Quantifying Impact of Directed Elements on [625878]

Quantifying Impact of Directed Elements on
Structural Reliability Indices of Electrical Power
System

Oleg Kotov, Sergey Gusev
Ural Federal University
Ekaterinburg, Russia
[anonimizat] Vladislav Oboskalov
Ural Federal University;
ITP of the Ural Branch of the Russian Academy of Science
Ekaterinburg, Russia

Abstract—In order to take into account distinctive features of
contingency and post-contingency states in structural reliability
calculations, researchers divide power system failures into two
types – “short-circuit” failures (SCF) and “cut-off” failures
(COF). These failures differ significantly in repair time and, as a
result, in analytical and mathematical models. The paper focuses
mainly on the “short-circuit” failures. It presents an approach for
structural reliability calculations, which takes into account the
direction of power flows. In addition, it allows calculating
additional structural reliability indices (SRI), for instance, SRI for
simultaneous failure of several buses. The paper shows that
probability of a short-circuit switching failure is one of the key
indices for calculations concerning SCF. The paper proposes an
integrated “exclusion-restoration” technique for SRI calculations
concerning SCF. The proposed technique has been tested on
several test systems.
Keywords—electrical power system, power grid, sheme
reliability, reliability indices, power system simulation, stochastic
simulation, short-circuit failure.
I. INTRODUCTION
Traditionally, reliability assessment for any technical system
is associated with the estimation of stochastic system parameters
which characterize connectivity of a certain design scheme
which, in turn, is more or less similar to the real system in its
behavior [1, 2]. In the framework of electrical power systems,
such approach defines the main idea for calculating the electrical
power system (EPS) structural reliability (SR) [3-5]. The idea
implies that the consumer bus is supplied if there is at least one
electrical path connecting the bus with generation and all the
path elements are operating properly [6, 7]. In this case, SRI
characterize load shedding when an element (or several
elements) of the path is being repaired after a failure. This
approach for SR calculations is referred to as the “cut-off”
failure model. In fact, the model ignores key electrical features
of an EPS, such as the Ohm law and the Kirchhoff law [8, 9].
Moreover, it does not consider permissible boundaries of state
parameters, a shunting impact of short-circuit failures, etc. In
addition, the COF model usually does not account for relay
protection and does not simulate its failure or incorrect
operation. In order to increase the accuracy of SR calculations,
researchers implement the COF model in combination with the
SCF model. The SCF model [10] considers the failures
associated with short circuiting of power system elements. Such
failures are eliminated by routine switching in automatic or
manual manner. Manual switching procedure has a certain
timeline – usually one hour or half of an hour. As a result,
researchers can simulate characteristic features of certain power
equipment, such as circuit breakers, disconnectors and relay
protection [5, 11, 12].
It should be noted that some EPS elements, for instance,
step-down transformers, are directed, i.e. they can transfer
power only in a certain direction. When the COF and the SCF
models consider the direction of the EPS elements, this leads to
a better (more adequate) simulation of transformers and directed
automatic transfer circuit breakers (ATCB). Moreover, directed
ties with a zero failure probability help to evaluate stochastic
parameters for complex failures, such as a simultaneous failure
of several selected buses [5, 12].
II. DIRECTED MODEL FOR STRUCTURAL RELIABILITY
CALCULATIONS
This paper shows the impact of directed ties on overall SRI
basing on a case study featuring a simple test system with two-
winding power transformers and directed ATCBs (Fig. 1). The
system has two power supplies – Bus 1 and Bus 2. It presents a
high-voltage power line with two-way power supply (Line 1)
connecting two customer substations (Substation 1 and
Substation 2). In normal operating conditions the line is
sectionalized by circuit breaker L1-5 at Substation 2. Moreover,
the system has a power line with one-way power supply (Line
2) sectionalized by directed ATCB L2-8 at Substation 1. Buses
7 and 8, as well as 9 and 10, are splitted by a non-directed ATCB.
For the sake of a better illustration, the case study doesn’t
simulate inherent failures of buses. All elements of the test
system have the failure probability of 0.01 and the failure rate of
1 yearିଵ. The probability of a circuit breaker’s failure to trip
short-circuit is considered equal to 0.1 for all circuit breakers
[13].
The work was supported by Act 211 Government of the Russian
Federation, contract № 02.A03.21.0006.

