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International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:20 No:02 13
I J E N S 2020 IJENS AprilIJENS © -IJMME-9696-402200
Reliability Analysis of Gas Turbine Power Plant Based on
Failure Data *and Marwa M. Ibrahim *, M. A. Badr*Amal El Berry
*Mechanical Engineering Department, Engineering Research Division, National Research Centre (NRC) 12622, Egypt Abstract-- To predict the reliability of a product or a system, life data from a representative sample of the system
performance is fitted to the suitable statistical distribution.
Reliability analysis techniques have been accepted as standard
tools for the planning, design, operation, and maintenance of
thermal power plants. Therefore, the parameterized
distribution can be used to estimate important life
characteristics such as reliability, or probability of failure at a
given time, mean life, and failure rate.
In today’s competitive environment reliability analysis is the
most important requirement of almost all types of systems,
subsystems, and complex systems; whether they are
mechanical, electrical, or electronic devices. To alleviate
failures and improve the performance and increase the
operational life of these components and systems, key
performance indicators such as: Failure Rate, Reliability,
Availability, and Maintainabilityare
investigated.Weibull++/ALTA is used to fit the available data
set concerning three sets of gas turbines (GT) operating in a
power plant to estimate the probability density function (PDF),
plant reliability, and failure rate of each set and for the whole
plant. In this study data of a gas turbines (GT) power plant
(three groups of GTs) is used. Two methods for parameter
estimation are applied in the data fitting stage: Maximum
Likelihood (MLE) and Rank Regression Analysis X –axis
(RRX).
Using Mean Time Between Failure (MTBF) data, the results
show that the system overall reliability is 97% at 413 hr while
using Down Time (DT) data the system reaches the same
reliability at 289 hr. Also at 800 hr, the reliability of Group-1 is
74% while the reliability of Group-2 and Group-3 is 83% and
45% respectively. Downtime losses and cost of maintenance of
the power plant can be minimized by implementing a proper
mix of maintenance and repair approaches on system
reliability failure rate.
Index Term-- Reliability, Gas Turbine, MeanTime between Failures, Failure Rate, Mean Time to Repair,
WeibullDistribution
ABBREVIATIONS
Aggregate Criterion DESV Maximum Likelihood MLE
Availability A Mean Time Between Failure MTBF
Combined Cycle Power Plants CCPP Mean Time To Failure MTTF
Cumulative Distribution Function CDF Mean Time To Repair MTTR
Condition Monitoring CM Median Ranks MED
Correlation Coefficient Test CC Non-Homogeneous Poisson Process NHPP
Down Time DT Probability Density Function PDF
Fisher Matrix Confidence Bounds FM Pseudo Failure Characteristic PFC
Gas Turbine GT Rank Regression Analysis X –Axis RRX
Kolmogorov-Smirnov Test K-S Reliability R(t)
Likelihood Value Test LHV
1. INTRODUCTION Reliability life data analysis refers to the analysis and
modeling of observed data over the product life to estimate
important features such as system (or component)
reliability, failure rate, or mean time to failure (MTTF).
Several studies for reliability assessment were; and still are,
conducted. Mechanical equipment reliability evaluation is
highly important in condition-based maintenance to lower
costs and increase equipment efficiency;which is the reason, that it an important research field for reliability analysis of
mechanical equipment and life prediction.
1.1 Reliability Approaches and Indices
Failed machine must be removed from service for either
repair or replacement; this occurrence is known as a failure
and may have a negative impact on the system's ability to
provide the load required and impact on the system
reliability. A general approach to system reliability
assessment is to determine one or a number of its reliability
indices that measure some aspects of system reliability
performance such as Mean time between failure (MTBF), failure rate (ƛ) and Mean time to repair (MTTR)
[1].Numerous studies have found empirical models that are
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focused on Weibull, exponential, uniform, and other
distributions.
