818 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 35, NO.

1, JANUARY 2020

Inverse Reliability Evaluation in Power Distribution Systems

Sajjad Sharifinia , Mohammad Rastegar , Member, IEEE, Mehdi Allahbakhshi ,
and Mahmud Fotuhi-Firuzabad , Fellow, IEEE

Abstract—In power distribution systems, sometimes, system re- The IRE in power distribution systems is more significant,
liability indices are available, where some components’ reliability due to the fact that distribution systems are the final link between
parameters are unknown. This letter presents the inverse relia- the bulk power system and customers, and the most customer
bility evaluation (IRE) problem in radial distribution systems to
find unknown components’ parameters from the known system interruptions occur due to failures in the distribution systems
reliability indices. To this end, a nonlinear system of equations is zone [2]. Therefore, as a new framework, this letter defines the
presented and solved. The solutions are analyzed in the RBTS bus IRE problem in distribution systems and analyzes the associ-
2 to verify the applicability of the proposed approach and to show ated solutions, as long as the structure of the system is radial.
the importance of the IRE problem. This letter also deals with the conditions of unique or infinite
Index Terms—Inverse reliability evaluation, reliability solutions for an IRE problem, so that the operator can determine
parameters, reliability indices. whether the failure rate of a specific component is attainable
according to the available data. The solutions, i.e., component
reliability characteristics, can be used for the distribution system
I. INTRODUCTION maintenance strategies or planning approaches in the future.
The reliability of power systems can be evaluated according
to the components’ reliability parameters such as failure rate
and repair time. Due to the data entry mistakes, data erasure, II. PROBLEM DEFINITION AND SOLUTIONS
and variation of reliability parameters, some components’ pa- The primary reliability indices such as failure rate, outage
rameters may not be available in practice. However, distribution duration, and annual outage time are fundamentally important,
system reliability indices, such as system average interruption but they do not always give a complete representation of the sys-
frequency index (SAIFI), system average interruption duration tem reliability. Therefore, to address the severity or importance
index (SAIDI), and energy not supplied (ENS) are usually of a system outage, customer- and energy-oriented reliability
known by the distribution system companies based on the indices, which are commonly used in the distribution systems
recorded number of interruptions, the duration of interruptions, [2], are used as defined in Table I:
and the amount of load interrupted in each load points. Finding According to the presented definitions of the reliability indices
the unknown parameters from the available reliability indices, in Table I, SAIFI, SAIDI, and ENS are three independent indices,
named inverse reliability evaluation (IRE), can be performed to and other indices can be calculated by a linear combination of
realize the condition of power system components for upcoming these indices. These three indices can be formulated by:
power system asset management.
An IRE in the composite generation and transmission systems
⎧    −1
is presented in [1], which considers the system loss of load ⎪ NLP NLP
⎪
⎪ SAIF I = N λ
i i
l
N i
probability, loss of load frequency, and expected energy not ⎪
⎪ i=1 i=1
⎪
⎪ NLP NC
supplied, to obtain the unknown failure rates and repair times. ⎪
⎪ = i=1 j=1 a1ij λij
⎪
⎪
The authors in [1] focus only on the method of solving nonlinear ⎪
⎪    −1
⎨ NLP NLP
equations, in which the number of unknowns is equal to the SAIDI = i=1 N U
i i i=1 N i
NLP NC , (1)
known reliability indices. ⎪
⎪
⎪
⎪ = i=1 j=1 a 2ij λ ij rij
⎪
⎪  
⎪
⎪ NLP NLP NC
Manuscript received April 14, 2019; revised August 18, 2019 and October 3, ⎪
⎪ EN S = L i U i = L i λ ij rij
2019; accepted November 4, 2019. Date of publication November 8, 2019; date ⎪
⎪ i=1 i=1 j=1
⎪
⎩  
of current version January 7, 2020. Paper no. PESL-00087-2019. (Correspond-
ing author: Mohammad Rastegar.)
= NLP NC a λ r
i=1 j=1 3ij ij ij
S. Sharifinia, M. Rastegar, and M. Allahbakhshi are with the School of Electri-
cal and Computer Engineering, Department of Power and Control Engineering,
Shiraz University, Shiraz 7134851154, Iran (e-mail: s.sharifinia@shirazu.ac.ir; where, incorporated parameters and variables are defined in
mohammadrastegar@shirazu.ac.ir; allahbakhshi@shirazu.ac.ir).
M. Fotuhi-Firuzabad is with the Center of Excellence in Power System Table II.
Control and Management, Electrical Engineering Department, Sharif University In addition, a1ij , a2ij , and a3ij in (1) are coefficients derived
of Technology, Tehran 11155-4363, Iran (e-mail: fotuhi@sharif.edu). from a simplification of the left-hand sides. Assume that the
Color versions of one or more of the figures in this letter are available online
at http://ieeexplore.ieee.org. system reliability indices are known, and some components’
Digital Object Identifier 10.1109/TPWRS.2019.2952518 reliability parameters are unknown. Therefore, there would be a

