A. Sittithumwat K. Tomsovic F. Soudi
School of Electrical Engineering and Computer Science
Washington State University
Pullman, WA 99164 tomsovic@eecs.wsu.edu
Abstract: This paper addresses the problem of finding appropriate levels of maintenance for each component in a distribution system network given approximate information on the equipment condition. Maintenance resource optimization is presented using a fuzzy linear programming technique. The objective of the optimization problem is to minimize the reliability indices, subject to system constraints such as financial resources or crew resources. This work extends earlier results by introducing fuzzy sets, through the quality of condition information, into the optimization.
Keywords: Distribution systems, distribution reliability, fuzzy number, fuzzy linear programming, preventive maintenance.
1. INTRODUCTION
Since a goal of any electric utility company is to supply reliable power to customers at low cost, prevention of power system failures is of paramount importance during the design and operation of the system.
On overhead distribution systems, a fault can be classified as temporary or permanent. Approximately
75 to 90 percent of the faults are of a temporary nature caused by trees, animals, lightning, high winds, flashovers, and so on [5]. If the temporary fault cannot be cleared by the automatic interrupting devices, it will
Distribution systems have the greatest impact on customer outage frequency and duration [1]. become a permanent fault that requires repairs by a crew. A study of distribution outage causes at Duke
Consequently, a proper preventive maintenance program in distribution systems can have a significant effect on the achievement of high reliability.
The preventive maintenance policies adopted by electric utilities are aimed either at detecting deterioration of the equipment before it fails or are based on the priori assumption that the equipment has deteriorated and requires replacement without proof of deterioration [2].
Power Company between 1987 and 1990 indicated that equipment failures contributed 14%, trees 19%, animals
18%, lightning 9%, and the remaining 40% was of an unknown nature [6]. Note that other utility companies may experience different conditions of distribution faults. Some of the failures cannot be avoided, such as lightning, whereas others can be prevented by proper maintenance plans.
In either case, there is a need to select the appropriate maintenance frequency and level. The timing of preventive maintenance involves tradeoffs between the requirement for reliable power supply at all times and the cost of performing maintenance activities. Recently, utilities have conducted pilot applications of the
Reliability Centered Maintenance (RCM) on various power plant systems [3]. RCM is a qualitative method for determining applicable and effective preventive maintenance procedures. Dai and Christie [4] discussed the optimal scheduling technique for tree trimming in transmission and distribution, and showed that the cost of the optimal maintenance schedule is significantly smaller than that of the constant maintenance interval schedule.
The maintenance activity taken by utility companies can involve equipment inspection, repair, replacement, tree inspection, tree trimming, installation of animal guard, and washing of insulators. Conventionally, decisions made as to when, where, and how to perform maintenance have been made by engineers using broad planning heuristics. To address maintenance activities systematically, a rational approach based on quantitative analysis (i.e., an optimization model) has been developed to provide a systematic framework that allows the consideration of all facets of the decision problem [7]. Still, this quantitative analysis of the maintenance is based on the assumption that the reliability parameters (e.g., failure rate, average outage time, etc.) are exact values. In practice, one difficult problem in reliability evaluation is the quantification of

the reliability parameters [8]. Even if they are available, e.g., from measurement, they are often inaccurate and thus, subject to uncertainty. The accuracy of these outage rates depends on the amount of testing, the completeness of field studies and other essential data
(e.g., operators’ experience or manufacturer’s specification). Even with the lack of precise parameters, it is imperative that the engineers strive to obtain quantitative results to identify the maintenance activity.
In this paper, reliability parameters will be described by where N i
is the number of customers in section i, l i
is the failure rate for section i, and N
T
is the total number of customers on the circuit. The numerator of (1) can be rewritten in terms of the total number of sections or protective devices as follows
å i l i
N i
= j
J
K
åå j
= = 1
)
A jk
(2) where J is the total number of a main feeder and laterals, and K j
is the number of protective devices on an interval of probabilities (i.e., a trapezoidal fuzzy number) and the binary programming optimization framework from [7] is used to determine a maintenance level for each component in a radial distribution network. This is accomplished by assigning a possible range of variation (owing to uncertainty) to failure rate and using a fuzzy programming technique.
