Access to this full-text is provided by MDPI.
Content available from Processes
This content is subject to copyright.
processes
Article
Reliability Study of Parameter Uncertainty Based on
Time-Varying Failure Rates with an Application to Subsea Oil
and Gas Production Emergency Shutdown Systems
Xin Zuo 1,*, Xiran Yu 2, Yuanlong Yue 1, Feng Yin 3and Chunli Zhu 3
Citation: Zuo, X.; Yu, X.; Yue, Y.;
Yin, F.; Zhu, C. Reliability Study of
Parameter Uncertainty Based on
Time-Varying Failure Rates with an
Application to Subsea Oil and Gas
Production Emergency Shutdown
Systems. Processes 2021,9, 2214.
https://doi.org/10.3390/pr9122214
Academic Editor: Alexey V. Vakhin
Received: 9 November 2021
Accepted: 2 December 2021
Published: 8 December 2021
Publisher’s Note: MDPI stays neutral
with regard to jurisdictional claims in
published maps and institutional affil-
iations.
Copyright: © 2021 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
1
Department of Automation, China University of Petroleum, Beijing 102249, China; yueyuanlong@cup.edu.cn
2College of Information Science and Engineering, China University of Petroleum, Beijing 102249, China;
2019211238@student.cup.edu.cn
3
CNOOC Research Institute, Beijing 100027, China; yinfeng@cnooc.com.cn (F.Y.); zhuchl@cnooc.com.cn (C.Z.)
*Correspondence: zuox@cup.edu.cn
Abstract:
The failure rate of equipment during long-term operation in severe environment is time-
varying. Most studies regard the failure rate as a constant, ignoring the reliability evaluation error
caused by the constant. While studying failure data that are few and easily missing, it is common
to focus only on the uncertainty of reliability index rather than parameter of failure rate. In this
study, a new time-varying failure rate model containing time-varying scale factor is established,
and a statistical-fuzzy model of failure rate cumulated parameter is established by using statistical
and fuzzy knowledge, which is used to modify the time-varying failure rate model. Subsequently,
the theorem of the upper boundary existence for the failure rate region is proposed and proved to
provide the failure rate cumulated parameter when the failure rate changes the fastest. The proposed
model and theorem are applied to analyze the reliability of subsea emergency shutdown system in
the marine environment for a long time. The comparison of system reliability under time-varying
failure rate and constant failure rate shows that the time-varying failure rate model can eliminate
the evaluation error and is consistent with engineering. The reliability intervals based on the failure
rate model before and after modification are compared to analyze differences in uncertainty, which
confirm that the modified model is more accurate and more practical for engineering.
Keywords:
failure rate; time-varying; reliability evaluation; parameter uncertainty; model modifica-
tion; subsea emergency shutdown system
1. Introduction
As the core part of subsea production system, subsea control system has important
functions of monitoring production status and manipulating control equipment [
1
–
3
]. The
mainstream type of subsea control system is multiplexed electro-hydraulic control system,
which has short response time, high redundancy and low cost of umbilical cable, and
can give consideration to both reliability and economy [
4
–
6
]. An indispensable part of
the multiplexed electro-hydraulic control system is subsea emergency shutdown (ESD)
system, which can prevent the occurrence of major production accidents to a great extent,
ensure the stable production of oil and gas fields, and protect the personal safety of field
personnel, production facilities, and marine environment [
7
]. Therefore, the reliability level
of subsea ESD system determines the safe operation of offshore oil and gas exploitation,
and its reliability evaluation is of far-reaching significance.
The failure rate of equipment during reliability evaluation is the essential data support.
In most reliability studies, the failure rate of each equipment is usually simplified to a
constant value due to the lack of basic data such as failure records of partial equipment.
Wang et al. [
8
] established a reliability model for the electrical control system of the subsea
control module by markov processes and multiple beta factor model using the constant
Processes 2021,9, 2214. https://doi.org/10.3390/pr9122214 https://www.mdpi.com/journal/processes
Processes 2021,9, 2214 2 of 19
failure rate (CFR) and its value range. When assessing the reliability and safety of sub-
sea Christmas tree, Pang et al. [
9
] converted the failure rate of hydraulic and electronic
components which obey an exponential distribution and mechanical components with
Weibull distribution into a constant. Bae et al. [
10
] referred to the CFR of equipment
in offshore and onshore reliability data (OREDA), and used the multi-objective design
optimization method to optimize the ESD system to ensure its high reliability and rea-
sonable cost.
Signorini et al. [11]
collected 106 subsea control module (SCM) field data
sets and compared them with OREDA for quantitative and qualitative reliability study of
SCM. None of these studies have considered the reliability evaluation error caused by the
CFR. Not only is the field of offshore oil research prone to ignore this error, but numerous
other fields tend to assume a CFR as well. Ismagilov et al. [
12
] proposed a combined
method of analysis the reliability indicators based on functional-cost analysis and failure
modes and effects analysis for the aviation electromechanical system, taking a CFR when
calculating the failure occurrence probability. When calculating the reliability of the battery
electric vehicle powertrain system, Tang et al. [
13
] simplified the failure rate into constant.
Tawfiq et al. [14]
used the block diagram technique to minimize the number of system
component rates after integrating multiple factors, either failure or repair. It is considered
that the failure rate unchanged during modeling.
The failure rate of the subsea ESD system varies over time after being affected by
factors such as equipment, environment, and operation. Considering the time-varying
characteristics of the equipment failure rate when establishing the reliability model, it can
not only reduce the evaluation error caused by the CFR, but also deeply explore the impact
of the time-varying failure rate (TFR) on reliability. Earlier Hassett et al. [
15
] used a general
polynomial to express the TFR and proposed a hybrid reliability and availability analysis
method combined with Markov chain analysis. In industrial applications, the failure rates
of most equipment vary with the service life of components. Retterath
et al. [16]
established
the TFR model of distribution system, then analyzed the impact of TFR on distribution
system by Monte Carlo simulation. Wang et al. [
17
] divided the equipment failure of
relay protection device into random failure and aging failure, and proposed the estimation
approach for the TFR of relay protection device. Abunima et al. [
18
] determined the TFR
of photovoltaic system by comprehensively considering weather conditions, PV system
architecture and components, interactions of PV components with the weather conditions.
Liu et al. [
19
] combined historical fault data with related weather forecasts to establish an
overhead transmission line reliability model considering the TFR, thereby proposing an
optimal inspection strategy. Zhang et al. [
20
] fitted the failure rate with the length of the
submarine cables, and proposed an improved dynamic reliability model of the TFR based
on the seasonal changes of the failure caused by fishing operations and anchor damage.
Li et al. [21]
established a multi-state Markov failure rate prediction model for transformers,
and used the aging failure model to modify it to accurately predict the real-time failure
rate. Liu et al. [
22
] used an exponential function that is more in line with the actual trend
to characterize the TFR of the solenoid valve power supply, which accurately reflects the
aging process of the power system. However, the parameter uncertainty of TFR model is
not studied in this work.
