Energy and Power Engineering, 2013, 5, 625-629
doi:10.4236/epe.2013.54B121 Published Online July 2013 (
The Study of Overhead Line Fault Probability Model
Based on Fuzzy Theory
Jing Li1, Lijie Chen1, Donghong Zhao2, Yadi Luo1
1China Electric Power Research Institute, Beijing, China
2China Wuzhou Engineering Group Co., Ltd., Beijing, China
Received January, 2013
In the background of the design and construction of Smart Grid Operation Supporting System for District Power Net-
works, this paper established the weighted fault probability model of the overhead line which is based on equipment
operating status, utility theory and fuzzy theory. In this model, the proper membership function is adopted to describe
the influence of lightning, wind speed, line ice and temperature, and the outage rate of overhead line, derived from his-
torical statistics, is amended. Based on this model, the power supply risk analysis software is developed to calculate the
online risk indicators of district grid, and provide real-time decision support information based on risk theory for sched-
uling operations personnel.
Keywords: Power System; Fault Probability Mode; the Overhead Line; Risk Assessment
1. Introduction
Regional power grid plays an important role in transpor-
tation and distribution of electric energy between high-
voltage transmission grid and power consumer. It is
closely linked with the country's economic development
and people's living standards. With the development of
social economics and power technology, the regional
power grid is appearing the characteristic of more com-
plex structures, more random load and more complex
operational condition[1,3,5]. So, it is of great signifi-
cance to study how to assess the security of district
power grid.
In this paper, the overhead line fault probability model
is established based on power system risk assessment
theory and combined with the knowledge of fuzzy theory,
and the forced outage rate of the lines is more in line
with the actual operating conditions.
2. Overhead Line Fault Probability Model
Based on Fuzzy Theory
During the operation of the power system, overhead lines
operation conditions are more complex and most se-
verely affected by uncertain factors such as climate en-
vironmental and so on, which have different influence
characteristics to the overhead lines[2,4]. In this paper, a
method of dealing with uncertainty information based on
the fuzzy theory was adopted of appointment, and com-
bined with the actual operation conditions of the power
system, the overhead line fault probability model is es-
2.1. Main Uncertain Factors
The overhead line unplanned outage causes in 2010(table
C-3), released by the Electricity Reliability Management
Center of the State Electricity Regulatory Commission,
presented that the unplanned outage times caused by nat-
ural disasters and climate factors is 632accounting for
59.7% of the total number of the unplanned outage times;
the unplanned downtime caused by natural disasters and
climate factors is 6758.02 hours, accounting for 70.97%
of the total time of non-planned outage. Therefore, the
study of climate environmental factors for the impact of
overhead lines and the prevention and control measures
has far-reaching significance to ensure the safe and stable
operation of the power system.
External environmental factors have greater impacts to
overhead line fault, The main reason is that: on the one
hand, the long distance of most of overhead lines, and
across different climate regions, increase the difficulty of
planning and research; on the other hand, most overhead
lines are direct contact with the external environment, the
impact of climatic and environmental conditions around
is the larger, and frequent lightning, line Icing, the higher
wind speed will caused the line outage rate rising. How-
ever, it is still a problem to known the influence to the
line reliability integrating various factors. This paper
Copyright © 2013 SciRes. EPE
main study the Uncertainties factors affecting line outage
rate as follows:
1) Temperature: The temperature changes will cause
expansion and contraction of the line, and thus cause the
lines sag and stress changes, and will affect the electrical
parameters of the line. Higher temperatures, due to ther-
mal expansion factors, the lines sag increase and the
length elongate, which impact the safe distance of the
wire-to-ground and cross across, and make the line resis-
tance increases, and thus increasing the power loss in the
line transfer power; When the temperature decreases, due
to the cold shrink effect, the line length becomes shorter,
the stress is increased, affecting the mechanical strength
of the wire.
