations installed in Dokan and Derbandikhan are (5

x 80 MW) and (3 x 83 MW) respectively, which are insufficient to meet the power demand [4] To improve the condition of power supply for these three governorates, a 29

MW Diesel power plant was installed in each governorate (Sulaimany, Erbil and Dohuk) [5]

The Sulaimani-Erbil 132 kV transmission system consists of 8

lines whose length varies between 25 and 99 km

III TIE SET AND CUT SET METHOD

A tie set of a network is a subset of edges (representing

components) that constitutes a path from input to output If all the components of the tie set operate, the overall system operates properly If no node is passed through by more than one time when tracing the tie set, such a tie set is called the minimal tie set In other words, if any one of the components of

a given minimal tie set is removed, the remaining set is no longer a tie set A cut set is a subset of system components which, when failed, causes failure of a system In terms of a reliability network, the definition can be interpreted as a set of components which must fail in order to disrupt all paths between the input and output of the reliability network The system reliability can be determined from the tie set and the cut set but the cut set method is more powerful than the tie set method in evaluating the reliability of a system for the following two reasons [6-7]:

1) It can be easily programmed with digital computer for the fast and efficient solution of any general network

2) The cut set is directly related to the modes of system failure and therefore it is easy to identify the distinct and discrete ways in which a system may fail

The minimum subset of any given set of components which can cause system failure is known as a minimal cut set A minimal cut set is a set of system components which, when failed, cause failure of the system but when none of the component of the set fails, the system will not be failure [7]

Asso R Majeed, Ghamgeen Izat Rashed, S.J.Cheng, Senior Member, IEEE

Reliability Modeling and Evaluation of Sulaimani – Erbil Electrical Power System

P

1-4244-0493-2/06/$20.00 ©2006 IEEE.

Table 1 Networks information

No of cut sets

st or

nd or

rd or

3 Fig.4 12 20 164 150 0 2 39 0.999797 0.000203

The following definition of minimal cut set is also appropriate:

If {A} is a cut set and no subset of {A} forms a cut set, then

{A} is a minimal cut set [8]

IV APPLICATION OF MINIMAL CUT SET

The following example is used to illustrate the algorithm

which can be used to obtain the minimal cut sets

Considering the bridge type network shown in Fig.(1-a)[7],the

minimal tie set is made up of

components: X1X3, X2X4, X1X5X4and X2X5X3 , as

shown in Fig.(1-b), which means that the minimal tie set can be

represented by Eqn.1:

) 1 (

)

( ) (

) (

)

( X1 X3 X2 X4 X1 X5 X4 X2 X5 X3

Similarly, the minimal cut set is made up of components,

3 5 2 4

5 1 4

3

2

Fig.(1-C) Thus, it can be represented by Eqn.2

) 2 ( )

( ) (

) (

)

( X1 X2 X3 X4 X1 X5 X4 X2 X5 X3

Therefore, employing the minimum-cut-set method, the

unreliability of the system is represented by Eqn,3:

) ( )

(

) ( ) ( ) ( ) ( ) (

3 1 2

1

4 3 2 1 4 3 2

1

E E P E

E

P

E P E P E P E P E E E

E

P

Q

∩

−

∩

−

+ + +

=

∪

∪

∪

=

) (

) (

) (

)

( E1 E4 P E2 E3 P E2 E4 P E3 E4

−

) (

) (

) ( E1 E2 E3 P E1 E2 E4 P E1 E3 E4

+

) 3 ) (

( ) ( E2 E3 E4 P E1 E2 E3 E4

+

where

2 1

1)

4 3

2)

4 5 1

3)

3 5 2

4)

5 4 2 1 4 3 2 1 3 5 2 4 5 1 4

3

2

X

Q = ′ ′ + ′ ′ + ′ ′ ′ + ′ ′ ′ − ′ ′ ′ ′ − ′ ′ ′ ′

) 4 (

2 1 2 3 4 5

4 3 5 2 4 3 5 1

5

3

2

X ′ ′ ′ ′ − ′ ′ ′ ′ − ′ ′ ′ ′ + ′ ′ ′ ′ ′

−

From this example, it is able to describe the algorithm used to

form the minimal cut set as follows:

1) Deduce all minimal paths

2) Construct an incidence matrix that identifies all component

in each path

3) If all elements of any column of the incidence matrix is

non-zero, the component associated with that column

forms a first order cut

4) Combine two columns to form a second order cut

Elimination any cut containing first order cuts to give the

second order minimal cuts

5) Repeat step (4) with three columns at a time to give third

order cuts and to eliminate any cuts containing first and

second order cuts; and

6) Continue this procedure until maximum order of cut has

been reached

Only the first, the second and the third order cut sets are

considered in the current investigation

Two basic approximations are used to deal with the evaluation of power system reliability by the minimal cut set: 1) The first approximation is that Eqn.3 is a precise representation of the minimal cut set However, as this is a very complicated equation, it is approximated by the following

simplified form:

) 5 ) (

( ) ( ) ( ) ( E1 P E2 P E3 P E4 P

For this particular case the following representation can be obtained:

) 6 (

3 5 2 4 5 1 4 3 2

Q

If the terms in the right hand side of Eqn.6 are identical, the system unreliability becomes:

) 7 (

2

2 Q2 Q3

2) The cut sets of high orders are neglected because the probability of their occurrence becomes relatively very small Different networks shown in Fig.2-4 are solved using the above mentioned method The results are given in table 1

5

(a)

(b)

1

2

3

4

1 4 5

2 3 5

E1 E2 E3 E4

(c)

Fig.(1) Reliability block diagrams showing bridge arrangement and its equivalents: a) bridge-type network; b) equivalent minimum-tie set

Read Input Data : 1-Number of Nodes

2 -Number of Branches 3-Incidence and Connection Matrix

Subroutine Program for finding all paths in the system

Subroutine Program for finding minimal paths in the system

Subroutine Program for constructing incidence matrix and from it minimal cutsets finding for the system

Subroutine Program for finding reliability of the system by using approximation method

Printing Results

Fig.5 Program Flowchart for Reliability

V

CONNECTION AND INCIDENCE MATRIX

The connection matrix is defined as an analytic

correspondence of the system configuration and has a size

ofk × k

The incidence matrix identifies all components between any

two nodes

VI SOFTWARE DEVELOPMENT

For the purpose of reliability evaluation, a software package

programmed in BASIC language is developed The flowchart

of the program is shown in Fig.5

The program consists of two parts The first part makes the

qualitative evaluation and second part makes the quantitative

evaluation

In the first part the software package, the following steps are

included:

1-Enter the number of nodes and the number of branches of the

system

2-Enter the connection matrix and the incidence matrix which

can be used to identify each element between two nodes

3-Establish the subprogram for finding all paths in the network

4-Establish the subprogram for finding all minimal paths of the

network from the paths obtained in step 3 by removing all

paths that have a path sub set

5-Construct the incidence matrix which can be used to identify

all components in each path

6-Form the minimal cut set from the incidence matrix obtained

in step 5

In the second part of the software package, the quantitative

steps are performed for reliability evaluation of the system

from the minimal cut set by use of the approximation method

mentioned above

VII Disjoint Technique

In a generalized network, the terminal pair reliability expression is usually derived from the logic diagram of the system by the following two steps [9]:

1) All minimal paths or cut sets are determined

2) The system success / failure function is changed into reliability expression using probability theory, Boolean algebra and graph theory

VIII EXCLUSIVE OPERATOR

Exclusive operator E is a kind of operation of Boolean expression which is defined as follows:

) 8 (

) ( Xi Xi

) 9 ) (

(

) ( ) ( )

( F1F2 Fm E F1 F1E F2 F1F2 Fm1E Fm

) 10 ) (

( )

( ) ( )

( F1 F2 Fm E F1 E F2 E Fm

For a particular case, if Fi = Xi, for all i, the above relationship can be simplified to the following form:

) (

) ( ) ( )

(X1X2 X m E X1 X1E X2 X1X2 X m1E X m

) 11 (

2 1

=

) ( )

( ) ( )

) 12 (

2

X

=

It can be seen from Eqn.11 that all conjunctive terms are mutually disjoint

IX Reliability Evaluation This method makes use of some of the elementary operators

of Boolean algebra The starting point can be either the system –success function or the system-failure function The choice between of these two depends on the number of paths or cut set The method consists in applying exclusive operator

terms being mutually disjoint [9]