The problem of parameter uncertainty is particularly prominent in reliability analysis,
which brings uncertainty to the system reliability evaluation and affects the accuracy. When
the reliability uncertainty exceeds a certain range, the result of reliability analysis will lose
practical significance. Zhang et al. [
23
] discussed the influence of the uncertainty of each
parameter on the output power performance, stability, and reliability by modeling the
randomness of the parameters affecting the output of a photovoltaic cell. Li et al. [
24
] used
non-sequential Monte Carlo simulation to establish the analytical expressions of variable
reliability parameters, so as to establish a system of nonlinear equations for solving the
reliability parameters of unknown equipment. Wang et al. [
25
] established the gray three-
parameter Weibull distribution model of the relay protection device, and estimated the
reliability parameters to ensure the calculation speed and accuracy.
Miranda et al. [26]
con-
Processes 2021,9, 2214 3 of 19
sidered the uncertainty of stochastic equipment data in power system expansion planning,
and calculated uncertainty by using interval arithmetic through the theory of imprecise
probabilities. The parameters of the software reliability model were estimated and pre-
dicted by Zhen et al. [
27
] using the hybrid WPA-PSO algorithm. Wang et al. [
28
] carried out
point estimation and approximate interval estimation of model parameters for Kijima type
Weibull generalized renewal processes models I and II, and proposed a calculation method
of the reliability indices for repairable systems with imperfect repair. The failure rates of
the repairable system modules obeying exponential distribution were estimated by Uprety
and Patrai [
29
] using the fuzzy triangle number obtained from the point estimation and
confidence interval. Yang et al. [
30
] focused on the reliability uncertainty of wind power
systems, and estimated the unknown parameters of the autoregressive integrated moving
average prediction model using bayesian estimation methods based on Markov chain
Monte Carlo. Wang et al. [
31
] utilized the statistical properties of the poisson binomial
distribution to develop analytical confidence intervals for failure probability estimation.
Hu et al. [
32
] regarded the state probability and performance rate of the multi-state device
as uncertain variables, and combined with probability theory and uncertainty theory to
introduce the uncertain universal generating function for reliability analysis of random
uncertain multi-state system with missing samples. Li et al. [
33
] established a life model
and uncertainty statistical method based on uncertainty theory considering life test type-I
and type-II censoring, precise and interval data of failure data. Wang et al. [
34
] aimed
at the uncertainty of the reliability data in the phasor measurement unit by combining
statistical methods and fuzzy Markov to establish parameter membership functions of
reliability indices, and evaluated the impact of parameter uncertainty. These studies only
focus on system reliability indexes and ignore the uncertainty of failure rate parameters.
The failure rate of subsea emergency shutdown (ESD) system is time-varying due to
long-term operation in complex marine environment, and its failure data has the character-
istics of small quantity, long collection period, high confidentiality (less access). In order
to solve the shortcomings that ignore the time-varying and simplify the failure rate to a
constant value, ignore the existence of uncertainty in the failure rate parameter identified in
previous studies, a new time-varying failure rate (TFR) model including time-varying scale
factor is established in this study. Firstly, a statistical-fuzzy model of the failure rate cumu-
lated parameter is established by combining statistical and fuzzy set knowledge, and the
TFR model is effectively modified using this model. Secondly, for the modified TFR model,
the theorem of upper boundary existence for failure rate region is proposed and proved. In
this study, the above theoretical method is used for the subsea ESD system. By comparing
and calculating the reliabilities of the systems under the TFR and CFR, it is shown that
the new TFR model can eliminate the evaluation error caused by constant failure rate
(CFR) and make the variation tendency of system reliability consistent with the industrial
application. The upper boundary existence theorem accurately provides the failure rate
cumulated parameter when the failure rate changes the fastest, and then uses TFR models
before and after the modification to compare the uncertainty of failure rate cumulated
parameter to the uncertainty of system reliability, the degrees of influence are different.
The modified TFR model effectively corrects the reliability interval containing uncertainty.
The rest of this paper is organized as follows: in Section 2, the TFR model and the
statistical-fuzzy model of failure rate cumulated parameter are established, and the TFR
model is modified with the model of failure rate cumulated parameter. The theorem of
upper boundary existence for the failure rate region is proposed and proved, and the
numerical example is given in Section 3. Section 4takes the subsea ESD system as the
analysis object. Firstly, the reliabilities of the systems under the TFR and the CFR are
compared and evaluated, and then the reliability simulations of the systems are compared
and analyzed based on TFR models before and after modification. Finally, conclusions are
given in Section 5.
Processes 2021,9, 2214 4 of 19
2. Modeling and Modification
2.1. Time-Varying Failure Rate Model
Failure rate is one of the most important indexes for calculating reliability. Typical
equipment failure rate is a curve with time of use as the abscissa and failure rate as the
ordinate. Its trend is high at the beginning, low in the middle, and high at the end, so it is
called the “bathtub curve” [35].
The bathtub curve is divided into burn-in phase, steady state phase, and wear-out
phase, which is shown in Figure 1. Due to defects of equipment raw materials and manufac-
turing, the failure rate in the burn-in phase is usually high. Then the equipment operation
after debugging and running-in tends to be normal, while the failure rate decreases. The
failure rate of the steady state phase is constant and the lowest. Over time, the failure rate in-
creases with the aggravation of equipment wear and fatigue, and enters the wear-out phase.
Burn-in phase
Failure rate
Time
Steady state phase Wear-out phase
Figure 1. Diagram of bathtub curve.
The equipment will complete factory acceptance tests to filter out early failures, so the
TFR modeling can skip the burn-in phase and only consider the steady state phase and
wear-out phase. The CFR in the steady state phase is usually used in reliability evaluation,
that is, the exponential distribution which is too ideal for equipment is selected. Therefore,
this study puts forward a time-varying scale factor and improves it in the wear-out phase
to comprehensively describe the TFR in the steady state phase and wear-out phase.
The time-varying scale factor varies over time during the equipment life cycle, and
the CFR is determined by the time-varying scale factor to increase or decrease. The TFR in
this paper is as follows:
λ(t) = ε(t)λ0, (1)
where
ε(t)
is the time-varying scale factor and
λ0
is the CFR of the equipment. The time-
varying scale factor of the steady state phase and the wear-out phase are given below,
respectively.
1.
Steady state phase. The failure distribution is consistent with the bottom of the
“bathtub curve” when the equipment is running normally, which means the failure
rate is constant. The time-varying scale factor is a constant value of 1 in this phase.
ε(t) = 1 (2)
2.
Wear-out phase. The wear-out phase is usually described by Weibull distribution,
because the shape parameter in Weibull distribution is excessive that will lead to a
rapid rising trend of failure rate [
22
], the time-varying scale factor of the wear-out
phase in this study adopts an exponential function to reasonably describe the rising
trend of “bathtub curve”.
