Reliability Analysis of a Power System Based on the
Multi-State System Theory
Chunyang LI
College of Mechatronics Engineering and Automation
National University of Defense Technology
Changsha, 410073, China
E-mail:
Xun CHEN, Xiaoshan YI
College of Mechatronics Engineering and Automation
National University of Defense Technology
Changsha, 410073, China Abstract—Reliability analysis of power systems using the
traditional system reliability theory usually can not represent the
real-life situation. The multi-state system theory is introduced to
analyze the reliability of a power system. States and
corresponding probabilities of the battery are defined. The
reliability of the power system is estimated by the multi-state
system theory. The results show that the system reliability
estimated by the traditional system reliability theory is
conservative, and the proposed method in this paper is better to
analyze the reliability of power systems.
Keywords- power system; multi-state system theory; reliability;
universal ge
nerating function
I. I
NTRODUCTION

A power system composed of battery pack provides en
ergy

multi-state system theory.
II. P
ROBLEM
F
ORMATION

The power system is composed of eight identical batteries.
A branch c
onsists of two batteries connected in series, and the
system consists of four branches connected in parallel as
depicted in Fig. 1. The required capacity of the power system is
not less than 22.8 Ah. To protect proprietary data, all
parameters have been scaled. This does not in any way affect
the validity of the method presented in this paper.

Figure 1. Structure of the power system
A test of 120 batteries shows that the capacities of the
batteries follow the s-normal distribution with mean 6000,
variance . In short, , where
G
is the
capacity of the battery.
2
150
()
2
~ 6000,150GN
To analyze the reliability of the system by the traditional
system reliability theory, we must gain the reliability of the
battery first. The power system has to provide the required

Equation (1) indicates that when the capacity of the system
is above 22.8 Ah, the system is reliable, though the capacity of
a battery is lower than 5700 mAh. Suppose the capacity of the
first branch is 5600 mAh and the capacity of other branches are
all above 5800 mAh, the system is reliable because the
required capacity is reached. But when we analyze the system
reliability using the traditional system reliability theory, the
system fails. So this problem will be solve by another method
— the multi-state system theory.
III. M
ULTI
-
STATE
S
YSTEM
T
HEORY

Assume that the component has
M
possible states, and the
performance is
{ }
12
,,,
M
g gg=g "
, with the corresponding
probability is
{ }

ors of the universal generating function are defined as
follows [2, 3]:
(3)
() ()
()
()
,
12
11
,
kl
MM
fg g
kl
kl
UzUz qqz
==
Ω=⋅⋅
∑∑
,

() () ( )()
()
() () () ()
()
11
11
,, , ,,
,, , ,,
kk n

⎣
""
""
)
⎤
⎦
The
( )
,
kl
f gg
is defined according to the structure of the
multi-state system.
When the performance of the system is eq
ual to the sum of
the performance of components, define the
π
operator:

() ()
()
12
11
,
kl
MM
g g
kl
kl
UzU z qqz

the situatio
ns.
The universal generating function for the system can be
obtained usi
ng simple algebraic operations over individual
universal generating function of components:
, (8)
() () ()
()
1
1
,,
i
m
M
G
in
m
Uz Uz U z qz
σ
=
==
∑
"
m

() () ()
()
1
1

. is the
performance of subsystem
i
; is the corresponding
probability;
m
G
m
q
( )
Uz
is the universal generating function of the
system; is the number of subsystems in the system;
N
sys
M
is
the number of possible states of the system;
s
G
is the
performance of the system;
s
q
is the corresponding probability.
Define the following
δ
operator over
( )
Uz

0, ,
s
ss
G
s
s
qGW
qz W
GW
δ
≥
⎧
=
⎨
<
⎩
where
W
is the required performance level of the system.
Then the reliability of the system is:

( ) { }()
()
Pr ,
s
sms
GW
RW G W UzW q
δ
≥

MM
ww
−
,
[
)
,
M
w ∞
,
11
0
M M
ww
−
w< << <"
. The states of the battery can be
defined as follows:
state 0:
1
0 Gw≤ <

state 1:
12
wGw≤ <

…
state
1
M

5850,6000
,
[
)
6000,6150
,
[
)
6150,6300
,
[
,
)
6300,6450
[
)
6450,6800
.
96
To obtain the lower bound of the system reliability, the
performance of each state is defined as the minimum capacity
of each interval, that is:
1
5200g =
, , , ,
2
5550g =
3
5700g =
4

{}
{}
2
7
Pr 5550 5700
Pr 6300 6450 0.02140 ,
qG
qG
=≤<
== ≤< =

{}
{}
3
6
Pr 5700 5850
Pr 6150 6300 0.13591,
qG
qG
=≤<
== ≤< =

{}
{}
4
5
Pr 5850 6000
Pr 6000 6150 0.34134 .
qG
qG


V.

R
ELIABILITY
A
NALYSIS OF THE
P
OWER
S
YSTEM

The reliability of the power system is analyzed using the
mult
i-state system theory. According to (2), the universal
generating fun
ction of the battery is:

( )
5200 5550 5700 5850
6000 6150 6300 6450
0.00135 0.02140 0.13591 0.34134
0.34134 0.13591 0.02140 0.00135 .
j
Uzzzzz
zzz
=+++
z
+ +++



C
ONCLUSIONS

The multi-state system theory is introduced to analyze the
reliability of the power system
in this paper, and is compared
with the traditional system reliability theory. The results show
that:
97
(1) The reliability of the power system obtained by the
traditional system reliability theory is always conservative.
(2) The power system is a multi-state system. The multi-
state sy
stem theory can define the relationship between
component performance and system performance, and the
reliability of the power system obtained by this method is much
better.
A
CKNOWLEDGMENT

The authors would like to thank the Graduate School of
National Univer
sity of Defense Technology for supporting this
research work.
R
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