The following assumptions are used in this method [9]:

n3

n1 1

2 3 5

6 7

n5

n4

n6

8

9

4

Fig.2 Network No.1

n2

n5 n1

n3

n6

n4

n7

n8 1

2

12

11 9

7

Fig.3 Network No.2

n12

n11

n3

n7 n6

n5

n8 n10

1 3

5 6 7

8 16 15

14 13 19

17

10

9

Fig.4 Network No.3

n2

n3

n4

n5

X4

X3 X1

X2

X5

X7

X6

Fig.6 A general non series parallel network

X1

x4 X3

x2

x5

Fig.7 bridge-type network

1- All nodes are perfectly reliable

2- Each branch of the overall network takes either of the

following two states: good or bad

3-The network is free from self-loops and directed cycles

Steps used for the calculation of the terminal pair reliability

are given below:

1) The system success function is written as:

) 13 (

2

T

where Ti represent the minimal paths of the network

Eq.13 is directly obtained by processing of determining

paths

2) For eachTi , 1 < i ≤ m, Fi is defined to be the union of

all predecessor terms T1, T2, , Ti−1in which any literal

that is presented in both Tiand any of the predecessor

terms is deleted from those predecessor terms, i.e

1 2

1∪ ∪ ∪ −

ofTi → 1 ( 14 )

In fact, the literals of Ti are assigned the Boolean value of 1

and this value is substituted in any predecessor term in which

they occur The resulting function Fi can be simplified by

using standard Boolean reduction identities as shown in

Appendix (A)

3) Using Exclusive operatorE, to obtain:

) 15 (

)

( int)

(

2

m

i

iE F T T disjo

=

=

4) All logical variables are changed into their analogue

probability variables to get the reliability expression (all

terms are mutually exclusive)

) 16 (

, int)

S

If source –terminal cut set is used instead of the paths in a

particular system, the system failure function is obtained and

can be processed similarly to derive system unreliability

expression

X Application of Disjoint Technique

Application One:

Consider the general non series parallel network shown in

Fig.6 [9]:

1- The cut set for the above network

is X1X2,X6X7,X4X5X6,X2X3X4,X1X3X5X6,X2X3X5X7,thus the system

unreliability function is give by Eqn 17:

) 17 (

7 5 3 2 6 5 3 1 4 3 2 6 5 4 7 6 2

X

By applying Eqn.14, the definition of exclusive operator and Eqn.15, Fi, E ( Fi) and S (disjo int) can be calculated as

follows:

The representation of the unreliability is given as follows:

+ +

+ +

+ +

= q1q2 q6q7( p1 q1p2) q4q5q6( p7p1 p7p2q1) q2q3q4( p1p6

Q

) 18 ) (

( ) 1 3 5 6 2 4 7 2 3 5 7 1 4 6 6

7 5

1P P q q q q q p p p q q q q p p p

Application Two:

To use system success function for finding the reliability expression of a system, consider the bridge shown in Fig.7:

The minimal paths for this bridge are:

3 5 2 5 4 1 4 2 3

can be expressed as:

3 5 2 5 4 1 4 2 3

X

By applying Eqn.14, the definition of exclusive operator and Eqn.15, Fi,E ( Fi) and S (disjo int) are found as follows: The reliability expression is as follows:

4 1 3 5 2 2 3 5 4 1 3 1 1 3 1 3

1p p p ( q p q ) p p p ( q q ) p p p q q p

If all components are assumed to be identical, the reliability expression is given by the followings:

2 3 2 3 3 2 2

q p q p q p q p p

Assuming the component reliability 0.99 and applying the two methods mentioned above, the system reliability for the solved two examples is obtained and given in table 2

i

) ( i

i E F T

2 1

2 X X

F = ′ ′ X1 ∪X1 ′X2 X6′X7′ (X1∪X1′X2)

7 2 1

3 X X X

F= ′ ′ ∪ ′ X7(X1∪X1′X2) X4 ′X5 ′X6 ′ (X7 (X1 ∪X1 ′X2 ))

)

6 1

F= ′ ∪ ′ ′ ∪ ′ X1X6 ∪X1X5X7X6 ′ X2′X3′X4′ (X1X6∪X1X5X7X6′ )

7 4 2

5 X X X

F = ′ ∪ ′ ∪ ′ X2X4X7 X1′X3′X5′X6′(X2X4X7)