Lack of reliability data leads to reduction of production,
excessive expenditure, equipment failure, and downtime. As
a result, reliability analysis techniques have increasingly
become adopted as standard tools for planning, constructing, running, and maintaining thermal power plants. The
efficiency of the generating system is subdivided into
adequacy and security [2], [3].
Reliability prediction approach depends upon the product
development stages and its related reliability metric [4].
Reliability prediction methods address application of
mathematical models and component data for the purpose of
estimating the field reliability of a system before failure data
are available for the system. Various reliability prediction
methods, their concepts of application, advantages, and
disadvantages were discussed by Thakur and Sakravdia[5].
The classical approach fits equipment failure rates to
statistical models[6]; while in the data-mining approach, it is
modeled using a data-mining algorithm; decision tree
instruction, establishing logical, mathematical, and
statistical relations between MTTF and its various factors of
impact (equipment conditions, failure history, etc.).
Component failure rates depend on time, and therefore can
be viewed as time series. Unplanned equipment failures and
their consequences have significant effect on the total
operating cost of the system.
Duane proposed the power law model on the failures of a
complex repairable system; where the accumulated MTBF
was linearly related to the operating time on log-log scale
[7]. On the other hand, Barabady and Kumar[8]used various
statistical distributions including Weibull, exponential,
normal, and log normal distribution to analyze the reliability
of a crushing plant, in order to identify the bottlenecks in the
system and to find the components or subsystems with low
reliability for a given designed performance.
To get a proper maintenance plan for individual components
in a complex system, Son et al [9] introduced Soft
Computing Methodology. They used a combination of neural network and evolutionary algorithm to discover the
relationship between individual parts of a complex system,
to improve their reliability.
Kuang[10]suggested a new model of reliability evaluation
based on quality loss and the development of quality
characteristics. Wang [11]showed that the limited intensity
procedure was appropriate for the reliability assessment of
degradation in machine tools with regular maintenance
behavior, while Li [12]examined the device reliability
assessment based on acoustic emissions signals. Another
research proposed a method of reliability assessment based
on the distribution of the degradation path related to the
signal characteristics [13]. The signal characteristics of the
machining process were used in this research to replace
traditional time data and fit equipment degradation model
with the characteristic of a pseudo failure.
The demand for reliable products and manufacturing processes with lower cost is persistently growing, especially
in the electronic industry. Factors, reliability, and cost
determine the warranty period allocation for electronic
equipment, Wu et al [14].
1.2 Reliability of Electric Components and Devices
A study reviewing the failure physics approach that is used
in developing highly reliable semiconductor devices was
presented [15]. The study summarized device failures in
fieldand discussed a failure rate prediction model. Pecht[16]
discussed the role of reliability prediction in design,
development, and deployment of electronic equipment;
overviews the history of reliability predictions for electronics.
The complete time series of end-of-life electronic products
for empirical failure rate can be used as an empirical
knowledge base of product reliability.Jónás et al developed
a novel approach focused on the application of both
analytical decomposition of the time series of empirical
failure rates and soft computational techniques to predict
bathub-shaped failure rate curves of consumer electronic
products [17]. Another method suggested by Perera[18]
provided an index of reliability for the estimation of mobile phone failure rates. However there was a significant
correlation between the reliability index and the failure rate.
1.3Reliability of Electric Power Generation System
Globally, the reliable availability of electricity is seen as an
effective and indispensable mechanism for the rapid
industrial and economic growth of any nation [19]. Types of
PV modules failure such as hot spot, diode failure and glass
breakage are highly dependent on the PV module design
technology and the installation site environmental
conditions [20]. Bravoet al. [21] used realistic operation and
maintenance data to estimate the failure rates, grouped by components and the relative effect of failures on the PV
plant's energy balance. Results showed that the impact of
failures in all evaluated PV plants energy losses are small,
reaching a maximum value of 0.96 percent of net energy
yield.