Processes 2021,9, 2214 5 of 19
ε(t) = eη(t−T), (3)
where
η
is the failure rate cumulated parameter of equipment, and
T
is the duration
of the steady state phase. Combined with the previous equations, the TFR model of
this paper is determined as:
λ(t) = ε(t)λ0=λ0t≤T
eη(t−T)λ0t>T(4)
2.2. Statistical-Fuzzy Model of the Failure Rate Cumulated Parameter
There is a degree of uncertainty in the failure rate cumulated parameter obtained
either by fitting failure data or by using expert experience. The fitting may suffer from
missing data and low precision, and expert experience is subjective. In this paper, the
interval value of the failure rate cumulated parameter is used instead of the single value,
which can not only express the uncertainty of the parameter quantitatively, but also enable
the reliability analysis result to cover this uncertainty. Since the concept of the cut set in
fuzzy membership function is consistent with a range [
34
], this study uses the parameter
estimation in mathematical statistics combined with expert experience to carry out interval
estimation for failure rate cumulated parameter. The Pseudo-Gaussian (PG) membership
function of the failure rate cumulated parameter is further obtained, and the statistical-
fuzzy model of the failure rate cumulated parameter is established to directly reflect the
uncertainty interval of the failure rate cumulated parameter.
The failure rate cumulated parameter
η
is estimated using the chi-square distribution
at a given significant level α0.
ηα0
L,ηα0
U=hχ2
1−α0/2(2r)/2Tr,χ2
α0/2(2r+2)/2Tri(r=1, 2, · · · ,n−1), (5)
¯
η=η0, (6)
where
χ2
1−α0/2(
2
r)
is the quantile when the integral from 0 to
χ2
1−α0/2(
2
r)
of chi-square
distribution is
α0/
2 under the degree of freedom 2
r
,
χ2
α0/2(
2
r+
2
)
is the quantile when
the integral from
χ2
α0/2(
2
r+
2
)
to
∞
is
α0/
2 under the degree of freedom 2
r+
2.
Tr
is the
statistical time, and η0is the expert experience value.
Considering the asymmetry when using the chi-square distribution, a PG membership
function of the failure rate cumulated parameter is established.
f(x,σ,c) =
exp−(x−c)2
σL2x≤c
exp−(x−c)2
σR2x>c
, (7)
where
x
represents the failure rate cumulated parameter, and
c
is the expert experience
value of the failure rate cumulated parameter,
σL
and
σR
are the scale parameters of the left
and right parts of the PG membership function, respectively.
The degree of membership is similar to the concept of significant level, both of which
can reflect subjective confidence degree. Therefore, the upper and lower bounds of the
failure rate cumulated parameter
η
in (5) correspond to the upper and lower boundaries at
the 1 −α0-cut of the PG membership function.
exp−(ηα0
L−η0)2
σL2=1−α0
exp−(ηα0
U−η0)2
σR2=1−α0
(8)
Processes 2021,9, 2214 6 of 19
σL
and
σR
of the PG membership function of
η
can be estimated after the transforma-
tion of (8).
σL=−(ηα0
L−η0)/p−ln(1−α0)
σR= (ηα0
U−η0)/p−ln(1−α0)(9)
The PG membership function of the failure rate cumulated parameter can be obtained
by substituting (9) into (7), which is a convex function in the real number field as shown in
Figure 2.
1
0
1
−
0
L
0
U
0
0
( , )f
Figure 2. Diagram of the PG membership function for failure rate cumulated parameter.
Extending the PG membership function f(η,αi)to the general form as:
f(η,αi) =
exp(η−η0)2ln(1−αi)
(ηαi
L−η0)2η≤η0
exp(η−η0)2ln(1−αi)
(ηαi
U−η0)2η>η0
, (10)
where 1 −αiis an arbitrary cut set.
2.3. Modification of Time-Varying Failure Rate Model
Different failure rate cumulated parameter
η
corresponds to different trends of TFR
curves, and the time-varying scaling factor characterizing by the exponential function
increases monotonically with
η
during wear-out phase, so
ηαi
L,ηαi
U
corresponds to a region
enclosed by innumerable TFR curves. The upper and lower boundaries of the region
are determined by the upper and lower boundaries of the interval, which intuitively
reflects the degree of uncertainty contained in system reliability analysis. In order to
make the reliability analysis of the system with uncertainty have practical significance, the
uncertainty of the failure rate cumulated parameter cannot exceed a certain range, and the
upper and lower boundaries of the region enclosed by the failure rate curves should be
focused on. Considering that the membership function can be used to describe the degree
to which the object belongs to a certain definition, this study proposes a new method to
modify the TFR model by using the statistical-fuzzy model of the failure rate cumulated
parameter. This method can more objectively and accurately specify the upper and lower
boundaries of the region enclosed by the modified failure rate curves.
Firstly, the upper and lower boundaries of
ηαi
L,ηαi
U
and
η0
are substituted into (4),
respectively, and the corresponding TFR models are as follows:
ληαi
L
(t) = εηαi
L
(t)λ0=(λ0t≤T
eηαi
L(t−T)λ0t>T(11)
Processes 2021,9, 2214 7 of 19
ληαi
U
(t) = εηαi
U
(t)λ0=(λ0t≤T
eηαi
U(t−T)λ0t>T(12)
λη0(t) = εη0(t)λ0=λ0t≤T
eη0(t−T)λ0t>T(13)
The TFR curve drawn from (11)–(13) is shown in Figure 3. It can be directly observed
that the failure rate curve corresponding to
ηαi
L
rises the slowest, and the failure rate
curve corresponding to
ηαi
U
rises the most rapidly. The failure rate curve corresponding
to
η0
is sandwiched between the upper and lower boundary curves. The failure rate
ληαi
L
<λη0<ληαi
Uat any same time point in the wear-out phase.
Time
( )
t
0
i
L
i
U
t
0
Figure 3. Time-varying failure rate curves under different failure rate cumulated parameter.
To consider the confidence attached to
η
in the failure rate model objectively, the
TFR model is modified using the statistical-fuzzy model of
η
. This method is only for the
wear-out phase where parameter uncertainty exists, and the modified model is as follows:
λmo (t) = λ(t)·f(η,αi) = eη(t−T)λ0·exp"(η−η0)2ln(1−αi)
(ηαi−η0)2#(14)
The upper and lower boundaries of
ηαi
L,ηαi
U
and
η0
are substituted into (14), and the
corresponding modified TFR models are as follows:
λmo,ηαi
L
(t) = (λ0t≤T
(1−αi)eηαi
L(t−T)λ0t>T(15)
λmo,ηαi
U
(t) = (λ0t≤T
(1−αi)eηαi
U(t−T)λ0t>T(16)
λmo,η0(t) = λ0t≤T
eη0(t−T)λ0t>T(17)
When the TFR model is modified by the statistical-fuzzy model of
η
, the size order of
failure rates
ληαi
L
,
λη0
and
ληαi
U
at any same time point in the wear-out phase will change,
and the upper and lower boundaries of the region enclosed by TFR curves will also change
synchronously. The degree of change is discussed in Section 3.
Processes 2021,9, 2214 8 of 19
3. The Existence Proof of Upper Boundary of Modified TFR Region
3.1. The Upper Boundary Existence Theorem and Proof
The membership function of the failure rate cumulated parameter and the failure rate
function of the wear-out phase both increase monotonically within the interval
ηαi
L,η0
.
From (14), it can be seen that the modified failure rate function
λmo,ηαi
L
(t)<λmo,η0(t)
always holds. Comparing (15) and (16), it can be seen that the modified failure rate function
λmo,ηαi
L
(t)<λmo,ηαi
U
(t)
always holds. This study focuses on the TFR model modified by the
PG membership function on the right half.