2) Wind: effects of wind on the overhead line are
mainly in three aspects: Firstly, it will increase the load
of wires and towers when the wind blows on the towers,
conductors and its accessories; Secondly, with the action
of the wind, the wire will deviate from the vertical plane,
which will change the ground distance of the live wire,
cross arm, towers, etc; Thirdly, the wire will vibrate and
dancing in the wind, and the vibration will cause the wire
fatigue, in severe condition it will cause broken stocks or
short line, dancing makes chaos between the upper and
lower rows of wire .
3) Lightning: line tripping caused by lightning can
reach 70% of the total number of line tripping. And it can
trigger a chain of reactions after being struck by lightning,
such as wires blown, insulator broken, switch trip, etc.
Being struck by lightning, over-voltage of overhead lines
will result in flashover accident of insulation breakdown.
4) Line Icing: Line Icing cause wire and towers form-
ing a vertical load, line load increasing, may cause the
disconnection and connection fittings destruction even
down rod accident; Otherwise, ice-shedding difference or
uneven may cause overhead lines jump, easily leading to
flashover between parallels or between the wire and the
lightning conductor, then burn wires or lightning conductor.
2.2. The Method of Fuzzy Theory to Deal with
Uncertain Factors
The fuzzy uncertain factors are different from random
factors, there is no exact probability distribution, and
classical probability statistical methods can not be used
to describe it. The fuzzy set theory introduced by Zadeh
Professor is a powerful tool to deal with and descript the
fuzzy uncertain factors. The fuzzy set allows for the de-
scription of concepts in which the boundary is not sharp.
Besides, a fuzzy set concerns whether an element be-
longs to the set and to what degree it belongs. It does not
consider the situations where elements do not belong to.
As a result, the range of fuzzy set is in [0,1]. A fuzzy set
is mathematically defined by Zadeh as:
 (1)
where is the membership function of in A, and X is the
universe of objects with elements x. In the case of the
classical “crisp” set A, membership of x in A can be
viewed as a characteristic function that can obtain two
discrete values:
() 0;
ifx A
xifx A
For the fuzzy set A, the value of the membership func-
tion can be anywhere between 0 and 1, making it differ-
ent from a crisp set. Membership function of a fuzzy set
expresses to what degree the value of x is compatible
with the concept of A.
There is a wide variety of forms for fuzzy numbers,
and triangular fuzzy numbers and trapezoidal fuzzy
numbers are the most widely used in practical applica-
tions. Trapezoidal fuzzy number is function based on left
expand function L(x) and right expand function R(x). As
shown in Figure 1, it is a L-R fuzzy numbers described
by the real parameters in (a, b, c, d), and the representa-
tion of its membership function as:
() (),
Lx axb
xRx cxd
where L(x) = (x-a) / (b-a) for [a, b] single increasing
function; R(x) = (d-x)/(d-c) of [c,d] within a single reduc-
tion function; the trapezoidal fuzzy numbers center value
is (b + c) / 2; a, d, respectively, is the left and right
borders of the fuzzy numbers.
Trapezoidal fuzzy numbers to characterize fuzzy fea-
tures of the value have better usability. In power systems,
the generator, load, and component failure status pa-
rameters can be described by the trapezoidal fuzzy num-
ber. For example, predict the maximum load of a system
within a year, the fuzzy predictive method may conclude
that: “the highest load will not be greater than 900 MW
or less than 750 MW, more possibly from 800 MW to
850 MW”, then it is more appropriate to indicates it adopt-
ing the trapezoidal fuzzy number, as Figure 1 shows.
Figure 1. Trapezoidal fuzzy function.