4 6 1

6 X X X

F = ′ ∪ ′ ∪ ′ X1X4X6 X2′X3′X5′X7′(X1X4X6)

i

3 1

2 3

4 1

It can be seen from table 2 that the approximation method

gives the upper bound value of the reliability since the

probability of the intersected events is ignored, while the

disjointed reliability expression gives more accurate value The

error is included in the original starting set of cut set but not in

the quantitative evaluation of the symbolic reliability

expression

Table 2 System reliability for the two solved examples

No Network Approximation

Method

Disjoint Method

1 Fig.6 0.99979798 0.99979801

2 Fig.7 0.99879900 0.99979805

XI Sulaimani-Erbil Electrical Power System Reliability

Evaluation Fig.8 shows the single line diagram of the 132 kV systems

for Sulaimani-Erbil electrical power system

For the purpose of reliability assessment, data were collected

for each transmission line for the period of 6 years [10] With

the relevant data, the reliability indices were found and the

reliability of each 132 kV transmission line is calculated for

two cases:

1-Only forced outages of the line are taken into account

2-Both the forced and scheduled outages of the line are taken

into account

The following assumptions are made:

data

2 The reliability of Dokan-Tasluja 132 kV transmission line

during the period 1996 to 2001 is evaluated in two parts:

from 1996 to 1998 the line is operated with double circuit

and from 1999 to 2001 the line is operated with single circuit

because one of the circuits is energized by 33 KV

3 Reliability of Dokan and Derbandikhan H/P are considered

to be 0.98 and 0.95 respectively [11]

4 Reliability of the 29 MW Diesel power station is assumed to

be 0.9

5 Reliabilities of the 33 kV and 11 kV transmission lines are

assumed to be 0.9

The reliability of each line is given in table 3 and table 4 for

the period 1996-2001

XII Reliability Modeling of the System

A simplified reliability model for regional power system is

shown in Fig.9, in which the following assumptions are made:

1 The line components are modeled as a single block also the

sending and the receiving ends are assumed fully reliable

2 The regional power stations are considered as a separate

blocks

3 All components are unidirectional except the components

that construct ring in the system

The detail of the coding for the component numbers is given in

table 5

XIII Representation of nodes

To represent nodes (branches) in the reliability network model, a general Terminal Numbering Convention (TNC) is used in this paper [12] In this convention the numbering of nodes (branches) begins at the source and continues in such away that the output terminal of each branch (node) is assigned

a number greater than the number assigned for its input terminal, taking further care that each node (branch) is assigned a specific number Using TNC, the first vertex n1

represents the source and the last vertex nkrepresents the sink where kis total number of the nodes

XIV Case Study and Results From the reliability block diagram of regional power system fourteen case studies are investigated The reliability of each case study is evaluated for the period of 1996-2001 for the following two states:

In state one, only the forced outages of the line are take into account and in state two, both the scheduled and forced outages

of the line are take into account

1 Case 1 to case 8 reliability of regional power system evaluated by evaluating minimal paths and cuts of the system from the network modeling and by using the program that is established for this purpose For each case study different S/S assumed to be the output of the system as:

a in case study no.1 Rizgary S/S is take as a sink node because this S/S is the main S/S in Sulaimani governorate and the main tie lines for Sulaimani region connected to this S/S

b in case study no.3 Tasluja S/S is take as a sink node because this S/S supplying Tasluja cement factory and it is considered an important substation for reconnection of the regional system to the national grid

c in case study no.5 Dokan S/S is take as a sink node because it supplies Dokan water pumping station

d in case study no.6 Derbandikhan S/S is take as a sink node because it supplies some factories in this area

e in case study no.7 and 8 Azadi and N.E S/S are taken as a sink node respectively These two S/S are the main substations in Erbil governorate and main tie lines for Erbil governorate connected with these two S/S

2 Case study 9 and 10 reliability of the regional power system evaluated, with 29 MW Diesel power station are taken into account for both Sulaimani and Erbil governorate

3 Case study 11 and 12 reliability of the regional power system

evaluated by disjoint technique and compared with the previous case studies

4 Case study 13 and 14 investigate the indices Annual Average Interruption Rate (AAIR), this indices indicated the expected number of days in a year that the specified outage for a given load point will happen and it’s evaluated from the following relation:

AAIR = Q * 365 = (1-R) * 365

XV Results of case Studies Table 6 shows the reliability for case 1 to 8 that is studied during the

period of 1996-2001 with different types of outages taken into account

Table 7 shows the system reliability for cases 9 and 10 when the 29 MW

Diesel power stations is take into account for both Sulaimani and Erbil

region in the year 2001

Table 8 shows the unreliability for cases 11 and 12 obtained by

deriving a symbolic equation using disjoint technique

Table 9 and table 10 show the results of AAIR evaluation for cases 13

and 14

XVI Conclusions This paper investigates the reliability of power system In the reliability

evaluation, power system is modeled by the (RBD) and two techniques,

cut sets and disjoint, are used

The investigation results show that both the cut sets and the disjoint

techniques can be used to evaluate the reliability of power system The

disjoint technique gives more efficient and accurate solution However,

it is more complex and consequently more time consuming As to the

Sulaimani-Erbil power system, following conclusions are obtained:

1-It is found that the 132 kV transmission line power system that

energized by 33 kV system reduces the reliability of the system

Therefore, in order to improve the reliability of the 132 kV power

systems, these lines must be restored to 132 kV level

2-The Reliability of the system will be increased if the 29 MW diesel

power station is taken into account for Sulaimani-Erbil region

3-As the outage of power plant greatly reduces the reliability of the

power system, it must be carefully programmed

4-The T tied line greatly effects on the reliability of the overall power

system

Table 4 regional 132 kV transmission line reliability data during the period 1996-2001

forced and scheduled outages are take into account Reliability Data (Forced and planning Outages Take into

Account) Calculated Name of the line

1996 1997 1998 1999

Tasluja-Rizgari

Azadi-N.E Tasluja-Azmer Azmer-Rizgari Derbandikhan-Rizgar

Table (4) Continue Reliability Data (Forced and planning Outages Take into Account) Calculated Name of the line

2000 2001

Assumed

Derbandikhan-Rizgari 0.981360049 0.983837519 Derbandikhan-Azmer 0.948315118 0.963759513

Table (5) Component Coding of the system RBD Component

Table 3 regional 132 kV transmission line reliability data during the period

1996-2001 only forced outages are take into account

Calculated Reliability index for all 132 kV Transmission lines

Calculated Name of the line

1996 1997 1998 1999

Tasluja-Rizgari

5

Azadi-N.E

Tasluja-Azmer

Azmer-Rizgari

Derbandikhan-Rizgar

Table 3 Continue

Calculated Reliability index for all

132 kV Transmission lines Calculated Name of the line

2000 2001

Assumed

Derbandikhan-Rizgari 0.994573467 0.990711568

Derbandikhan-Azmer 0.991442775 0.98391172

Table 6 reliability results for case study 1-8 during the period 1996-2001

Reliability Results for each case study obtained from the program ( Only Forced Outages Take into Account)

Years

Case

Study

Numbers

Case

Study

Numbers

Reliability Results for each case study obtained from the program ( Scheduled and Forced Outages Take into

Account)

Table 8 Unreliability and Reliability Results for case study 11 and 12

Case Study

Reliability results from disjoint method

Reliability results from approximation method Only Forced Outages Taken into

Account

1996 0.998748062 0.99874341

1997 0.998822864 0.99881959

1998 0.998857484 0.99885482

1999 0.998764729 0.99875963

2000 0.99885846 0.99885577

11

2001 0.998838939 0.99883592

1996 0.994950121 0.99485034

1997 0.998671602 0.99866533

1998 0.998769628 0.99876583

1999 0.998619351 0.99861377

2000 0.998681822 0.99867743

12

2001 0.997801362 0.99778998

Table 8 Continue

Case Study

Reliability results from disjoint method

Reliability results from approximation method Scheduled and Forced Outages Taken into Account

11

12

Table ( 6)Continue Reliability Results for each case study obtained from the program ( Only Forced Outages Take into Account)

Years

Case Study

Numbers

Case Study

Numbers

Reliability Results for each case study obtained from the program ( Scheduled and Forced Outages Take into Account)

Table 7 Reliability Results For Case Study 9 and 10

Taken into Account)

9 0.99979740

10 0.99935985

Outage Taken into Account)