2.1 Maintenance Optimization Model which has an automatic reclosing device as a backup
The objective of this maintenance optimization problem protection against temporary faults, only the permanent is to minimize the System Average Interruption
Frequency Index (SAIFI) by determining the level of so that each zone can be isolated automatically or manually from the remaining circuit if a fault occurs are applied. maintenance for each component. The constraint of this optimization is financial and/or manpower resources. arrangement with protection for each zone, or section,
The total failure rate for each line section can be divided
Distribution systems generally have a radial into fixed failure rates and adjustable failure rates. The fixed failure rates represents the failure mode that cannot be affected by maintenance. The types of adjustable failure rates for each section depend on a within the zone. Consequently in our framework, the number of interrupting devices is equal to the number of failure rate is applied, and for a fuse with no backup devices, both the permanent and temporary failure rate section’s components that can be impacted by maintenance, e. g., breakers, reclosers and line sections zones. The following assumptions are also used in the close to tree limbs. Hence, the total failure rate can be stated as model:
2. PROBLEM FORMULATION
1. A triangular or trapezoidal fuzzy set is employed to
A jk
can be expressed as l
)
)
A jk
= l
) jk
)
N jk
(3) where
)
N jk jk
is the total failure rate for the section jk with
customers. This total failure rate is determined based on the type of the protective devices. For a circuit breaker and a recloser, the permanent failure rate of all components within the zone is applied. For a fuse, l
) jk
= l
) jk 1
+
L
å l = 2 l
) jkl q
Q l å
= 1 s lq x jklq
(4) represent the failure probability range of a component and a failure rate multiplier for any maintenance impact.
2. The failure probability range of each component is
where l
) jk 1
is a fixed failure rate of the section jk, Q l
is known. the number of maintenance levels for failure rate type l,
L is the number of types of failure rates, s lq
is the
3. The failure rate multiplier range for each maintenance level is given. (Failure rate multiplier indicates variation of failure rates corresponding to maintenance quality for multiplier of the maintenance level q for failure type l, and if the variable x jklq
=
1 , maintenance level q for a given time period.)
2.1.1 Objective function
The SAIFI index is defined as
SAIFI
=
å l i
N
T
N i
(1) failure type l will be performed, and otherwise x jklq
=
0 . Substituting (3) and (4) into (2) yields j
K
J
j
=
1 k
=
1
é
ê
æ
ççè l
) jk 1
+ l
L
=
2 l
) jkl
Q l q
=
1 s lq x jklq
ö
÷÷ø
)
N jk
ù
ú (5)
This is the objective function for our problem since minimizing (5) is equivalent to minimizing (1).

2.1.2 Cost constraints
The availability of financial resources for maintenance is generally limited. The cost constraints can be written as j
J
K j
L Q l
1 k 1 l 2 q
=
1 c jklq x jklq
£
C
R
(6) where c jklq
is the cost of maintenance level q for failure type l on section jk and
2.1.3 Manpower constraints
There is limitation on the crew to perform maintenance.
These constraints can be expressed as
J
K j
L Q l
j 1 k 1 l 2 q
=
1 h jklq x jklq
£
H
R
(7) where h jklq
is the man-hours required to perform maintenance level q for failure type l on section jk and
2.1.4 Maintenance level constraints
Only one level of maintenance is allowed for each component, i. e.,
Q l q
å
=
1 x jklq
= 1 (8)
Note, the failure rates and failure rate multipliers of all components are fuzzy numbers. As a result, fuzzy manipulation, i. e., arithmetic operations on fuzzy numbers is applied in determining the optimization model. The optimization model obtained in this work is in the form that the coefficients of the objective function are fuzzy numbers; thus fuzzy linear programming techniques discussed in the following section will be used but modified for binary programming.
2.2 Linear Programming with Fuzzy Objectives
In classical linear programming, the coefficients of problems are assumed to be deterministic and fixed in value. In fuzzy programming problems, the constraints and the goals are viewed as fuzzy numbers with known membership functions [9]. In this paper, we consider linear objective functions with fuzzy coefficients as follows:
Subject to x
Î
X
=
{ x | Ax
£ b and x
³
0
}
(9) where c
Î
[ c
L
, c
U
] is an interval-valued coefficient vector. To solve (9), first define: z
Min
( x ) = c L x and z
Max
( x ) = c U x which are the two extreme cases. Rommelfanger et al.