Copyright © 2013 SciRes. EPE
J. LI ET AL. 627
2.3. Modeling Overhead Line Fault Probability
Considering Fuzzy Uncertain Factors
1) Uncertainties membership function choice
The method of establishing the membership function
include weighted method, fuzzy statistics, expert scoring
method, interpolation ,standard function method and so
on. There is strong uncertainty to the impact of climate
change for overhead lines running. In this paper, based
on the long-term experience of dispatcher to judge for
these types of environmental factors and determine the
membership function.
a) The membership function of lightning impact
The density of lightning is an important indicator to
determine the lightning degree of a region. Lighting Lo-
cation System (LLS) can automatically measure and re-
cord the density of lightning. Lightning protection design
standards also adopt lightning density as a reference. The
membership function of the lightning effects identified
here as shown in Figure 2:
The membership function of lightning disasters impact
on overhead lines running as:
() ,
where in (4) a and b is the lightning density threshold
determined according to the experiences of the dispatch-
ing personnel, In other words, it does not affect while the
lightning density is less than the lower limit threshold
value a, and the influence coefficient is 0, otherwise,
higher than the high limit threshold value b is considered
a greater impact, influence coefficient is 1.
b) the membership function of wind speed and line Ic-
Wind speed can be obtained by the meteorological
department forecast; while ice thickness for the line, air
humidity, temperature and wind size the extent of ice
damage has a larger impact, not yet theoretical or em-
pirical model to predict the extent of ice cover based on
meteorological conditions, we use the actual ice thick-
ness measurement indicators to assess the severity of the
ice storm. Wind speed and line of ice thickness with the
fall line health density similar to lighting, the same form
of the membership function μ2(x), μ3(x), shown in Fig-
ure 3:
For the wind speed, in μ2 (x), A is the impact threshold
value determined according to the experiences of the
dispatching personnel, b is the critical value determined
catastrophe occur; Line Icing μ3 (x), a and b are respec-
tively the upper and lower critical value of ice thickness
c) the membership function of temperature impact on
overhead lines
The temperature forecast information can be obtained
by contact with the meteorological department. Within
the normal temperature range, the temperature did not
affect the line running, so the value is set at 0; when the
temperature is too low or too high, its influence is large,
and the function value is set to 1. The membership func-
tion shown in Figure 4 below:
The membership function of temperature impact on
overhead lines as:
() 0,
Figure 2. The membership function of the lightning effe c ts.
Figure 3. The membership function of the wind speed and
line Icing.
Figure 4. The membership function of air temperature.
Copyright © 2013 SciRes. EPE
where in formula 5, a, b, c and d determined according to
the experiences of the dispatching personnel are the im-
pact threshold value that air temperature impact on over-
head lines running.
2) overhead line fault possibility model
For any transmission lines, Generally, the span of
overhead lines is larger, and different segments of the
line in different climatic and environmental conditions.
So first according to geographic information system (GIS)
informationwith the most complex of climatic factors
that lines through as criteria for overhead line segment
description, assume that the cross-regional temperature
distribution of the most complex and divided into n stag-
es, the definition:
[,, ,,]
ll llL (6)
where is segmented vector.
123 n
Any overhead line segmentation, the fuzzy number
vector composed by the membership function of various
impact factors is determined as follows
,,, ,lll l
[,, ,]
iii i
rrrrr (7)
where ri1, ri2, ri3 and ri4 successively corresponds the
fuzzy numbers that lightning, wind speed, line icing and
temperature affect the segment i line outage probability ,
then the fuzzy number matrix as:
11 121314
21 222324
1234nnn n
rrr r
where, rij represents the fuzzy number of the j-th impact
factors impact on the i-th segment line. Definition B =
[B1, B2, B3, B4] as the weight coefficients of line fault
outage rate considering four influence factors, then:
[, , , ]
aa aARB (9)
where, ai is the impact factor of the i-th overhead lines
considering the four influential factors, thus four factors
for the average impact factor of the whole of the line as:
ii i
Set the outage rate of overhead lines acquired from
historical statistics is λ´, and after correction by the in-
fluencing factors of overhead line outage rate is λ, then:
(1 ),(1 )1
1, (1)1
 
3. Analysis of Examples
According existing research results as defined in Section
2, this section defines the membership function coeffi-
cient and the critical value of the influencing factors for a
hypothetical transmission lines to create a its outages
model. Line forced outage rate under normal operating
conditions adopted “220 kV transmission line in 2010
forced outage rate”, released by the State Electricity
Regulatory Commission, and it is 0.247. This section
assumes the line running condition to correct its forced
outage rate.