Reliability of generation system is mainly dependent on the generators reliability. Xu Zhang et al. [22]presented a
reliability analysis of floating wind turbines to overcome the
high cost of searching failure causes.Evaluation of floating
wind power system is based upon its structure and function,
which provide explicit internal relation of system and the
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requirement of failure modes analysis using dynamic fault
tree analysis. Failure rate of an offshore wind turbine
gearbox was estimated based on the data available for
similar onshore wind turbine systems [23].
Techno-economical decisions ofpower plant equipment maintenance were based on the reliability modeling of the
combined cycle power plants and steam turbine power
plants, Sabouhi[24]. The author proposed reliability-
oriented sensitivity indices to identify the plant critical
components.
As gas turbine (GT) is considered acrucial component of
electric power and aerospace industries, it had prompted a
great number of researches in the fields of material,
mechanical, and electrical engineering to increase their
efficiency. Some gas turbine components work in an
extreme environment of high temperatures which impacts the maintenance cycle, and performance of the turbine.
Some available statistical techniques such as Pareto
analysis, Weibull probability density function, and
calculation of MTBF and Laplace test can be used to
develop failure and reliability analysisand provide an
accurate diagnosis [3].
System failure events and maintenance actionsof a GT were
derived from condition monitoring (CM) data and were
fitted to a non-homogeneous Poisson process (NHPP) using
maximum likelihood estimation (MLE)[25], [26]. The
modified CM data set was used to estimate the parameters of the system reliability models. These models represent the
failure levels of the gas turbine fordifferent life cycle
intervals.
GT power plant reliability is a function of failure rate,
maintenance which in turn depends on the equipment or
systems MTBF and MTTR. Other factors affecting GT
reliability are turbine or system design complexity, rank,
and age. Aneke et al [27] attempted to find the crucial
component in the GT power plant, determine the
relationship between the failure rate and the availability of
GT power plants, and consequently its reliability. Another
research examined the performance indices of selected Nigerian GT power stations [28].
In the same context, Chang evaluated the effect of high
thermo mechanical fatigue on the GT lifetimeduring a
steady-state operation [29]. The study results showed that
the generating units were underused because of inadequate
routine maintenance and fault development of the
equipment.
The above reviewed literature exhibits the importance of
estimating the failure rate and reliability of all types of
systems or components that require data availability over
reasonably long period of time. As for GT power plant
reliability estimation depends on availability of MTBF and
MTTR data. In the current work two data fitting techniques;
maximum likelihood estimation (MLE) method, and rank
regression analysis (RRX) are used.The performance
distributions are then evaluated using three forms goodness
of fit tests to compare the resulting distributions.To select the best-fitted distribution, the aggregate ranking criterion is
used.
2. METHODOLOGY As stated above data gathering, analysis, and fitting plays an
important role in reliability study. The parameters of the
fitted data distributionsare used to analyze the failure rate,
reliability, availability, and maintainability of gas turbine
power plants. The success of such research work depends on
the availability of statistical data from a target company; a
case study, beside the knowledge of reliability theories and
fitting statistical models.To evaluate system (or component)
different reliability functions such as failure rate, availability, etc are calculated; the following subsections
present different tools that are used to estimate the reliability
and maintainability of any mechanical or electric
component/or system.
2.1 Basic Concepts and Approaches for Reliability
Analysis
The techniques of reliability analysis were increasingly
accepted as standard tools for the planning, design,
operation and maintenance of various mechanical or
electrical systems[27] for;
Ability to fulfill basic needs
Efficiency to make effective use of the energy supplied
Reliability to start or continue operating
Maintainability of return to service quickly after one failure
2.1.1 Mean Time between failures (MTBF)
This is a measure of how long the equipment will; on
average, function as defined before an unplanned failure occurs. This can be determined by testing the system for
a total time period T during which N-faults occur. The
fault is repaired, and it puts the system back on test
when the repair time is removed from the total check T
period. The MTBF index is given by equation (1)[27],
[30]:
MTBF = 𝑇
𝑁 =
1
𝐹 (hours), F = expected failure rate. (1)
This error would allow for assumption from the gain. All
things are identical, the system with the biggest MTBF
is considered to be the most effective.