0885-8950 © 2019 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.
See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 35, NO. 1, JANUARY 2020 819

TABLE I TABLE III

CUSTOMER/ LOAD- AND ENERGY-ORIENTATED INDICES TWO POSSIBILITIES FOR THE PRESENTED LINEAR EQUATIONS IN (3)

If the repair/replacement times are assumed to be known,

therefore we have a system of linear equations as follows:
⎧ N ⎫
⎪ 
⎪
⎪
⎪ c1n λn = SAIF I ⎪
⎪
⎪
⎪
⎪
⎪ n=1 ⎪
⎪
⎨ N ⎬
c2n λn = SAIDI ≡ {C3×N XN ×1 = B3×1 } (3)
⎪
⎪ n=1 ⎪
⎪
⎪
⎪ ⎪
⎪
⎪
⎪
N ⎪
⎩ c3n λn = EN S ⎪ ⎭
n=1

where, X is the vector of unknown parameters. There are two

possibilities in solving this system of equations, as concluded in
Table III [3]. In case 1, matrix C is square and invertible, so the
system of equations has a unique solution. In addition, if N < 3
and matrix B is in the column space of C, one solution can be
obtained. In case 2, there are three non-parallel planes, which
intersect in a line. Hence, there are infinite solutions, which are
obtained from the sum of particular solution xp and null space
vectors xs , as follows [3]:

TABLE II x = xp + xs (4)
PARAMETERS AND VARIABLES OF EQUATION (1)

where, rankC is the maximum number of linearly independent

column vectors in matrix C.
It should be noted that the presented system of equations
could only be formed in radial distribution systems, considering
that the restoration time of a load point is either equal to the
repair time of the component or the switching time to isolate
the faulty section of the system. In addition, during the period
that reliability indices are measured, the distribution system
may experience several reconfigurations. If a reconfiguration
led to a new permanent structure for the distribution system
system of equations as follows: and the system was not returned to the original structure after a
while, the proposed IRE problem may lead to misleading results.
⎧N N ⎫ Therefore, the proposed method in this letter is mainly applicable
⎪
⎪ LP  C
⎪
⎪
⎪
⎪ a1ij λij = f1 (λn ) = SAIF I ⎪
⎪ to networks, which are returned to their original structure after
⎪
⎪ i=1 j=1 ⎪
⎪
⎪
⎨ N ⎪
⎬ reconfiguration procedures. To tackle this issue, one may eval-

LP N C
a2ij λij rij = f2 (λn , rn ) = SAIDI , (2) uate reliability indices for short periods, considering reliability
⎪
⎪ i=1 j=1 ⎪
⎪
⎪
⎪ ⎪
⎪
implications in the truncated data.
⎪
⎪ N 
LP N ⎪
a3ij λij rij = f3 (λn , rn ) = EN S ⎪
C
⎪
⎩ ⎪
⎭
i=1 j=1
III. RESULTS AND DISCUSSIONS
where, f1 , f2 and f3 are algebraic functions of λn and rn as the The proposed problem is examined on feeder 1 of the RBTS
unknown parameters, and n ∈ {1, 2, .., N }. N is the number bus 2 with the input data reported in [4]. It is assumed that the
of unknowns. This nonlinear system of equations may lead to system is designed with disconnectors, fuses, alternative supply,
infinite solutions. In this case, the region of solutions can be and transformers. SAIFI, SAIDI and ENS are reported by the
limited to a bounded interval by other data available for the operator as 0.248 (interruptions/cust.yr), 3.62 (hrs/cust.yr) and
operator. 13.172 (Mwh/yr), respectively.