[10] reduced the infinite objective functions of (9) to a search for solutions of the multiple objective linear programming problem: z
*
Min
= z
Min
( x
*
Min
) = Min x
Î
X z
Min
( x )
(10)
By this procedure, the set of compromise solution is not restricted to the "extreme" points of X . To solve (10),
Zimmermann’s approach [11] is used to first find:
Min x
Î
X
[ z
Min
( x ), z
Max
( x )] z
*
Max
= z
Max
( x
*
Max
) = Min x
Î
X z
Max
( x )
, and z
¢
Min
= z
Min
( x
*
Max
) and z
¢
Max
= z
Max
x
*
Min
The membership functions of the objectives of (10) can be established as follows: m z
Min
( x ) =
ì
ïï
ï z
¢
Min z
¢
Min
-
1 z
Min
-
0 z
(
*
Min x ) if if if z
Min
( x ) £ z
*
Min z
*
Min
£ z
Min
( x ) £ z
¢
Min z
Min
( x ) ³ z
¢
Min and similarly for z . Finally, a compromise solution
Max can be calculated by solving the LP model as follows:
Max l
Subject to m z
Min
( x )
³ l
, and m z
Max
( x )
³ l x
Î
X (11)
Now, if the decision makers express their ideas about the imprecise objective coefficients in the form of fuzzy sets: c
=
{
( c , m
( c )) | c
Î
[ c
L
, c
U
]
} then the problem arises to take into account the additional information contained in the possibility distribution m
(c ) as opposed to simply the interval.
Further, if the possibility distributions are convex, then a finite set of degrees of possibility ({ a
1
, a
2
,..., a r
}) can be taken with a i
levels r objectives at the corresponding
, i = 1, 2,…, r . Consequently, this problem becomes:
Min x Î X
( c a
1 x , c a
2 x ,..., c a r x )

where c a i are interval-valued as shown in Figure 1.
Similar to the interval concept of (10) for each objective function, the following auxiliary problem is obtained as:
Max l
Subject to m i , z
Min
( x ) m i , z
Max
( x ) x Î X
³ l
³ l , and i
=
1 , 2 ,..., r
(12) where m i , z
Min
( x )
=
ï
î
ì
ï z i
¢
, Min z i ,
¢
Min
-
1 z i , Min
( x )
-
0 z i ,
*
Min if if if z i , Min
( x )
£ z
* i , Min z i ,
*
Min
£ z i , Min
( x )
£ z i ,
¢
Min z i , Min
( x )
³ z
¢ i , Min and similarly for z
Max
. The subscript i represents the relevant a i
- levels. Our extension of this approach to binary variables is omitted for brevity.
3. EXAMPLES
The simple overhead radial system shown in Figure 2 was analyzed with the component data in the form of trapezoidal and triangular fuzzy numbers as shown in
Table 1, 2, and 3. For this example, the total failure rate of each section is divided into three categories. The fixed failure rate cannot be affected by maintenance.
The tree failure rate can be minimized by tree trimming.
The recloser failure rate can also be impacted by preventive maintenance.
There are three levels of maintenance for both tree trimming and recloser maintenance: extensive, minimal, and no maintenance. In these tables, the value of the multiplier represents the expected failure rate change from nominal over one year time period. A recloser is installed at each location except at location 11, in which a breaker with its protective devices is installed. The available budget for this network is $13,000. Solutions are shown in Tables 4 - 7 providing the optimal solutions and fuzzy values for SAIFI. m
(c i
)
1 a i c i c L a i c U a i
Figure 1 The possibility distributions of c i
The solutions of Table 5 was calculated by using a
level = {0, 0.25, 0.5, 0.75}. From tables, the result obtained by the conventional approach suggests minimal maintenance for almost components. The fuzzy approach leads to more varied suggestions. To show the advantage of the fuzzy solution, values
F m z i , k
( x
*
F
)
that correspond to the level of satisfaction of the attained objective values z i , k
( x *
F
) for all extreme objective functions z i , k
are determined as follows: m z i , k
( x
*
F
)
= z i ,
¢ k
z i
¢
, k z i , k
( x
*
F
)
z i
*
, k
, where k is either Min or Max. The membership values m z i , k
( x
E
) where x
E
is the expected solution and m z i , k
( x *
F
) are shown in Tables 8 and 9 respectively.