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2.1.2 Frequency of Failure or Failure Rate (F)
This index is sensitive to sampling errors, as the method
is being tested for a single sample of its total life. This
error would allow for removal from the result the system
with the highest MTBF, therefore is considered the most efficient. This is a very major deficiency; because there
may be cases where it is more beneficial to have short
repair times than high MTBF. A better measure of
reliability is therefore needed which takes into account
the repair time.
2.1.3 Mean time to Repair (MTTR)
This is a measurement of how long it will take on average
to get the equipment back to normal service status if it
fails, as shown in the following equations [27], [30].
MTTR = 𝜑𝑡
𝜑𝑛 (2)
Where: φt= total outage hours per year.
φn= No. of failure per year
Also, MTTR = 1
𝜇 (3)
Where μ =expected repair rate.
2.1.4 Availability (A)
This is a measurement of the percentage of time that
equipment is able to produce the end product at a certain
acceptable level defined. For a turbine in a power plant, availability is a function of the fraction of time that the
nominal power output is being generated It is calculated
by dividing the whole time in a given period into two
categories that are:
a) 'Up Time', UT: 'when the machine is in operation'.
b) 'Down Time', DT: Where the machine is defective or
failed to fix. The total period is then UT + DT and
availability exhibited in equations 4&5[27]:
A=𝑈𝑇
𝑈𝑇+𝐷𝑇 (4)
A=𝑀𝑇𝐵𝐹
𝑀𝑇𝐵𝐹+𝑀𝑇𝑇𝑅 (5)
2.1.5 Reliability (R(t))
Reliability is considered and identified by Kuo et al.
[31], [32] as the capability of the equipment to perform
its required task satisfactorily under defined conditions
over a given time period. It can also be said that reliability is the possibility that the equipment will
work without fail over time t as shown in the equation
below [27], [32].
R(t)= 𝑒𝑡
𝑀𝑇𝐵𝐹 (6)
Using equation (1) in equation (6), we have
R(t)=𝑒−𝐹𝑡 (7)
Where; t = specified period of failure-free operation
2.2 Data fitting and Parameters’ Estimation
Also these data are commonly referred to as Weibull's reliability life data results. Life data from a representative
sample of units is fitted to the correct statistical distribution
to estimate the life of all items within the population. To fit
into a statistical model, it is important to estimate the
parameters of the statistical distribution which will make the
equation closely fit the data. The function with probability
density (pdf) is the mathematical function representing the
distribution. The pdf can be interpreted mathematically or on
a plot where the x-axis represents time. The pdf of the
statistical total distributions is shown in the following
subsections.
2.2.1 Weibull Distribution
The 3-parameter Weibull pdf is given by[33], [34]:
𝑓(𝑡) =𝛽
ƞ(
𝑡−𝛾
ƞ)𝛽−1𝑒
−(𝑡−𝛾
ƞ)𝛽
(8)
Where:f(t) ≥ 0, t ≥ 0 or 𝛾, β > 0, ƞ > 0, -∞ 0. An
exponential random variable with mean = 1/¸ represents the waiting time until the first event to occur, where
events are generated by a Poisson process with mean ¸
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while the gamma random variable X represents the
waiting time until the athevent to occur. Therefore,
X = ∑ 𝑌𝑎𝑖 (10)
Where Y1, …. ; Yn are independent exponential random
variables with mean= 1/.