λmo (t) = eη(t−T)λ0·exp"(η−η0)2ln(1−αi)
(ηαi
U−η0)2#(18)
The TFR function is shifted
T
units to the left ignoring the steady state phase, and the
modified TFR model is simplified as:
λmo (t) = eηtλ0·exp"(η−η0)2ln(1−αi)
(ηαi
U−η0)2#(19)
Theorem 1. The upper boundary existence theorem.
Assuming that parameters
t
,
αi
,
η0
,
λ0
are given, there is a definite upper bound-
ary of the region bounded by
λmo (t)
under the domain
ηα0
L,ηα0
U
of
η
, if and only if
η=minnηαi
U,η0−(ηαi
U−η0)2t/2 ln(1−αi)o.
Proof of Theorem 1.
Given that
t
,
αi
,
η0
,
λ0
, so
ηαi
U
is also a definite value, then only
η
is
the independent variable in
λmo (t)
. Rewrite
λmo (t)
as
λmo (η)
, and take the derivative of
η
in λmo (η).
dλmo (η)
dη=λ0eηt·exp"(η−η0)2ln(1−αi)
(ηαi
U−η0)2#·"t+2(η−η0)ln(1−αi)
(ηαi
U−η0)2#(20)
Let (20) be rewritten as:
dλmo (η)
dη=g(η)·h(η), (21)
where:
g(η) = λ0eηt·exp"(η−η0)2ln(1−αi)
(ηαi
U−η0)2#(22)
h(η) = t+2(η−η0)ln(1−αi)
(ηαi
U−η0)2(23)
According to the properties of failure rate and the exponential function,
λ0>
0,
eηt>
0,
exp(η−η0)2ln(1−αi)
(ηαi
U−η0)2>
0 always hold, therefore
g(η)>
0 always holds. The
following focuses on h(η).
h(η) = Kη+R(24)
Combining (23) and (24), the following can be obtained:
K=2 ln(1−αi)
(ηαi
U−η0)2(25)
Processes 2021,9, 2214 9 of 19
R=t−2η0ln(1−αi)
(ηαi
U−η0)2(26)
αi∈[0, 1]
,
t>
0,
η0>
0 is known, so
K<
0,
R>
0 always hold. It can be seen that
h(η)
is a straight line decreasing monotonically that intersects the positive vertical axis.
Suppose the intersection point of
h(η)
and the horizontal axis is
η0
, it follows that
η0>
0.
Furthermore, it is known that:
h(η)>0η<η0
h(η)≤0η≥η0(27)
Combining (21), (22), and (27) can be obtained:
dλmo (η)
dη>0η<η0
dλmo (η)
dη≤0η≥η0(28)
Therefore,
λmo (η)
increases first and then decreases progressively, and there must be a
local maximum value in the real number field. Given
h(η0) =
0, the local maximum point is:
η0=η0−(ηαi
U−η0)2t
2 ln(1−αi)>η0(29)
Given
η∈ηα0
L,ηα0
U
, to further obtain the maximum point of
λmo (η)
, compare the
size of η0and ηαi
Uby subtracting the two:
η0−ηαi
U=(η0−ηαi
U)2 ln(1−αi)−(η0−ηαi
U)t
2 ln(1−αi), (30)
where η0−ηαi
U<0, 2 ln(1−αi)<0.
According to (30), comparing the size of
η0
and
ηαi
U
means discussing the sign of
2 ln(1−αi)−(η0−ηαi
U)t.
ηαi
U>η0−2 ln(1−αi)/tη0>ηαi
U
ηαi
U<η0−2 ln(1−αi)/tη0<ηαi
U
(31)
Substitute ηαi
U=χ2
α0/2(2r+2)/2Trinto (31).
(χ2
α0/2(2r+2)>2Trη0−4Trln(1−αi)/tη0>ηαi
U
χ2
α0/2(2r+2)<2Trη0−4Trln(1−αi)/tη0<ηαi
U
, (32)
where
χ2
α0/2(
2
r+
2
)
is the quantile of the chi-square distribution table, which can be
obtained by referring to the table.
Equation (32) shows that there are mathematical conditions to determine the size of
η0
and
ηαi
U
. Since
λmo (η)
increases first and then decreases, the maximum point in domain
η∈ηα0
L,ηα0
Uis ηαi
Uwhen η0>ηαi
U, and the maximum point is η0when η0<ηαi
U.
To sum up: Assuming that parameters
t
,
αi
,
η0
,
λ0
are given, there is a definite upper
boundary of the region bounded by
λmo (t)
under the domain
ηα0
L,ηα0
U
of
η
, if and only if
η=minnηαi
U,η0−(ηαi
U−η0)2t/2 ln(1−αi)o. Theorem proving completed.
3.2. Numerical Example
A long-running pressure vessel safety instrument system [
36
] is composed of pressure
transmitter (PT), programmable logic controller (PLC), valve 1 (V1) and valve 2 (V2). The
PT measures the pressure in the vessel and feeds back to the PLC, and the PLC will shut
down V1 and V2 when the pressure exceeds the warning value. Figure 4is reliability block
diagram of the system, in which V1 and V2 are connected in parallel and then connected in
series with PT and PLC.
Processes 2021,9, 2214 10 of 19
PT
V1
V2
PLC
Figure 4. Reliability block diagram of the pressure vessel safety instrument system.
According to the reliability evaluation method of hybrid structure system, it is ob-
tained that:
RSI S(t) = RPT (t)RP LC (t)RV1(t) + RPT(t)RPLC (t)RV2(t)−RPT(t)RPLC (t)RV1(t)RV2(t), (33)
where
RSI S(t)
represents the reliability of the pressure vessel safety instrument system,
RPT (t)
,
RPLC (t)
,
RV1(t)
, and
RV2(t)
represent the reliability of PT, PLC, V1, and V2 respec-
tively.
The reliability model of this system before and after modification is obtained by
substituting (4) and (14) into (33).
In the numerical example of the pressure vessel safety instrument system,
α0,i
is taken
as 10%, the failure rate
λ
of the equipment in the steady state phase, the single value
η0
and the interval value
ηα0
L,ηα0
U
of the failure rate cumulated parameters, and the local
maximum value
η0
are shown in Table 1.
λ
refers to the OREDA database,
η0
is the expert
experience value, the interval value
ηα0
L,ηα0
U
is estimated by chi-square distribution for
η
,
and
η0
is the local maximum value calculated by the upper boundary existence theorem.
The interval value of
η
is substituted into the system reliability model before and after
the modification, and the system reliability is analyzed by MATLAB. The simulation time
is selected as 10 years, in which the first 6 years is in the steady state phase and the last
4 years enter the wear-out phase, and the value of Tis 52,560 h.
Table 1.
Failure rates and failure rate cumulated parameter of equipment in pressure vessel safety
instrument system.