9 0.99977642

10 0.99878043

Dokan H/P

Dokan S/S

2

Tasluja S/S

Chamchamal S/S

To Kirkuk

Rizgary S/S

5 Azmar S/S

6

Old Kirkuk

3*83 MW

9

10 Kifri S/S

Hamrin H/P

Azadi S/S 11

17

12

N.E S/S

13 18 19

Erbil Park S/S

14

Khalifan S/S

Soran S/S

15 Kalar S/S

To Dibs G/P

Fig.8 Single Line Diagram of 132 kV Power System For Sulaimani-Erbil Region

Table 9 AAIR evaluation for case study 13

Years

Only Forced Outages Taken into Account

Scheduled and Forced Outages Taken into

Account

Table 10 AAIR evaluation for case study 14

Years

Only Forced Outages Taken into Account

Scheduled and Forced Outages Taken into

Account

6

15 14

9

10

18 5

Fig.9 Reliability Block Diagram for Sulaimani-Erbil Electrical Power System

Appendix A BOOLEAN ALGEBRA 1- Commutative Laws:

a) a + b = b + a b) a × b = b × a

2-Distributive Laws:

a)a+ (b×c) = (a+b) × (a+c)

b)a×(b+c)=(a×b)+(a×c)

3-Identity Laws:

a) a + 0 = a b) a ×1 = a

4-Complement Laws:

a) a + a ′ = 1 b) a × a ′ = 0

5-Idempotent Laws:

a) a + a = a b) a × a = a

6-Boundedness Laws:

a) a + 1 = 1 b) a × 0 = 0

7-Absorption Laws:

a) a + ( a × b ) = a b) a × ( a + b ) = a

8-Associative Laws:

a) (a+b)+c=a+(b+c) b) (a×b)×c=a×(b×c)

9-Involution Law: ( a ′ ) ′ = a

10-DeMorgan’s Laws:

a) ( a + b ) ′ = a ′ × b ′ b) ( a × b ) ′ = a ′ + b ′

11-Disjoint set: a + b = a + a ′ b

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[2] A.E.Green and A.J.Bourne “ Reliability Technology”, John

Wiley & Sons Ltd., 1972

[3] Feasibility study on the options for the addition of

generation capacity in the northern governorates of Iraq

Final report, prepared by SMEC Nov 1999

[4] Distribution Construction Manual, Revesion 2:February

2002, Distribution sector UNDP –ENRP

[5] Electricity Network Development plan Sulaimany

governorate Revesion 1: February 2002,UNDP-ENRP,

Distribution sector

[6] T Gönen “ Electric Power Transmission System

Engineering Analysis and Design”, John Wiley and Sons,

1988

[7] R Billinton and R N.Allan “ Reliability Evaluation of

Engineering Systems”, Pitman Advanced Publishing

Program, 1983

[8] G.B.Jasmon and K.W Foong “ A method for evaluating all

the minimal cuts of a graph”, IEEE Transactions on

Reliability, Vol.R-36, No.5, December 1987

[9] S Rai and K.K Aggarwal “ An efficient method for

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[10] G.I.Rashed, A.R.Majeed and S J.cheng “ Determination

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in Sulaimani-Erbil Network”, Asian network for scientific

information, Information Technologu Journal, Vol.(4), No.2, pp.106-113, Jan 2005

[11] T.M.Tahir ”Load forecasting and power system reliability evaluation”, MSC.,Thesis, University of Technology, Electrical Engineering Dept., Nov 1994

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Asso R Majeed,(E-mail: drassomajeed@hotmail.com)

received his Ph.D in electrical engineering from Baghdad university, Iraq Recently he is head of electrical engineering department in Sulaimani university His area is power system reliability

Ghamgeen I Rashed,(E-mail:gh197493@yahoo.com)

received his bachelor degree in electrical engineering from Salahaadin University- Iraq, in 1995, and his M.sc in University of Sulaimani-Iraq in 2003 Recently he is Ph.D student in Huazhong University of Science and Technology, China

Shijie Cheng, senior member IEEE,(E-mail:Sjcheng@hust.edu.cn) Got his ph.D degree

in Canada in 1988 He is a life professor of the Huazhong University of Science and Technology, China In recent years

he has been engaged in the areas of power line communication, intelligent control, stabilization control of power system and superconducting power technology