The probability density function of Gamma distribution is
given by[33]:
𝑓(𝑥; 𝛼, 𝛽) =1
Г(𝛼)𝛽𝛼 𝑒−𝑥 𝛽⁄ 𝑥𝛼−1,𝑥 > 0, 𝛼 > 0, 𝛽 > 0(11)
Where 𝛼 is the shape parameter, β is the scale parameter, and Γ is the gamma function which has the formula
2.2.3 G-Gamma Distribution
The generalized gamma X (α, β, y) is used to imply that
the generalized gamma distribution of the random
variable X has real positive parameters α, β, and y. In
equation 12 [33], a generalized gamma random variable
X with a scale parameter α and form parameters β has the
following probability density function.
𝑓(𝑥) = 𝛾𝑥𝛾𝛽−1𝑒−(𝑥 𝛼)⁄
𝛾
𝛼𝛾𝛽Г(𝛽), x>0 (12)
3. CASE STUDY
In this section a case study describing the reliability analysis
of gas turbine power plant as subsystems and overall is
presented. To investigate reliability and failure modes of
electricity generation system that is based on gas turbines,
data are obtained from a previous study of a power plant in
literature [27]. The plant power is generated from three
groups of gas turbines (GT). These data were collected over
a time period of 10years (from 2005 to 2015). The10-years
datafor group-1are exhibited in Table I, while the total set of
data are shown in appendix A.
Table I
Case study GT, Group-1 published data [27]
Year 2005 2007 2008 2009 2010 2011 2012 2013 2014 2015
No. of Failures 45 75 48 87 48 30 51 20 36 36
MTBF (h) 891.1 871.3 932.5 632.5 2540.5 1736.4 1608.0 370.2 632.7 1574.2
Downtime (h) 1415.3 1331.7 2754.9 1247.0 603.9 650.0 1621.1 1382 693.2 2934.4
MTTR (h) 283.74 221.19 1053.27 147.99 164.85 244.5 470.3 695.2 418.8 1090.1
3.1 Application of Weibull++ ALTA Package
The aim of life data analysis is to apply a statistical
distribution to fault time data in order to understand a
product's reliability performance over time or to make
predictions of future behavior. Several life features can be
derived from the study, such as probability of failure,
reliability, mean life, or failure rate. A quantitative
accelerated life testing data analysis is conducted where the
fault behavior of the product within normal conditions could
be extrapolated in a shorter time to obtain reliability
information about a product (e.g., mean life, probability of failure, etc.). Weibull++ ALTA package provides lifetime
distributions and analytical methods as follows:
"1, 2 and 3 parameter Weibull" "1 and 2 parameter Exponential" "Normal and Lognormal" "Gamma and Generalized Gamma" "Logistic and Log logistic" "Gumbel" "Bayesian-Weibull (with prior knowledge of the
Weibull shape parameter)"
"2, 3 and 4 subpopulation Mixed Weibull" (for situations when there are different trends in the data
and distinct failure mode for each data point can’t be identified)
All of the above distributions were applied in the mean time
between failures (MTBF) and down time data (DT) to get
the best fit, as shown in the section on goodness of fit
section.
3.2 Goodness of Fit Tests
Using goodness-of-fit test the fitted distributions are
determined. There are several ways to determine goodness-
of-fitness. Chi-square, among the most popular methods
used in statistics, "Kolmogorov-Smirnov test", "Anderson-Darling test", and the "Shapiro-Wilk
test"[33].Weibull++/ALTA package; used in this analysis,
provides three "fitness tests" in order to rate the fit
distributions to determine the best fit; these tests are:
"Kolmogorov-Smirnov (K-S)"; tests for the statistically significant correlation between the
expected results and those obtained from the
distribution fitted.
"Correlation coefficient (CC)"; analyses how well the plotted match a straight line.
https://www.itl.nist.gov/div898/handbook/eda/section3/eda363.htmhttps://www.itl.nist.gov/div898/handbook/eda/section3/eda364.htm
International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:20 No:02 18
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"Likelihood Value (LHV)"; estimates the log-likelihood value, given the distribution parameters.