Fig. 1. The directed design scheme.
ATCB failure probability is defined by (1):
ܳ஺்஼஻ൌܳ஼஻൅ܳ஺்ௌെܳ஼஻ܳ஺்ௌ, (1)
where ܳ஼஻ is a failure probability of a circuit breaker; ܳ஺்ௌ is a
failure probability of an automatic transfer switch.
In accordance with (1) and [13] ATCB failure probability
equals to ܳ஺்஼஻ൌ0.01൅0.12െ0.01∙0.12 ൌ0.1288.
The test system has two design schemes – directed and non-
directed. The first scheme has no constraints as to the power
flow direction for step-down transformers and the circuit breaker
that connects line “2” and bus “8”. The second scheme, in turn,
has five directed ties. Arrows in Fig. 1 show the permissible
power flow direction for these ties.
All calculations in the paper are made by software
«STRUNA» which considers both the SCF and the COF failure
models, as well as directed ties. Basic mathematical models and
approaches of the software are described in detail in [5].
The case study considers the substation buses (buses 3-10).
According to the non-directed design scheme, failure
probabilities and failure rates of these buses characterize load
shedding when failed system elements are being repaired (the
COF model). When directed EPS elements are ignored, SRI for
the buses correspond to the maximal topological structure. It is
possible to estimate the indices by traditional approaches which
are used in the reliability theory of technical systems. Failure probabilities and failure rates for the analyzed buses
were calculated according to the COF model. The indices are
shown in Table I (columns 2-3). The short-circuit failure model
provides failure rates (Table I, column 4) which characterize
load shedding that is caused by a reconstructive switching
routine. The case study assumes that the duration of the
switching routine is 1 hour.
In practice, short-circuit failures do not increase the resulting
bus failure probability due to their relatively short duration.
However, they significantly affect the overall failure rate (Table
I, column 5). This issue is extremely important for the first
category consumers which should not be blacked-out longer
than duration of automatic transfer switching.
Table II shows resulting reliability indices for the directed
design scheme. The scheme considers power transformers and
ATCB 8-L2 as directed elements. SRI calculating procedure
implies that directed ties have different failure probabilities in
forward and backward directions, i.e. in the backward direction
the failure probability is q=1. Due to this fact, all the test system
buses (with the exception of Bus 7 and Bus 8) have worse
connectivity with the power supply buses in comparison with a
non-directed design scheme. Consequently, the resulting failure
probabilities and failure rates for these buses are higher.
Table III and Fig. 2 compare the resulting reliability indices
for the directed and the non-directed design schemes. The
difference is shown in percent, basing on the non-directed
reliability indices.
The horizontal axis in Fig. 2 presents the analyzed buses,
while the vertical axis shows the difference in the resulting
reliability indices for diverse design schemes and failure models.
The comparison shows that the largest increase in the failure
rate in the SCF model is concerned with Bus 9 and Bus 10. In
the COF model, the failure probability and the failure rate for
these buses increases also. This is due to their location. The
buses are supplied only by Bus 2 and the supply path has a radial
part. In the COF model, the failure probabilities and the failure
rates for Bus 3, Bus 4, Bus 5 and Bus 6 grow significantly when
directed EPS elements are taken into consideration. In the SCF
model, the failure rates for these buses are decreasing.
Reliability indices for Bus 7 and Bus 8 do not change. This is
because directed ties do not change connectivity of Bus 7 and
Bus 8.
TABLE I. RELIABILITY INDICES FOR THE COF AND THE SCF MODELS (THE NON-DIRECTED DESIGN SCHEME )
Bus COF model SCF model Overall failure rate,
(year-1) Layout of overall failure rate, %
Failure probability, (p.u.) Failure rate, (year-1) Failure rate, (year-1) COF SCF
3 0.001553 0.1888 3.495 3.68 5 95
4 0.001852 0.248 3.862 4.11 6 94
5 0.001852 0.248 2.762 3.01 8 92
6 0.001553 0.1888 3.386 3.57 5 95
7 0.0054 0.5968 2.961 3.56 17 83
8 0.001553 0.1888 4.072 4.26 4 96
9 0.001553 0.1888 4.044 4.23 4 96
10 0.001553 0.5968 2.934 3.53 17 83
Line 1
Line 2
Bus 4
Bus 5
Bus 6
Bus 7
Bus 8
Bus 9
ATCB
ATCB
ATCB
Substation 1
Substation 2
L2-8
L1-5
L1-4
Bus 3
Bus 1
Bus 2
Bus 10
L2-9