Abbreviation λ(h)η0[ηα0
L,ηα0
U]η0
PT 8.13 ×10−70.00005 [0.000004, 0.000072]0.00013
PLC 5.84 ×10−80.00011 [0.00003, 0.00014]0.00026
V1 2.45 ×10−70.00008 [0.000016, 0.0001]0.00015
V2 2.45 ×10−70.00008 [0.000016, 0.0001]0.00015
Figure 5shows the reliability curves of the pressure vessel safety instrument system
under single value
η0
and interval value
ηα0
L,ηα0
U
based on the TFR model before and
after modification. The reliability curve under
η0
is the current curve, and
λ[ηα0
L,ηα0
U](t)
and
λmo[ηα0
L,ηα0
U](t)
are the author-proposed curves. Before model modification, the reliability
curve corresponding to
ηα0
L
decreases the slowest, the reliability curve corresponding to
ηα0
U
decreases the fastest, the reliability curve corresponding to
η0
is sandwiched between
the upper and lower boundary curves, and at any same time point after the sixth year,
there is
RSI S,ηα0
L
>RSI S,η0>RSIS,ηα0
U
. According to the upper boundary existence theorem,
the parameter corresponding to the upper boundary of the modified failure rate region
is
ηα0
U
, there is
RSI S,ηα0
L
>RSI S,η0>RSIS,ηα0
U
at any same time after the sixth year. Before
and after the modification, the reliability intervals of the system in the tenth year are
[0.7825, 0.9213]
and
[0.7986, 0.9246]
respectively, indicating that the reliability curve de-
creases at a slower rate after the modification, and the end values of the reliability interval
including uncertainties both increase. The results show that the method of TFR model
modification changes the reliability interval by changing the upper and lower boundaries
of the region enclosed by the modified failure rate curve. This is because the confidence
Processes 2021,9, 2214 11 of 19
of
η
is attached to the modified TFR model, which increases the end value of the system
reliability interval containing uncertainty, and the reliability interval obtained after the
modification is more accurate. The proposed upper boundary existence theorem provides
guidance for obtaining
η
when the failure rate changes fastest and discussing the degree of
change in the reliability interval.
012345678910
0 . 7 5
0 . 8 0
0 . 8 5
0 . 9 0
0 . 9 5
1 . 0 0
R e l i a b i l i t y b a s e d o n λη0( t )
R e l i a b i l i t y b a s e d o n λ[ηLα0,ηUα0]( t )
R e l i a b i l i t y b a s e d o n λm o [ηLα0,ηUα0]( t )
R e lia b ility
T i m e ( y e a r )
Figure 5.
Comparison of reliability curves of the pressure vessel safety instrument system models
before and after modification.
4. Study Case
As the mainstream type of subsea control system, multiplexed electro-hydraulic
control system runs in deep water environment for a long time, so the equipment failure
rates of this system have time-varying characteristics. Figure 6shows the layout of the
valves and instruments of the multiplexed electro-hydraulic control system located in the
tree and downhole.
PMV PWV
AMV
AAV
PTTT
02
XOV
SSIV
SCSSV
PCV
DCIV02
DCIV01
CIMV01
CICV01
CIMV02
CICV02
VX LSIV
HC4
HC3
HC5
HC26
HC6 ROV EC02
ROV EC01
LCP
VX USIV
from MQC
HOTSTAB01
HOTSTAB02 HOTSTAB03
UCP
PSIV
AWV
UCP
PSIV HC26
PTTT
01
PTTT
03
DHPT
Figure 6. Layout of valves and instruments of the control system located in the tree and downhole.
Processes 2021,9, 2214 12 of 19
The subsea emergency shutdown system is an important part of the multiplexed
electro-hydraulic control system. In order to intensively study the influence of TFR on
reliability, this paper takes the subsea emergency shutdown system as the analysis object.
Firstly, the system reliability is evaluated under TFR by using a single value of the failure
rate cumulated parameter, and the evaluation error is obtained by comparing with the
reliability evaluation under CFR. Secondly, combining the upper boundary existence
theorem and the interval value of the failure rate cumulated parameter, the influence of the
parameter uncertainty intervals on the uncertainty intervals of the system reliability are
compared based on the TFR models before and after the modification.
4.1. Reliability Model of Subsea Emergency Shutdown System Based on Time-Varying Failure Rate
Figure 7shows the composition and function of the subsea ESD system [
37
], which
performs the functions of both basic process control system (BPCS) and safety instrumented
system (SIS) during oil and gas production. During normal production, the master control
station (MCS) gives control instructions, and the offshore equipment opens the surface-
controlled subsurface safety valve (SCSSV) through the SCM. The production master valve
(PMV) and the production wing valve (PWV) remain normally open, and the subsea oil and
gas are transmitted through the main loop pipeline. In case of production emergency, the
temperature or pressure measured by the pressure transmitter and temperature transmitter
(PTTT) will exceed the maximum limit. At this time, the hydraulic directional control valve
controlled by SCM will lose power, forming oil return. The hydraulic oil pressure in the
driver chamber of the PMV will decrease, then the reset spring closes the PMV. If the PPTT
is still feeding back data, it will continue to close the PWV. In the event of pipeline leak or
fire at production facility that cannot be prevented by the shutdown of the PMV and the
PWV, the SCSSV will be closed to avoid a blowout and other accidents. When the SCSSV
cannot be closed, the hydraulic power unit (HPU) will act directly to return the hydraulic
oil to close the SCSSV.
MCS DCV2
PMV
PWV
SCSSVDCV3
DCV1
PTTT
DHPT
HPU
SCM
Figure 7. Function diagram of subsea emergency shutdown system.
When only considering the SIS function of the subsea ESD system shown in Figure 7,
the reliability block diagram model of the system is established:
The subsea ESD system in Figure 8is a hybrid system containing both series and
parallel structures. The reliability model of this system is obtained by using the reliability
calculation method of the series-parallel structure.
Processes 2021,9, 2214 13 of 19
Rsys =[(R1+R2−R1R2)R3+R4−(R1+R2−R1R2)R3R4]·R5·[R6R7+R8R9
−R6R7R8R9+R10R11 −(R6R7+R8R9−R6R7R8R9)R10R11)], (34)
where
Rsys
represents the reliability of the subsea ESD system, and
Ri
represents the
reliability of the equipment PTTT, DHPT, MCS, HPU, SCM, DCV1, PMV, DCV2, PWV,
DCV3, and SCSSV of the system in turn.
MCS
HPU
DCV1 PMV
DCV2 PWV
DCV3 SCSSV
PTTT
DHPT
SCM
Figure 8. Reliability block diagram of subsea emergency shutdown system.
1.
Equipment reliability model based on time-varying failure rate model before modifica-
tion. The TFR model for each equipment of the subsea ESD system before modification
is obtained using (4), then the reliability model of the equipment is as follows:
Ri(t) =
e−λ0,itt≤T
expλ0,i−λ0,ieη0,i(t−T)
η0,i+e−λ0,iT−1t>T, (35)
When
i=
1, 2,
· · ·
, 11,
Ri
,
λ0,i
and
η0,i
, respectively, represent the reliability, failure
rate in the steady state phase and the single value of failure rate cumulated parameter
of the equipment PTTT, DHPT, MCS, HPU, SCM, DCV1, PMV, DCV2, PWV, DCV3,
and SCSSV in this system.
2.