3.3 Parameter Estimation
Determining the best fit distribution, reliability is then
estimated using the reliability function of the fitted distribution. There are several methods of parameter
estimation that can be used to estimate the distribution
parameters such as: the maximum likelihood estimation
(MLE) method, rank regression analysis, median ranks
(MED), and Fisher matrix confidence bounds (FM).
In order to obtain the distribution parameters, the regression
line is applied to the data points on the plot when the
parameters are determined using a rank regression analysis.
Therefore, the plot can be used to determine the extent to
which the distribution fits a given set of data. If the line of
regression closely follows the points on the plots the fit is stronger.
MLE method on the contrary, obtains the line solution using
probability function, not by plotting the data points.
Therefore the line is not supposed to follow the points of the
plot;hence the plot should not be used in this case to
determine the fit of a distribution.
4. RESULTS AND DISCUSSION After estimating the parameters, the best fitted distribution
is determined; as follows in sub-section 4.1. System reliability is then determined using the reliability function of
the fitted distribution.
4.1 Best Fit Distribution (Rank & Weight) Method
Using "Weibull++/ ALTA", MTBFGas Turbine data;shown
in table 1, are fitted using both MLEand RRX, then the
output distributions aretested using K-S goodness of fit test,
Correlation Coefficient (CC) test and Likelihood Value
(LHV) test.
To select the best-fitted distribution, the aggregate ranking
criterion is used. This method is based on calculating an
aggregate criterion (referred to as DESV) using the three
rankings values and weights assigned to the individual
criteria using equation (13)[33], [35]. The method assumes
that the lowest DESV value corresponds to the best-fitting theoretical distribution.
DESV= (K-S Rank × K-S Weight) + (CC Rank × CC
Weight) + (LHV Rank × LHV Weight) (13)
Performing goodness-of-fit statistics; for the three criteria,
ranks of different probability distributions are obtained. To
assign weights to the criteria, the default values of weights
selected by the software package are used in the current case
study. Finally, using the described DESV aggregate
criterion shown in Equation 13, the final ranking of the
eleven theoretical distributions was obtained. As previously stated, the distribution with the lowest DESV value was
identified as the best-fitting according to the aggregate
criterion, and was assigned number 1 in the ranking.
Theobtained lowest value of the DESV statistic was 3P-
Weibull distribution for both parameter estimation methods;
MLE and RRX as illustrated in tables II, III.
Implementing this method, the results of the ranking
procedure of gas turbine data (MTBF) for Group-1; (Table-
I), are summarized in Table-II for MLE method and Table-3
for RRX method while the results for Group 2& 3 are exhibited in appendix B.The first column exhibits the type
of the probability distribution, and the second shows the
probability of rejection of the working hypothesis for the
Kolmogorov–Smirnov (K-S) statistic. The third column
displays “Correlation coefficient”(CC) which gives the
mean absolute deviation of the theoretical Cumulative
Distribution Function (CDF) from the empirical CDF. The
fourth column exhibits theLikelihood Value (LHV) which
measures the goodness of fit determined using the log-
likelihood criterion. The value of calculated DESV is shown
in the fifth column.
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Table II
DESV Results of Group-1fitted Data using MLEMethod
Distribution K-S CC LHV DESV
Rank Weight Rank Weight Rank Weight
1P- Exponential 11 62.329 10 9.434 10 -83.212 1040
2P- Exponential 5 6.722 8 7.579 1 -77.794 330
Normal 7 11.161 9 7.589 9 -82.535 820
Lognormal 3 4.863 2 5.435 4 -80.013 340
2P-Weibull 6 6.998 4 6.052 7 -80.982 630
3P-Weibull 1 0.337 1 5.270 3 -78.644 200
Gamma 9 12.626 5 6.429 6 -80.479 710
G- Gamma 8 12.028 7 7.138 2 -78.109 490
Logistic 4 5.379 6 6.690 8 -82.318 620
Log-logistic 2 4.142 3 5.865 5 -80.354 360
Gumble 10 25.294 11 10.078 11 -84.639 1060
Table III
DESV Results of Group-1fitted Data using RRX Method
From tablesII & III, the presented analysis of MTBF of
Group-1 of gas turbines plant shows that the best-fitted
distribution, according to the aggregate criterion, is 3P-
Weibull. It should also be noted that with successive failures, the aggregate method may indicate a different best-fit
distributionfornewly gathered data, if there is
significantdifference from that previously analyzed.