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820 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 35, NO. 1, JANUARY 2020

Fig. 3. Mapping from λ coordinates to δ coordinates.

λ2 and λ3 . If the network operator determines the target values

for the reliability indices, any points on the line can be chosen
Fig. 1. Solution area for the system of nonlinear equations with four unknown to achieve the desired values for reliability indices.
parameters λ2 , λ3 , r2 and r3 .
In addition, assume that λ1 , λ2 and λ4 are unknown; so the
system of equations is
⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤
1 0.3221 1 λ1 0.1379 0.2480
⎣3.5767 1.6104 2.2945⎦⎣ λ2 ⎦ + ⎣ 3.2693 ⎦ = ⎣ 3.6200 ⎦.
7.9250 2.6750 8.0490 λ4 12.2890 13.1720
(8)
This condition is based on Case 1 in Table III.
Hence, a unique solution will be obtained as [ λ1 λ2 λ4 ] =
[ 0.0488 0.0390 0.0488 ]. These values are the same as the values
of the parameters reported in [4]. In this case, assume that the
network operator decides to decrease SAIFI, SAIDI, and ENS
for the next year by, respectively, δ1 %, δ2 %, and δ3 %. According
Fig. 2. Line of solutions for three unknown parameters λ1 , λ2 and λ3 .
to the mapping from λ coordinates to δ coordinates shown in
Fig. 3, the range of indices’ changes can be 0% ≤ δ1 ≤ 44.61%,
0% ≤ δ2 ≤ 9.69%, and 0% ≤ δ3 ≤ 6.74%. For instance, if the
λ2 , λ3 , r2 , and r3 are assumed to be the unknown parameters.
network operator wants to decrease SAIFI, SAIDI, and ENS, by
Thus, the system equations are:
⎧ 17%, 4% and 3%, [ λ1 λ2 λ4 ] should be [ 0.03 0.02 0.03 ].
⎪
⎨0.3221λ2 + 0.3221λ3 + 0.2187 = 0.2480
⎪
IV. CONCLUSION
0.3221λ2 r2 + 0.3221λ3 r3 + 3.4718 = 3.6200 (5)
⎪
⎪
⎩0.5350λ r + 0.5350λ r + 12.9286 = 13.1720 In this letter, the IRE problem in radial distribution systems
2 2 3 3
was presented. It was assumed that the network reliability indices
Using the available historical information for failure rates are known for the operator, and some of the failure rate and repair
and repair times of similar components, bounded intervals are time of components are unknown. To solve the problem, a system
selected for unknowns as of equations were presented. If the number of unknowns were
 less than or equal to the number of equations, there would be a
0.01 < λ2 , λ3 < 0.07(f /yr)
(6) unique solution for unknown parameters. Otherwise, a region
1 < r2 , r3 < 7(hr) for solutions is defined. The proposed IRE problem and the
Considering (5) and (6), infinite solutions are found as shown associated solutions were also used for approaching a partic-
in Fig. 1. Therefore, the network operator cannot accurately ular reliability index by choosing appropriate components. The
determine the value of unknown parameters. results showed the allowable ranges of components reliability
If λ1 , λ2 and λ3 are the unknown parameters, a system of parameters to achieve the targeted system reliability index.
linear equations according to Case 2 in Table II can be derived.
Hence, rankC = 2 and REFERENCES
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⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎣3.5767 1.6104 1.6104⎦ ⎣λ2 ⎦ + ⎣ 3.2991 ⎦ = ⎣ 3.6200 ⎦ Syst., vol. 33, no. 6, pp. 6569–6578, Nov. 2018.
[2] R. Billinton and R. N. Allan, Reliability Evaluation of Power Systems. New
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