Equipment reliability model based on time-varying failure rate model after modifica-
tion. The modified TFR model (14)–(17), for a given α0,i, when ηi<η0,i,
Ri,mo (t) =
e−λ0,itt≤T
exp exp (ηi−η0,i)2ln(1−α0,i)
(ηα0,i
L,i−η0,i)2!·λ0,i−λ0,ieηi(t−T)
ηi!+e−λ0,iT−1t>T(36)
When ηi>η0,i,
Ri,mo (t) =
e−λ0,itt≤T
exp exp (ηi−η0,i)2ln(1−α0,i)
(ηα0,i
U,i−η0,i)2!·λ0,i−λ0,ieηi(t−T)
ηi!+e−λ0,iT−1t>T(37)
Substituting the reliability model before and after modification of the equipment
(35)–(37) into (34), the reliability models of the system based on the TFR model before
and after modification are obtained.
4.2. Reliability Comparison and Analysis Based on Time-Varying Failure Rate and Constant
Failure Rate Models
As shown in Table 2, the failure rate of each equipment
λ0,i(h)
in the subsea ESD
system in the steady state phase refers to the OREDA, and the failure rate cumulated
parameter is the expert experience value. The simulation is carried out in MATLAB, and
the simulation time is 15 years, which is 131,400 hours. When considering the time-varying
characteristics of failure rate, the first ten years is the steady state phase, and the failure
Processes 2021,9, 2214 14 of 19
rate increases with the passage of time in the last five years. Therefore, the
T
value in
Equation (35) is 87,600 h.
Table 2.
Failure rate and failure rate cumulated parameter of each equipment in subsea emergency
shutdown system.
Abbreviation λ0,i(h)η0,ihηα0,i
L,i,ηα0,i
U,ii
PTTT 3.88 ×10−80.000097 [0.000026, 0.00012]
DHPT 1.85 ×10−80.000084 [0.000021, 0.00011]
MCS 1.239 ×10−10 0.000055 [0.0000083, 0.000078]
HPU 8.345 ×10−90.00007 [0.000012, 0.000089]
SCM 2.511 ×10−90.00008 [0.000017, 0.000099]
DCV1 5.3 ×10−70.0001 [0.000036, 0.00014]
PMV 5.66 ×10−80.00015 [0.000053, 0.00017]
DCV2 5.3 ×10−70.0001 [0.000036, 0.00014]
PWV 5.66 ×10−80.00015 [0.000053, 0.00017]
DCV3 5.3 ×10−70.0001 [0.000036, 0.00014]
SCSSV 6.8 ×10−70.00018 [0.000064, 0.00019]
The dot-dash line and the straight line in Figure 9respectively represent the reliability
curves of the subsea ESD system under TFR and CFR. When the time-varying characteristics
of failure rate is considered, the system is in the steady state phase in the first ten years,
and enters the wear-out phase in the tenth year. The aging of the components leads to an
increase in the failure rate, and the system reliability decreases at a faster rate. The system
reliability drops to 0.7995 in the fifteenth year when the failure rate cumulated parameter
of each equipment is
η0,i
. When the failure rate is constant, the system reliability drops
to 0.9712 in the tenth year and 0.9524 in the fifteenth year. The comparison of reliability
curves shows that the reliability of the system decreases too slowly when using the CFR
and relatively faster when using the TFR. The reliability of the subsea ESD system drops to
about 0.8 by the fifteenth year in engineering experience. The simulation result shows that
the reliability evaluation of the system based on the TFR model is basically consistent with
the engineering experience. It is too idealized to simplify the failure rate into a constant,
which brings reliability evaluation error compared with TFR, and this error will get bigger
and bigger over time. The TFR model in this paper eliminates this error.
4.3. Reliability Comparison and Analysis of Time-Varying Failure Rate Models before and
after Modification
The interval value
hηα0,i
L,i,ηα0,i
U,ii
of the failure rate cumulated parameter is given in
Table 2, where
α0,i
is taken as a small range of 5%, which is estimated by the chi-square
distribution for
η0,i
. The upper and lower bounds of this interval are substituted into
the TFR models before the modification during simulation, and the region of the system
reliability curve which changes with the value of
η
is obtained using Equation (35). In the
simulation of the modified TFR model, the value of
η
corresponding to the upper boundary
is quickly obtained through the upper boundary existence theorem determined by the
failure rate region. Based on Table 2and the upper boundary existence theorem, the upper
boundary of the modified failure rate region corresponds to the parameter ηα0,i
U,i.
Processes 2021,9, 2214 15 of 19
0 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5
0 . 7 8
0 . 8 0
0 . 8 2
0 . 8 4
0 . 8 6
0 . 8 8
0 . 9 0
0 . 9 2
0 . 9 4
0 . 9 6
0 . 9 8
1 . 0 0
1 . 0 2 R e l i a b i l i t y b a s e d o n λ0 , i
R e l i a b i l i t y b a s e d o n λ( t )
R e lia b ility
T i m e ( y e a r )
Figure 9.
Reliability curves comparison diagram based on time-varying failure rate and constant
failure rate.
The straight line and the dot-dash line in Figure 10, respectively, represent the reliabil-
ity curves of the subsea ESD system based on the TFR models before and after modification.
In the first ten years, the system is in the steady state phase, and enters the wear-out phase
in the tenth year. The difference from the simulation in Section 4.2 is that the uncertainty of
parameter
η
is considered, and the reliability analysis result changes from a single value to
an interval containing uncertainty, which is intuitively reflected in the expansion of the
reliability curve from a single line to a region. The reliability interval of the system in the
fifteenth year is
[0.9433, 0.6188]
based on the TFR model before modification. Based on the
modified TFR model, the reliability interval of the system becomes
[0.9451, 0.6412]
in the fif-
teenth year. The comparison shows that the reliability curve decreases more slowly and the
interval end values increase after the modification, which is because the TFR model before
the modification ignores the confidence of
η
. The modified model solves this problem and
makes the reliability interval containing uncertainty more accurate. The change trend of the
curves in Figure 10 is consistent with that in Figure 5, which further reflects the precision
of the model modification method and the upper boundary existence theorem. In fact, the
lower bound of the system reliability interval cannot be infinitesimal or too small. It should
be ensured that the reliability is meaningful in engineering applications. From the partial
enlarged view, the lower bound of the reliability interval after modification in the partial
enlargement diagram has a larger change range than the upper bound, and the lower
bound is higher overall, indicating that the modification method fits the actual situation.
Processes 2021,9, 2214 16 of 19
0 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5
0 . 6 0
0 . 6 5
0 . 7 0
0 . 7 5
0 . 8 0
0 . 8 5
0 . 9 0
0 . 9 5
1 . 0 0
1 . 0 5 R e l i a b i l i t y b a s e d o n λ( t )
R e l i a b i l i t y b a s e d o n λm o ( t )
R e liab ility
T i m e ( y e a r )
Figure 10.
Reliability curves comparison diagram based on time-varying failure rate models before
and after modification.
5. Conclusions
The failure rate of most equipment varies with time in long-term operation. In order
to solve the problem of reliability evaluation error caused by constant failure rate, a new
time-varying failure rate model is established after comprehensively considering the time-
varying characteristics of the failure rate in the steady state phase and wear-out phase.