4.2 Effect of Each Group on System Reliability
Fig. 1 exhibits the probability density function of the three
groups of gas turbines, (Fig. 1-a) for MLE while (Fig. 1-b)
for RRX. Similarly, Fig.2&3 show the failure rate and
reliability distributions for the two fitting methods.
Distribution K-S CC LHV DESV
Rank Weight Rank Weight Rank Weight
1P- Exponential 11 68.598 10 10.186 10 -83.226 1050
2P- Exponential 5 0.0143 8 4.424 1 -79.882 200
Normal 7 11.394 9 7.597 9 -82.532 780
Lognormal 3 2.682 2 5.115 4 -80.096 420
2P-Weibull 6 15.326 4 6.961 7 -81.267 750
3P-Weibull 1 0.004 1 4.293 3 -78.832 100
Gamma 9 0.306 5 5.479 6 -80.689 500
G- Gamma 8 0.023 7 4.858 2 -80.116 330
Logistic 4 16.042 6 7.612 8 -82.505 870
Log-logistic 2 5.123 3 5.234 5 -80.409 550
Gumble 10 31.255 11 10.384 11 -88.457 1050
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a) MLE fitting
MLE Group 1: 3P-Weibull
MLE Group 2: 3P-Weibull
MLE Group 3: 3P-Weibull
b) RRX Fitting
RRX Group 1: 3P-Weibull
RRX Group 2: Gamma
RRX Group3: 3P-Weibull
Fig.1. Probability Density Function of the Three Gas Turbine Groups for MLE & RRX Methods
a) MLE fitting
MLE Group 1: 3P-Weibull
MLE Group 2: 3P-Weibull
MLE Group 3: 3P-Weibull
b) RRX Fitting
RRX Group 1: 3P-Weibull
RRX Group 2: Gamma
RRX Group3: 3P-Weibull
Fig. 2. Failure Rate of the ThreeGas Turbine Groups for MLE & RRX Methods
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a) MLE fitting
MLE Group 1: 3P-Weibull
MLE Group 2: 3P-Weibull
MLE Group 3: 3P-Weibull
b) RRX Fitting
RRX Group 1: 3P-Weibull
RRX Group 2: Gamma
RRX Group3: 3P-Weibull
Fig. 3. Reliability of the Three Gas Turbine Groups for MLE & RRX Methods
From fig. 1, 2, and 3, it is clear that all the3 groups best
fitting distributions are 3P-Weibull distribution except group 2 of RRX method; which is Gamma distribution.
Also, figures (1, 2, and 3) show that the parameters of each
group have the same values for both methods.
From fig. 1, PDF values for each group using MLE & RRX
are almost equal, and the same applies on fig. 2, 3.This leads
to the conclusionthat parameter estimation (MLE or RRX)
method doesn't affect the resulting values. From fig. 3 it
could be seen that at time = 871 hr, the reliability of Group-
1 reaches around 74% for both parameter estimation
methods; MLE & RRX, also for Group-3 at 760 hr, the reliability is 64.5% using both methods.
Form fig. 2 in case of MLE method, Groups 1& 3 failure
rate decreasedfrom 0.003 to 0.001 in about 4000hrs while
for RRX method the failure rate reached 0.00035at the same
time. For group-2, the failure rate highly increased to reach
0.0022 at 4000hrfor RRX method and >0.003 at the same
time(4000 hr)for MLE method. Hence, Groups (1&3)have
lower failure rates compared with Group-2.