TABLE II. RELIABILITY INDICES FOR THE COF AND THE SCF MODELS
(THE DIRECTED DESIGN SCHEME )
Bus COF model SCF model
Failure probability (p.u.) Failure rate (year-1) Failure rate (year-1)
3 0.010 1.000 2.310
4 0.020 2.000 3.300
5 0.020 2.000 2.200
6 0.010 1.000 2.200
7 0.005 0.597 2.961
8 0.002 0.189 4.072
9 0.017 1.881 5.632
10 0.016 1.751 4.532
TABLE III. COMPARISON OF SRI FOR THE DIRECTED AND THE NON –
DIRECTED DESIGN SCHEMES
Bus Difference in failure
probability for the COF
model, % Difference in failure
rate for the COF
model, % Difference in failure
rate for the SCF
model, %
3 84.47 81.12 -51.30
4 90.69 87.60 -17.03
5 90.69 87.60 -25.55
6 84.47 81.12 -53.91
7 0.00 0.00 0.00
8 0.00 0.00 0.00
9 90.90 89.96 28.20
10 65.89 65.92 35.26

Fig. 2. Difference in resulting reliability indices.
Table IV presents the layout of the overall failure intensity
for the COF and SCF models when directed ties are taken into
consideration.
The results show that the directed design scheme increases
a part of the COF model in the layout of the overall failure rate.
In fact, the failure rates in the COF model in the non-directed
design scheme vary from 4% to 17%, while in the directed
scheme they vary from 4% to 48%. The SCF model has the
opposite trend.

TABLE IV. LAYOUT OF THE OVERALL FAILURE INTENSITY FOR THE COF
AND THE SCF MODELS
Bus Overall failure rate, (year-1) Layout of the overall failure rate, %
COF model SCF model
3 3.310 30 70
4 5.300 38 62
5 4.200 48 52
6 3.200 31 69
7 3.558 17 83
8 4.261 4 96
9 7.513 25 75
10 6.283 28 72
III. VERIFICATION OF THE RESULTS
«STRUNA» results can be verified by manual calculations.
For instance, let’s consider Bus 3 (Fig. 1). SRI calculations in
the COF model lead to series-parallel network reductions in
accordance with the following relationships:
ܳௌ௘௥௜௘௦ൌܳଵ൅ܳଶെܳଵ∙ܳଶ; (2)
ߣௌ௘௥௜௘௦ൌߣଵ൅ߣଶ; (3)
ܳ௉௔௥௔௟௟௘௟ൌܳଵ∙ܳଶ; (4)
ߣ௉௔௥௔௟௟௘௟ൌሺߣଵ∙ሺ1െܳଵሻ∙ܳଶ൅ߣଶ∙ሺ1െܳଶሻ∙ܳଵሻ
ሺ1െܳଵ∙ܳଶሻ, (5)
where ܳ is failure probability; ߣ – failure rate.
First stage of network reduction of the non-directed design
scheme is shown in Fig. 3. It relies on (2) – (3). SRI at the first
stage of network reduction are presented in Table V.