From Tables 8 and 9, the expected solution leads to a high degree of satisfaction for some a
-cuts but has inferior results for other a
-cuts. Thus, the fuzzy approach provides a more robust solution. That is there are still possibilities for the worst case scenarios of power interruptions that will be even more severe under the conventional approach. From the planning point of view, one should not overlook these situations.
4. CONCLUSION
This work develops techniques to enhance maintenance and condition monitoring tools, which should be helpful in making quantitative maintenance decisions in distribution systems with limited or noisy information.
The advantage of fuzzy expression is that the subjective information obtained from engineers or experts can be used in conjunction with the information obtained from probability techniques in the distribution reliability assessment.
11
21
12
31
41
13
51
Figure 2 Simple load point radial network

Section Miles l
) jk
(f/yr) Fixed
Failure
Tree Failure
[0.186, 0.193, (0.2)
*
,
0.207, 0.214]
[0.221, 0.2305,
(0.24)
*
, 0.2495, 0.259]
[0.192, 0.196, (0.2)
0.204, 0.208]
*
,
[0.23, 0.235, (0.24)
*
,
0.245, 0.25]
[0.38, 0.39, (0.4)
*
,
0.41, 0.42]
[0.096, 0.098, (0.1)
0.102, 0.104]
*
,
[0.38, 0.39, (0.4)
0.41, 0.42]
*
,
0.13
0.145
0.16
0.19
0.3
0.08
0.3
[0.032, 0.036, (0.04)
0.044, 0.048]
[0.04, 0.045, (0.05)
*
0.055, 0.06]
[0.032, 0.036, (0.04)
0.044, 0.048]
[0.04, 0.045, (0.05)
*
0.055, 0.06]
[0.08, 0.09, (0.1)
0.11, 0.12]
0.022, 0.024]
[0.08, 0.09, (0.1)
0.11, 0.12]
*
,
[0.016, 0.018,(0.02)
*
*
,
,
,
*
,
*
,
,
Table 1 Component data for the system
*
Expected value
Breaker or Recloser
Failure
[0.024, 0.027, (0.03)
*
,
0.033, 0.036]
[0.036, 0.0405,
(0.045)
*
, 0.0495, 0.054]
N jk
1600
2000
0 1400
0 800
0 400
0 200
0 200
ACKNOWLEDGEMENTS
This work was partially supported under NSF contract
ECS-9527302.
REFERENCES
[1] R.N. Allen and R. Billinton, “Power System
Reliability and Its Assessment: Part 3 Distribution
Systems and Reliability Costs”, Power Engineering
Journal, vol. 7, no. 4, August 1993, pp. 185-192.
[2] G.J. Anders, Probability Concepts in Electric
Power Systems, New York: John Wiley & Sons,
1990, pp. 415-453.
[3] J.P. Jacquot, “A Survey of Research Projects in
Maintenance Optimization for Electricite de France
Power Plants”, Proceedings of 1996 ASME
Pressure Vessels and Piping Conference, pp. 83-88,
Montreal, Canada, July 21-26, 1996.
[4] L. Dai and R.D. Christie, “The Optimal Reliability-
Constrained Line Clearance Scheduling Problem”,
Proceedings of the Twenty-sixth Annual North
American Power Symposium, vol. 1, pp. 244-249,
Manhattan, KS, USA, September 26-27, 1994.
[5] T. Gönen, Electric Power Distribution System
Engineering, New York: McGraw-Hill, 1986, pp.
500-576.
[6] M.Y. Chow and L.S. Taylor, “Analysis and
Prevention of Animal-Caused Faults in Power
Distribution Systems”, IEEE Transaction on Power
Delivery, vol. 10, no. 2, April 1995, pp. 995-1001.
[7] F. Soudi, “Towards Optimized Protection Design and Maintenance in Electric Power Distribution
Systems”, Ph.D. Dissertation, Washington State
University, 1997, pp. 70-81.