The time-varying scale factor in the model included the failure rate cumulated parameters
that influenced the curve trend. Due to the uncertainty of the failure rate cumulated
parameter, a statistical-fuzzy model is established based on the interval of the failure rate
cumulated parameter estimated by the parameter estimation combined with statistics and
fuzzy knowledge. In addition, the TFR model is modified by using statistical-fuzzy model,
which covers the confidence of the failure rate cumulated parameter, and changes the upper
and lower boundaries of the region enclosed by the failure rate curves. To further explore
the range of boundary variation, the upper boundary existence theorem for the failure rate
region is proposed and demonstrated, so as to obtain the failure rate cumulated parameter
when the failure rate changes fastest, and the theorem is applied to numerical example.
In this paper, the subsea emergency shutdown system which has been in marine
environment for a long time is selected as the research object, and the reliability model is
established by using time-varying failure rate model and system reliability block diagram.
When the failure rate cumulated parameter is a single value, the reliability of the systems
under time-varying failure rate and constant failure rate are compared and analyzed. When
the failure rate cumulated parameter is an interval, combined with the upper boundary
existence theorem for the failure rate region, the system reliability based on the time-
varying failure rate models before and after the modification are compared and analyzed.
The following conclusions are drawn.
1.
Compared with the constant failure rate, the system reliability with the time-varying
failure rate decreases faster and reaches 0.7995 in the fifteenth year. The reliability in
the fifteenth year in engineering experience is about 0.8, so the time-varying failure
rate model proposed in this paper is consistent with the actual situation and can
eliminate the reliability evaluation error caused by the constant failure rate.
Processes 2021,9, 2214 17 of 19
2.
Compared with the model before the modification, the modified time-varying failure
rate model has the confidence of
η
attached, which increases the end value of the
system reliability interval containing uncertainty, and the reliability interval obtained
after the modification is more accurate and realistic.
In practical engineering, the system requires equipment maintenance based on relia-
bility evaluation, so the reliability interval obtained from the modified failure rate model
proposed in this study can theoretically provide data support for maintenance strategy and
make it more flexible, and this work will be completed in the future.
Author Contributions:
Conceptualization, X.Z. and X.Y.; methodology, X.Y.; software, X.Y.; valida-
tion, X.Z., X.Y. and Y.Y.; formal analysis, Y.Y.; investigation, X.Y.; resources, F.Y.; data curation, F.Y. and
C.Z.; writing—original draft preparation, X.Y.; writing—review and editing, X.Z.; visualization, X.Y.;
supervision, Y.Y.; project administration, X.Z.; funding acquisition, X.Z., F.Y. and C.Z. All authors
have read and agreed to the published version of the manuscript.
Funding:
This work is supported by National High-tech Ships from Ministry of Industry and
Information Technology: Research on Integral Reliability Analysis Technology of Subsea Control
System (2018GXB01-03-004), Science Foundation of China University of Petroleum, Beijing (No.
2462020YXZZ023).
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement: Not applicable.
Conflicts of Interest: The authors declare no conflict of intrerst.
Abbreviations
ESD Emergency shutdown
CFR Constant failure rate
TFR Time-varying failure rate
OREDA Offshore and onshore reliability data
SCM Subsea control module
PG Pseudo-gaussian
PT Pressure transmitter
PLC Programmable logic controller
V1 Valve 1
V2 Valve 2
BPCS Basic process control system
SIS Safety instrumented system
MCS Master control station
HPU Hydraulic power unit
PTTT Pressure transmitter and temperature transmitter
DHPT Downhole pressure and temperature transmitter
PMV Production master valve
PWV Production wing valve
SCSSV Surface controlled subsurface safety valve
DCV1 Directional control valve 1
DCV2 Directional control valve 2
DCV3 Directional control valve 3
References
1.
Zhang, Y.; Tang, W.; Du, J. Development of subsea production system and its control system. In Proceedings of the 2017 4th
International Conference on Information, Cybernetics and Computational Social Systems (ICCSS), Dalian, China, 24–26 July 2017;
pp. 117–122. [CrossRef]
2.
lyalla, I.; Arulliah, E.; Innes, D. A critical analysis of open protocol for subsea production controls system communication. In
Proceedings of the OCEANS 2017, Aberdeen, UK, 19–22 June 2017; pp. 1–7. [CrossRef]
Processes 2021,9, 2214 18 of 19
3.
Bai, Y.; Bai, Q. (Eds.) 7—Subsea Control. In Subsea Engineering Handbook, 2nd ed.; Gulf Professional Publishing: Boston, MA, USA,
2019; pp. 173–202. [CrossRef]
4.
Hua, Y.; Kecheng, S.; Xin, D.; Chao, Y.; Yuanlong, Y. Adaptability Analysis of SubseaOil and Gas Control System in Bohai Sea
Region. Mech. Electr. Eng. Technol. 2021,50, 60–63, 137.
5.
Yangfeng, F.; Guochu, C. Application of subsea electro-hydraulic control system in deepwater gas field project. China Pet. Chem.
Stand. Qual. 2020,40, 125–126.
6. Lu, W.; Weizhenq, A. Comparison and Analysis of Subsea Production Control System. Petrochem. Ind. Technol. 2018,25, 17–19.
7.
Nolan, D.P. (Ed.) Chapter 11—Emergency Shutdown. In Handbook of Fire and Explosion Protection Engineering Principles for Oil,
Gas, Chemical, and Related Facilities, 4th ed.; Gulf Professional Publishing: Waltham, MA, USA, 2019; pp. 215–225. [CrossRef]
8.
Wang, X.; Jia, P.; Lizhang, H.; Wang, L.; Yun, F.; Wang, H. Reliability and Safety Modelling of the Electrical Control System of the
Subsea Control Module Based on Markov and Multiple Beta Factor Model. IEEE Access 2019,7, 6194–6208. [CrossRef]
9.
Pang, N.; Jia, P.; Wang, L.; Yun, F.; Wang, G.; Wang, X.; Shi, L. Dynamic Bayesian network-based reliability and safety assessment
of the subsea Christmas tree. Process. Saf. Environ. Prot. 2021,145, 435–446. [CrossRef]
10.
Bae, J.H.; Shin, S.C.; Park, B.C.; Kim, S.Y. Design Optimization of ESD (Emergency ShutDown) System for Offshore Process Based
on Reliability Analysis. MATEC Web Conf. 2016,52, 10. [CrossRef]
11.
Signorini, G.; Rigoni, E.; Rodrigues, M. Reliability Analysis Methodology for Oil and Gas Assets: Case Study of Subsea Control
Module Operating in Deep Water Basin at Brazilian Pre-Salt. In Proceedings of the 2020 International Petroleum Technology
Conference (IPTC), Dhahran, Saudi Arabia, 13–15 January 2020. [CrossRef]
12.
Ismagilov, F.; Vavilov, V.; Karimov, R.; Yushkova, O.; Timofeev, A. Combined Method of Technical Analysis to Optimize the
Aviation Electromechanical Systems Reliability Indicators. In Proceedings of the 2021 28th International Workshop on Electric
Drives: Improving Reliability of Electric Drives (IWED), Moscow, Russia, 27–29 January 2021; pp. 1–4. [CrossRef]
13.