Fig. 3. First stage of network reduction.
TABLE V. SRI AT THE FIRST STAGE OF NETWORK REDUCTION
Element q λ
element 3-7 0,02901 3
element 4-8 0,02901 3
element 8-9 0,15467 3
element 5-9 0,02901 3
element 6-10 0,02901 3
-50-30-101030507090110
3 4 5 6 7 8 9 10Qото
Lото
LоткзQCOF
LCOF
LSCF
Bus 1
Bus 3
Bus 4
Bus 5
Bus 6
Bus 7
Bus 9
Bus 10
Bus 2
element 3-7
element 8-9
element 4-8
element 5-9
element 6-10
Bus 8

Second stage of network reduction of the non-directed
design scheme is shown in Fig. 4. SRI at the second stage of
network reduction are presented in Table VI.

Fig. 4. First stage of network reduction.
TABLE VI. SRI AT THE SECOND STAGE OF NETWORK REDUCTION
Element q λ
element 3-7-8 0,53233 4
element 3-4-8 0,3439 4
element 9-5-6 0,3439 4
element 9-10-6 0,53233 4
Third stage of network reduction of the non-directed design
scheme is shown in Fig. 5. It bases on (4) – (5). SRI at the third
stage of network reduction are presented in Table VII.

Fig. 5. First stage of network reduction.
TABLE VII. SRI AT THE THIRD STAGE OF NETWORK REDUCTION
Element q λ
element 3-8 0,183068 2,49761
element 9-6 0,183068 2,49761

At the final stage of SRI calculations, the series circuit
connecting Bus 3 and Bus 2 is reduced. Resulting element forms
a parallel circuit with element 3 – 1 (Fig. 5). Futher reduction
results in the following SRI: ܳଷ஼ைிൌ0.00164; ܮଷ஼ைிൌ
0.209 yearିଵ.
According to the directed design scheme, Bus 3 is supplied
only through series circuit by Bus 2. As a result, it has the
following SRI: ܳଷ஼ைிൌ0.01; ܮଷ஼ைிൌ1 yearିଵ.
Failure rate of Bus 3 in the SCF model defines by:
ܮଷ஼ைிൌ෍ሺ௜௝
∀௜௝൅൫௝௔ௗ௝൅௝൯∗௝௜, (6)
where ij is a tie adjacent to Bus i; ௝௔ௗ௝ is an overall failure rate
of all ties adjacent to Bus j (with an exception to tie ij); λj –
failure rate of Bus j, ௝௜ is a rate of short circuit trip failure of tie
ji. In result, the non-directed design scheme has failure rate
ܮଷௌ஼ிൌ3,2 yearିଵ. The directed design scheme, in turn, has
ܮଷ஼ைிൌ2,2 yearିଵ. Therefore, the manual calculations support
«STRUNA» results with calculation error within 5 – 9%.
IV. CONCLUSIONS
The structural reliability analysis should consider specific
EPS features, such as directed elements and short-circuit failures
in order to increase the accuracy and adequacy of the resulting
reliability indices.
All failures can be classified into two types, depending on
their duration: the short-circuit failures which are eliminated by
reconstructive routine switching (usually within 0.5-1 hour) and
the cut-off failures which are eliminated by reconstructive
maintenance (more than 1 hour). As a result, the SR calculations
require both the SCF and the COF models.
Directed ties significantly change the resulting SRI and
should be considered in the SR calculations, especially when an
EPS has the first category consumers. Directed elements
substantially increase the failure probability and failure rate in
the COF model. In the SCF model, the resulting SRI show a
multidirectional change. Certain SRI are increasing, while
others are decreasing.
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Bus 1
Bus 3
Bus 6
Bus 9
Bus 2
element 1-3
element 8-9
element 3-4-8
element 9-5-6
element 9-10-6
element 6-2
element 3-7-8
Bus 8
Bus 2
element 3-8
Bus 1
Bus 3
Bus 6
Bus 9
element 1-3
element 8-9
element 9-6
element 6-2
Bus 8