[8] A.M. Oliveira, A.C.G. Melo and L.M.V.G. Pinto,
“The Impacts of Uncertainties in Equipment Failure
Parameters on Composite Reliability Indices”,
Proceeding of the Twelfth Power Systems
Computation Conference, vol. 1, pp. 574-580,
Dresden, Germany, August 19-23, 1996.
[9] G.J. Klir and B. Yuan, Fuzzy Sets and Fuzzy Logic
Theory and Applications, New Jersey: Prentice-
Hall, 1995, pp. 390-417.
[10] H. Rommelfanger, R. Hanuscheck and J. Wolf,
“Linear Programming with Fuzzy Objectives”,
Fuzzy Sets and Systems, vol. 29, 1989, pp. 31-48.
Linear Programming with Several Objective pp. 45-55.
Approaches to Equipment Condition Monitoring and Diagnosis”, Electric Power Applications of
Fuzzy Systems, ed. M. El-Hawary, IEEE Press,
1998, pp. 59-84.
[13] F. Soudi and K. Tomsovic, “Optimal Trade-Offs in
Distribution Protection Design”, IEEE Transactions on Power Delivery
, in press.
Maintenance quality
Multiplier Cost Maintenance per Mile quality
Extensive [0.94,
*
*
, 1.14] $1000
No maintenance
[1.1, (1.3)
*
Table 2 Tree trimming data
*
Expected value and a fuzzy value at a
- cut = 1
Multiplier Cost per
Unit
(0.93)
*
*
1.13] $500
No [1.05, (1.25) maintenance
*
Table 3 Breaker or line recloser maintenance data
*
Expected value and a fuzzy value at a
- cut = 1

Tree trimming 11
Breaker 11 maintenance
Tree trimming 12
Recloser 12 maintenance
Action maintenance
X Tree trimming 11
Extensive Minimal No
maintenance
X
X Breaker 11 X
X maintenance
Tree X trimming 12
X Recloser 12 maintenance
X
Tree trimming 13
X Tree trimming 13
X
Tree trimming 21
Tree trimming 31
Tree trimming 41
X
X
Tree trimming 21
Tree
X
X trimming 31
X Tree trimming 41
X
Tree X trimming 51
Tree trimming 51
X
Total Cost
$13,000
Total Cost
$12,500
SAIFI Index
[0.4523,0.4895, 0.522, 0.5705]
(interruptions/customer yr)
Table 4 Optimal solution for example using expected value for calculation
SAIFI Index
[0.4589, 0.4906, 0.5234, 0.5638]
(interruptions/customer yr)
Table 5 Optimal solution for example using fuzzy number for calculation a i a
1 a
2 a
3
=
=
0 .
0
0 .
25
=
0 .
50 a
4 a
5
=
=
0 .
75
1 .
0 m z i , k
( x
E
)
-
m z i , k
( x
*
F
) z z
0 .
75 , Min
0 .
75 , Min
SAIFI at a i
levels
[0.4523, 0.5705]
[0.4611, 0.5579]
[0.4702, 0.5456]
[0.4797, 0.5336]
[0.4895, 0.5220]
Table 6 Fuzzy values for SAIFI of optimal solution for example using expected value for calculation z z
0 .
75 , Max
0 .
75 , Max z z
0 .
5 , Min
0 .
5 , Min z z
0 .
5 , Max
0 .
5 , Max a i z
0 .
25 z
0 .
25
-
a
1 a
2
=
=
0 .
0
0 .
25 a a
3
4 a
5
=
=
=
0 .
50
0 .
75
1 .
0
SAIFI at a i
levels
[0.4589, 0.5638]
[0.4665, 0.5533]
[0.4743, 0.5431]
[0.4823, 0.5331]
[0.4906, 0.5234]
Table 7 Fuzzy values for SAIFI of optimal solution for example using fuzzy number for calculation
, Min
, Min z z
0 .
25 , Max
0 .
25 , Max z z
0 .
0 , Min
0 .
0 , Min z z
0 .
0 , Max
0 .
0 , Max
1.0 0.0 1.0 0.0 1.0 0.0 1.0 0.0
Table 8 The membership value of satisfaction of optimal solution using expected value for calculation.
0.3399 0.3882 0.3587 0.5383 0.3671 0.5636 0.3718 0.5740
Table 9 The membership value of satisfaction of optimal solution using fuzzy number for calculation.