1

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

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.

order
3
rd
order
Reliability
Unreliability
1 Fig.2 6 9 13 13 0 2 17 0.999797 0.000203
2 Fig.3 8 12 26 26 0 3 35 0.999695 0.000305
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:
3524514231
X and ,, XXXXXXXXX
, as
shown in Fig.(1-b), which means that the minimal tie set can be
represented by Eqn.1:
) 1 .() ()()()(
3524514231
XXXXXXXXXXS ∩∩∪∩∩∪∩∪∩=
Similarly, the minimal cut set is made up of components,

)()(
)()()()()(
3121
43214321
EEPEEP
EPEPEPEPEEEEPQ
∩−∩−
+++=∪∪∪=
)()()()(
43423241
EEPEEPEEPEEP
∩−∩−∩−∩−
)()()(
431421321
EEEPEEEPEEEP
∩∩+∩∩+∩∩+
)3) (()(
4321432
EEEEPEEEP
∩∩∩−∩∩+

where
211
)( XXEP
′
∩
′
=

432

′′′
+
′′′
+
′′
+
′′
=

)4 ( 2
54321435243515321
XXXXXXXXXXXXXXXXX
′′′′′
+
′′′′
−
′′′′
−
′′′′
−
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

+=

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.
1 3
2 4
5(a)

1 3
2 4
1 45
2
5 3
(b)

1
2
3
4
1
4
5

system by using
approxima tion metho d
Printin g Resul ts Fig.5 Program Flowchart for Reliability
Evaluation by using cut set method

V.
CONNECTION AND INCIDENCE MATRIX

The connection matrix is defined as an analytic
correspondence of the system configuration and has a size
of
kk ×
.
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.

expression which is defined as follows:
)8 ()(
ii
XXE
≡

) 9 ) (( )()() (
12121121
mmm
FEFFFFEFFEFFFE
−
∪∪∪≡

)10) (() ()() (
2121 mm
FEFEFEFFFE ≡∪∪∪

For a particular case, if
ii
XF
=
, for all
i
, the above
relationship can be simplified to the following form:
)( )()() (
12121121 mmm
XEXXXXEXXEXXXE
−
∪∪∪=


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
on
)( functionsucesssystemS −−
, which results in all its
terms being mutually disjoint [9].
The following assumptions are used in this method [9]:

4n2
n3
n1
1
2
3
5
6
7
n5
n4

n9 n4
n11
n2
n1
n3
n7
n6
n5
n8
n10
20
12
11
2
1
3
5
6
7
8
16
15
14
13
19
17
10
9

Fig.4 Network No.3

21 m
TTTS ∪∪∪=

where
i
T
represent the minimal paths of the network.
Eq.13 is directly obtained by processing of determining
paths.
2) For each
i
T
,
mi ≤<
1
,
i
F
is defined to be the union of
all predecessor terms
121
, ,,
−i
TTT
in which any literal
that is presented in both
i
T
and any of the predecessor
terms is deleted from those predecessor terms, i.e.

m
i
i
FETTdisjoS
U
=
=

4) All logical variables are changed into their analogue
probability variables to get the reliability expression (all
terms are mutually exclusive).
)16 (,int)(
iiii
qXpXdisjoSR →
′
→=

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

int)(disjoS
can be calculated as
follows:
The representation of the unreliability is given as follows:
++++++=
61432127176542117621
()()( ppqqqqppppqqqpqpqqqqQ

)18) (()
641753274265316751
pppqqqqpppqqqqqPPP ++

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:
3525414231
,,, XXXXXXXXXX
and the system reliability
can be expressed as:
3525414231
XXXXXXXXXXS ∪∪∪=

By applying Eqn.14, the definition of exclusive operator and
Eqn.15,
i
F
,
)(

XXF
′′
=

211
XXX
′
∪

)(
21176
XXXXX
′
∪
′′

7213
XXXF
′
∪
′′
=

)(
2117
XXXX
′
∪

))((

′
∪
′
∪
′
=

742
XXX

)(
7426531
XXXXXXX
′′′′

4616
XXXF
′
∪
′
∪
′
=

641
XXX

)(
6417532
XXXXXXX

XXF ∪=

23
XX
′′

)(
23541
XXXXX
′′

414
XXF ∪=

41
XX
′′

)(
41352
XXXXX
′′

5
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


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

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:


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
Dokan-asluja
0.99853075 0.99993133 0.999996942 0.994440639
Tasluja-Rizgari
Dokan-Azadi
0.918537492 0.965886606 0.976601979 0.965865677
Dokan-N.E.
0.917067016 0.912359209 0.962621766 0.944446347
Azadi-N.E.
Tasluja-Azmer
Azmer-Rizgari
Derbandikhan-Rizgar
i
0.970818154 0.968302892 0.976721842 0.945987443
Derbandikhan-Azmer
0.96340126 0.778076484 0.88245624 0.961244292

Table (4) Continue
Reliability Data (Forced and planning
Outages Take into Account)
Calculated

Name of the line
2000 2001

16 Derbandikhan H/P
17 Derbandikhan S/S
18
Derbandikhan –Rizgari 132 KV Transmission line
19 29 MW Diesel Power Station
20 11 KV Transmission Line
21 29 MW Diesel Power Station
22 Industrial S/S
23 33 KV Transmission Line

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
Dokan-asluja
0.998776941 0.999999536 0.999999965 0.997707382
Tasluja-Rizgari
Dokan-Azadi
0.959333257 0.98684551 0.993571157
0.99865106
5
Dokan-N.E.
0.955166591 0.995681126 0.996767504 0.99434551
Azadi-N.E.
Tasluja-Azmer
Azmer-Rizgari
Derbandikhan-Rizgar
i


8 0.99485034 0.99866533 0.99876583
Case
Study
Numbers
Reliability Results for each case study obtained from the
program ( Scheduled and Forced Outages Take into
Account)
1 0.99870563 0.99865115 0.99876189
2 0.99735302 0.99671572 0.99748737
3 0.99884993 0.99880522 0.99889511
4 0.99886429 0.99863273 0.99882758
5 0.94479823 0.93976295 0.94440871
6 0.97479832 0.96976298 0.97440875
7 0.98807019 0.99411255 0.99685073
8 0.98799670 0.99143618 0.99615175

Table 8 Unreliability and Reliability Results for case study 11 and 12
Case Study
Numbers
Years

Reliability
results from
disjoint method Reliability results
from
approximation
method

results from
approximation
method Scheduled and Forced Outages
Taken into Account
1996 0.998711296 0.99870563
1997 0.998661428 0.99865115
1998 0.998767499 0.99876189
1999 0.998420777 0.99840850
2000 0.998767479 0.99876273
11
2001 0.998709733 0.99870372
1996 0.98835124 0.98799670
1997 0.991604149 0.99143618
1998 0.996203793 0.99615175
1999 0.994234003 0.99412298
2000 0.997079661 0.99705076
12
2001 0.993418684 0.99334556

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

Case Study
Numbers
1999 2000 2001

9 0.99977642
10 0.99878043

8 Dokan H/P
5*8 0 M W
1
Dokan S/S
2
Tasluja S/S
34
Chamchamal
S/S
To Kirkuk
Rizgary S/S
5
Azmar S/S
6
Old Kirkuk
7

Only Forced Outages Taken into Account
1996 0.00125661 0.45866265
1997 0.00118041 0.43084965
1998 0.00114517 0.41798705
1999 0.00124035 0.45272775
2000 0.00114424 0.4176476
2001 0.00116409 0.42489285

Scheduled and Forced Outages Taken into
Account
1996 0.00129435 0.47243775
1997 0.00134884 0.4923266
1998 0.00123808 0.4518992
1999 0.0015915 0.5808975
2000 0.00123728 0.4516072
2001 0.00129627 0.47313855

Table 10 AAIR evaluation for case study 14
Q
S
AAIR (Days /Year)
Years
Only Forced Outages Taken into Account
1996 0.00514967 1.87962955
1997 0.00133465 0.48714725
1998 0.00123419 0.45047935
1999 0.00138625 0.50598125
2000 0.00132259 0.48274535
2001 0.00221002 0.8066573


9

Appendix A
BOOLEAN ALGEBRA
1- Commutative Laws:
a)
abba +=+
b)
abba ×=×

2-Distributive Laws:
a)
)()()( cabacba +×+=×+

b)
)()()( cabacba ×+×=+×

3-Identity Laws:
a)
aa =+ 0
b)
aa =×1

4-Complement Laws:
a)
1=
′
+ aa
b)
0=


9-Involution Law:
aa =
′′
)(

10-DeMorgan’s Laws:
a)
baba
′
×
′
=
′
+
)(
b)
baba
′
+
′
=
′
×
)(

11-Disjoint set:
baaba
′
+=+

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.
[12] S.S.Yau, Y.S. Tang “ An efficient algorithm for generating
complete test sets for combinational logic circuit”, IEEE
Trans. Computers, Vol.C-20, pp 1245-1251, Nov. 1971.

Asso R. Majeed,(E-mail: )
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:)
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.