Reliability analysis of a power system based on the multi state system theory

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 Ch

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: lichunyang.nudt@163.com

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 generating function

I INTRODUCTION

A power system composed of battery pack provides energy

for other systems, and the reliability of this power system is

very important To analyze the reliability of the power system

using the traditional system reliability theory, the reliability of

the battery should be gained first, and then the system

reliability can be computed according to the structure of the

system [1] This method is simple, but it can not be applied to

power systems with required capacities The power system will

fail when the system capacity is less than the required capacity

even though the batteries of the system are all working

The traditional system reliability theory defines the power

system and the batteries are binary, but they are all multi-state

actually The performance of the batteries can degrade, which

results in performance degradation of the power system So

there can be several states of degradation Compared with

binary system, the multi-state system can perform its task with

many different performance levels except failed and working

[2, 3] The research on multi-state systems began in the 1970s

[4, 5], and gained a lot of researchers’ attention Many papers

have been devoted to estimating the reliability of the

state system [6-8] and optimizing the structure of the

multi-state system [9-11]

The reliability of the power system will be analyzed by the

multi-state system theory in this paper The procedure of

applying the multi-state system theory to the reliability analysis

of the power system is studied The relationship between the

performance of the system and the performance of the batteries

is analyzed The results obtained by the traditional system

reliability theory are compared with the results obtained by the multi-state system theory

II PROBLEM FORMATION

The power system is composed of eight identical batteries

A branch consists 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

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 capacity, and the system will fail when the required capacity is not fulfilled According to the structure and the required capacity of the system, the reliability of the battery is defined

as the probability that the capacity of the battery is not less than

5700 mAh Then the reliability of the battery is:

{ }

When the capacity of one battery is less than 5700 mAh, the required capacity of the power system may be not fulfilled

So the reliability of the power system estimated by the traditional system reliability theory is:

st

R =R =

This method can solve the problem, but the result is

conservative Because actually the reliability of the power

system is:

(1)

{

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 MULTI-STATE SYSTEM THEORY

Assume that the component has M possible states, and the

performance is g={g g1, , ,2 " g M}, with the corresponding

probability is q={q q1, , ,2 " q M}, where q l Pr{G l},

is the performance of the component, Then the

universal generating function of the component is:

g

M

1, ,

l= "

( )

1

l

M g

l

=

To obtain the universal generating function of the system,

the operators of the universal generating function are defined as

follows [2, 3]:

(3)

( ) ( )

1 1

M M

f g g

k l

k l

= =

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

+ +

Ω

(5)

( ) ( ) ( ) ( )

(

( ) ( )

+

+

Ω

⎡

)

⎤⎦

The f g g is defined according to the structure of the ( k, l)

multi-state system

When the performance of the system is equal to the sum of

the performance of components, define the π operator:

( ) ( )

1 1

M M

g g

k l

k l

= =

When the performance of the system is equal to the

minimum of the performance of components, define the

σ operator:

( ) ( )

1 1

M M

g g

k l

k l

σ

= =

Of course, other operators can also be defined according to

the situations

The universal generating function for the system can be obtained using simple algebraic operations over individual universal generating function of components:

, (8)

( ) ( 1( ) ( ) )

1

M G

m

=

( ) ( 1( ) ( ) )

1

M G N

s

=

where U z is the universal generating function of subsystem i( )

; is the number of components in subsystem i;

number of possible states of subsystem i is the performance of subsystem i ; is the corresponding probability;

m

G

m

q

( )

U z is the universal generating function of the

system; N is the number of subsystems in the system; M sys is the number of possible states of the system; G s is the performance of the system; q s is the corresponding probability

Define the following δ operator over U z : ( )

, (10)

( )

,

(11)

,

G s

s

q z W

δ = ⎨⎧ ≥ <

⎩

where W is the required performance level of the system

Then the reliability of the system is:

( ) Pr{ } ( ( ), )

s

G W

≥

IV STATES DEFINITION AND PROBABILITIES ESTIMATION

Suppose that the capacity of the battery is divided into 1

M+ intervals: [0, w1) , …, [w M−1,w M) , [w M,∞ , )

0<w <"<w M− <w M The states of the battery can be defined as follows:

state 0: 0 G w≤ < 1 state 1: w1≤ <G w2 …

state 1M− : w M−1≤ <G w M

state M : G w≥ M

M can be determined by the analytical precision In this

paper, the capacity of the battery is divided into eight intervals, that is:

[5200,5550 ,) [5550,5700 , ) [5700,5850), [5850,6000 , )

[6000,6150 , ) [6150,6300 , ) [6300,6450), [6450,6800 )

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 5200

g = , g2 =5550, g3 =5700, g4 =5850,

5 6000

g = , g6 =6150, g7 =6300, g8 =6450

State probability is defined as q l =Pr{G g= l}, and then

the corresponding state probabilities of the battery are:

1

8

2

7

3

6

4

5

Because Pr{G<5200}=Pr{G≥6800}=4.82130 10× − 8 ,

the intervals [0,5200) and [6800,∞ are not considered )

Then we can get the state performance of the battery is

,

{5200,5550,5700,5850,6000,6150,6300,6450

=

and the corresponding probability is

{0.00135,0.02140,0.13591,0.34134,

0.34134,0.13591,0.02140,0.00135}

=

q

V RELIABILITY ANALYSIS OF THE POWER SYSTEM

The reliability of the power system is analyzed using the multi-state system theory According to (2), the universal generating function of the battery is:

6000 6150 6300 6450

j

z

Based on the operators defined in (7) and (8), the universal generating function of the branches can be obtained, and then the universal generating function of the power system can be computed by (6) and (9) According to (12), the reliability of the power system is:

{ }

The results show that when the required capacity of the power system is 22.8 Ah, the result gained by the multi-state system theory is larger than the result gained by the traditional system reliability theory Fig 2 is the reliability obtained by these two methods in different capacities

From Fig 2 we know that the results obtained by the traditional system reliability theory are always conservative For example, when the required capacity is 23.4 Ah, the reliability of the system obtained by the traditional system reliability theory is only 0.25107, but the reliability of the system obtained by the multi-state system theory is 0.55963

Figure 2 The results obtained by the two methods

VI CONCLUSIONS

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:

(1) The reliability of the power system obtained by the

traditional system reliability theory is always conservative

(2) The power system is a state system The

multi-state system 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

ACKNOWLEDGMENT

The authors would like to thank the Graduate School of

National University of Defense Technology for supporting this

research work

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