Tang, Q.; Shu, X.; Zhu, G.; Wang, J.; Yang, H. Reliability Study of BEV Powertrain System and Its Components—A Case Study.
Processes 2021,9, 762. [CrossRef]
14.
Tawfiq, A.A.E.; El-Raouf, M.O.A.; Mosaad, M.I.; Gawad, A.F.A.; Farahat, M.A.E. Optimal Reliability Study of Grid-Connected PV
Systems Using Evolutionary Computing Techniques. IEEE Access 2021,9, 42125–42139. [CrossRef]
15.
Hassett, T.; Dietrich, D.; Szidarovszky, F. Time-varying failure rates in the availability and reliability analysis of repairable
systems. IEEE Trans. Reliab. 1995,44, 155–160. [CrossRef]
16.
Retterath, B.; Venkata, S.; Chowdhury, A. Impact of time-varying failure rates on distribution reliability. In Proceedings of
the 2004 International Conference on Probabilistic Methods Applied to Power Systems, Ames, IA, USA, 12–16 September 2004;
pp. 953–958. [CrossRef]
17.
Wang, R.; Xue, A.; Huang, S.; Cao, X.; Shao, Z.; Luo, Y. On the estimation of time-varying failure rate to the protection
devices based on failure pattern. In Proceedings of the 2011 4th International Conference on Electric Utility Deregulation and
Restructuring and Power Technologies (DRPT), Weihai, China, 6–9 July 2011; pp. 902–905. [CrossRef]
18.
Abunima, H.; Teh, J. Reliability Modeling of PV Systems Based on Time-Varying Failure Rates. IEEE Access
2020
,8, 14367–14376.
[CrossRef]
19.
Zhang, Q.; Wang, X.; Du, W.; Zhang, H.; Li, X. Reliability Model of Submarine Cable Based on Time-varying Failure Rate. In
Proceedings of the 2019 IEEE 8th International Conference on Advanced Power System Automation and Protection (APAP).
Xi’an, China, 21–24 October 2019; pp. 711–715. [CrossRef]
20.
Li, Q.; Wang, Z.; Zhao, H.; Liu, H.; Tian, L.; Zhang, X.; Qiu, J.; Xue, C.; Zhang, X. Research on Prediction Model of Insulation
Failure Rate of Power Transformer Considering Real-time Aging State. In Proceedings of the 2019 IEEE 3rd Conference on Energy
Internet and Energy System Integration (EI2), Changsha, China, 8–10 November 2019; pp. 800–805. [CrossRef]
21.
Jian, L.; Feng, G.; Ming, Z.; Liuning, C.; Weiyao, L. Research on Optimal Inspection Strategy for Overhead Transmission Line
Based on Smart Grid. Procedia Comput. Sci. 2018,130, 1134–1139. [CrossRef]
22.
Liu, H.; Wang, Z.; Han, M. Reliability Analysis of Solenoid Valve Power Supply Based on Time-Varying Fault Rate. In Proceedings
of the 2019 4th International Conference on Mechanical, Control and Computer Engineering (ICMCCE), Hohhot, China, 25–
27 October 2019; pp. 154–1543. [CrossRef]
23.
Zhang, F.; Wu, M.; Hou, X.; Han, C.; Wang, X.; Liu, Z. The analysis of parameter uncertainty on performance and reliability of
photovoltaic cells. J. Power Sources 2021,507, 230265. [CrossRef]
24.
Fan, L.; Lübin, P.; Tao, N.; Bo, H.; Kan, C.; Kunpeng, Z. A method for obtaining unknown reliability parameters of components
based on simulated annealing algorithm. Electr. Meas. Instrum. 2021,58, 1–10.
25.
Wang, J.; Xu, Y.; Peng, Y.; Ye, Y. Estimation of Reliability Parameters of Protective Relays Based on Grey-three-parameter Weibull
Distribution Model. Power Syst. Technol. 2019,43, 1354–1360. [CrossRef]
26.
Laure Miranda, F.; Willer de Oliveira, L.; Henriques Dias, B.; Chaves de Resende, L.; Geraldo Nepomuceno, E.; José de Oliveira, E.
Composite Power System Reliability Evaluation Considering Stochastic Parameters Uncertainties. IEEE Lat. Am. Trans.
2020
,
18, 2003–2010. [CrossRef]
27. Li, Z.; Yang, L.; Wang, D.; Zheng, W. Parameter Estimation of Software Reliability Model and Prediction Based on Hybrid Wolf
Pack Algorithm and Particle Swarm Optimization. IEEE Access 2020,8, 29354–29369. [CrossRef]
28.
Wang, Z.; Pan, R. Point and Interval Estimators of Reliability Indices for Repairable Systems Using the Weibull Generalized
Renewal Process. IEEE Access 2021,9, 6981–6989. [CrossRef]
Processes 2021,9, 2214 19 of 19
29.
Uprety, I.; Patrai, K. Fuzzy Reliability Estimation Using Chi-Squared Distribution. In Proceedings of the 2016 3rd International
Conference on Soft Computing Machine Intelligence (ISCMI), Dubai, United Arab Emirates, 23–25 November 2016; pp. 169–173.
[CrossRef]
30.
Yang, X.; Yang, Y.; Liu, Y.; Deng, Z. A Reliability Assessment Approach for Electric Power Systems Considering Wind Power
Uncertainty. IEEE Access 2020,8, 12467–12478. [CrossRef]
31.
Wang, Z.; Shafieezadeh, A. On confidence intervals for failure probability estimates in Kriging-based reliability analysis. Reliab.
Eng. Syst. Saf. 2020,196, 106758. [CrossRef]
32.
Hu, L.; Yue, D.; Zhao, G. Reliability Assessment of Random Uncertain Multi-State Systems. IEEE Access
2019
,7, 22781–22789.
[CrossRef]
33.
Li, X.Y.; Chen, W.B.; Li, F.R.; Kang, R. Reliability evaluation with limited and censored time-to-failure data based on uncertainty
distributions. Appl. Math. Model. 2021,94, 403–420. [CrossRef]
34.
Wang, Y.; Li, W.; Zhang, P.; Wang, B.; Lu, J. Reliability Analysis of Phasor Measurement Unit Considering Data Uncertainty. IEEE
Trans. Power Syst. 2012,27, 1503–1510. [CrossRef]
35.
Rausand, M. (Ed.) Chapter 2—Failure Model. In System Reliability Theory: Models, Statistical Methods, and Applications, 2nd ed.;
National Defense Industry Press: Beijing, China, 2010; pp. 6–37.
36.
Xianhui, Y.; Haitao, W. Chapter 5—Reliability Model and Failure Data. In Functional Safety of Safety Instrumented Systems; Tsinghua
University Press: Beijing, China, 2007; pp. 78–95.
37.
Bodsberg, L. Application of IEC 61508 and IEC 61511 in the Norwegian Petroleum Industry; The Norwegianoil Industry Association:
Oslo, Norway, 2004; pp. 81–109.
Available via license: CC BY 4.0
Content may be subject to copyright.