3.1. The Membership Function
The critical value of the membership function of light-
ning, wind speed, line Icing and air temperature impact
on overhead lines is shown in Table 1.
3.2. Line Segmentation
It is assumed that the total length of an overhead line is
100 km, and the wind speed change are maximum in its
span area. The line is divided into four sections, namely
l1 = 20 km, l2 = 30 km, l3 = 30 km, l4 = 20 km, its
weather experienced can be shown as Table 2, this line
can be seen mainly effected by wind during the running.
3.3. The Correction Overhead Lines Forced
Outage rate
The fuzzy numbers of various influencing factors ob-
tained are shown in Table 3:
Table 1. The critical value of the membership function.
The Influencing
Factors The Critical Value Of The Membership
(number /(km2•a)) a1 = 1.05; b1 = 6.30
wind speed (m/s) a2 = 6; b2 = 15
line Icing (mm) a3 = 5 mm; b3 = 20 mm
Temperature () a4 = -20; b4 = 40; c4 = -40; d4 = 60
Table 2. The specific values of the overhead lines each seg-
ment external factors.
(number/(km2•a)) Wind Speed
(m/s) Line Icing Temperature
(mm) ()
l1 0 4 0 16
l2 0 8 0 18
l3 1.80 12 0 20
l4 1.80 13 0 20
Copyright © 2013 SciRes. EPE
Copyright © 2013 SciRes. EPE
4. Conclusions
Table 3. The fuzzy numbers of various influencing factor s.
The Fuzzy
Numbers Lightning Wind Speed Line Icing Temperature
l1 0 0 0 0
l2 0 0.222 0 0
l3 0.143 0.333 0 0
l4 0.143 0.444 0 0
Based on power system risk assessment theory and com-
bined with the knowledge of fuzzy theory, the overhead
line fault probability model is established, by selecting
the proper membership function to describe the influence
of lightning, wind speed, line ice and temperature. In this
model, the outage rate of overhead line, derived from
historical statistics, is amended, and the forced outage
rate of the lines is more in line with the actual operating
0 0.22200
0.143 0.333 0 0
0.1430.444 0 0
[1] CIGRE Task Force 38.03.12. “Power System Security
Assessment,” Electra, Vol. 175, 1997, pp. 49-77
[2] H. Wan, J. D. Mccalley and V. Vittal, “Increasing Ther-
mal Rating by Risk Analysis,” IEEE Transaci on Power
Systems, Vol. 14, No. 3, 1999, pp. 815-828.
Define the weight coefficient vector of four impact
factors for line fault outage rate is: B = [0.2,0.4,0.2,0.2] T,
[0,0.088 8,0.1618,0.2062]ARB (13) [3] W.-H. Fu, J. D. McCalley and V. Vittal, “Risk Assess-
ment for Transformer Loading,” IEEE Transacti on
Power Systems, Vol. 16, No. 3, 2001, pp. 346-353.
By (10), four factors for the average impact factor of
the whole line is μ = 0.1164.
By the formula (11) can be obtained the line forced
outage rate after the correction:
[4] Y. Q. FengW. C. Wu, H. B. Sun, et a1., “A Preliminary
Investigation on Power System Operation Risk Evalua-
tion in the Modern Energy Control Center,” Proceedings
of the CSEE, Vol. 25, No. 13, 2005, pp. 73-79.
(10.1164) 0.2470.276
  (14)
As can be seen the line running is affected by exter-
nal environmental factors, the forced outage rates is sig-
nificantly increased compared to normal circumstances
statistics value, so the in overhead lines outage modeling
process can not overlooked the impact of environmental
[5] N. Ming, J. D. McCalley, V. Vittal, et a1., “Online
risk-based security assessment,” IEEE Transacti on
Power Systems, Vol. 18, No. 1, 2003, pp. 258-265.