The value of Weibull distribution shape parameter () has an effect on failure calculation [36].Xie et al. stated
thatWeibull distribution showed to fit the failure characteristics of equipment at different stages of its life, by
merely changing the value of the shape parameter
appropriately. Shape parameter < 1 represents decreasing
failure rate stage, = 1 represents constant failure rate and
> 1 represents increasing failure rate stage. This explains
the decreasing failure rate of Groups 1&3 (Fig. 2) as < 1
for both cases. As for Group-2, = 1.6312 (i.e. > 1) that is why the failure rate highly increased.
Form fig.3,it could be seen that for both MLH or RRX
methods, the reliability of Group-1 reached 93% after 632 hr while Groups 2&3 reached the same reliability value after
794 and 413 hr, respectively. This means that Group-3 has
the minimum reliability at a specific time compared to
groups 1&2, while group 2 has the maximum reliability at
the same time.
4.3 Reliability Performance of Overall System
Fig.4 illustrates gas turbines overall system failure rate
using MLE and RRX methodswhile fig. 5 exhibits
thesystem reliability.
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Fig.4. Gas TurbineOverall System Failure Rate
From fig. 4, it is clear that system failure rate using MLE is higher than RRX method. In the beginning of system
operation, the two methods have the same trend of failure
rate till 0.0006 then the rate ofincrease of MLE curve is higher thanRRX. After 5000 hrMLE failure rate reaches
0.0015 whileRRX reaches0.0009.
Fig.5. Gas TurbineOverall System Reliability Using MTBF Data
As shown in figure5, it was found that value of system
overall reliability by MLE and RRX is almost the same. At
1800 hr, reliability is around 30% using MLH or RRX
methods. Similarly, at 3600 hr the reliability of MLE is 5%
while it is 7% of RRX method. Also, it could be seen that the
system reliability reaches 90% at around 400hrs.
All the above figures; MTBF data were used. Downtime
(DT) data werealso used to investigateGT system reliability
and it is compared with the results of MTBF data. Figure 6
illustrates reliability of GT overall system using DT data.
RRX
MLE
RRX
MLE Data Points
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Fig.6. Gas Turbine Overall System Reliability Using DT Data
From fig. 6, it is clear that system overall reliability; using
DT data, reaches 97%at 289hrcompared to413 hr using
MTBF (fig. 5). The reason is that downtime is the total
timethe machine isnotworking whether it is due to failure, maintenance, or schedule,etc, while MTBF is the time due
to failure only.
5. CONCLUSIONS Gas Turbine power plant reliability is a function of the
failure rate, which in turn depends on the equipment or
systems' Mean Time between Failures (MTBF) and
Downtime (DT). Those also depend on the complexity of
the design, the environment, the age of the equipment or
system and the availability of spare parts to some extent.
The failure rate is a main measuring index for system
availability.Data fitting is the first step in reliability estimation, in this study two curve fitting methods are used
MLE and RRX. The obtained results show that:
Both pdf parameters have the same value using both investigated curve fitting methods.
Group-1 reliability reached 93% at 632 hr while groups 2&3 reached the same reliability level at 794
and 413 hr, respectively using MLH or RRX method.
Group-3 has the highest failure rate in the power plant, while Group-2 has the highest reliability.
System overall reliability was calculated using MTBF & DT data. The results showed that the system
reliability reaches 97% at around 413 hr in case of
MTBF and 289 hr in case of DT data.
6. CONFLICT OF INTEREST The authors declare that there is no conflict of interest.
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AppendixA
10 Year Reliability indices of Transcorp Power Plant, UgheliDelta State Nigeria[27]
II Group 1, III Group 2, IV Group 3
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Appendix B
Group-1 Distributions Weight (MLE)
AVGOF: K-S AVPLOT: CC LKV: LHV