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555
Gas Turbine Power Plant System: A Case Sudy of Rukhia Gas
Thermal Power
Plant
1Asis Sarkar, 2Dhiren Kumar Behera
1Department of Mechanical Engineering, NIT Agarthala, India -799055 2Department of Mechanical Engineering, IGIT,Sarang,Odisha-759146
1sarkarasis6@gmail.com, 2 dkb_igit@rediffmail.com
Abstract—The reliability of the GTPPS were analyzed based on a five and half -year failure database. Such reliability has been estimated by selecting Proper model and different models for repairable system analysis were discussed. The reliability estimation by using Nonhomogenious process and Homogenous renewal Process are explained. Non Homogeneous Process was further divided into Power law Process and Log Linear model. The analysis showed that combustion chamber compressor and Generator of gas turbine unit follow Power law process and Turbine unit follow log linear model and Electrical system follow the Renewal Process.. Reliability Pattern at different Operating interval was drawn and the behavior was analyzed. The behavior shows abnormality in some component level. Finally Reliability Patern at system level was analyzed by the reliability pattern at different operating interval. From the Pattern of the Graph it can be concluded that Parallel system reliability with two unit standby is better than the Reliability with one unit standby. .All the units showed consistent reliability improvement in different operating intervals. Few components and the whole system showed abnormal trend in Reliability. This has to be further investigated The management of Power Plant has option either to keep one unit as stand by or two units as stand by or running all the units Parallaly. The component Performances are also determined by this way and one can have option how to run the plant in most efficient way.
Keywords: Gas Turbine Power Plant (GTPP), reliability, modeling, renewal, methodology
I. INTRODUCTION
In any real-life situation, the operation of a
repairable system, consisting of a number of
subsystems and components, is affected by a
number of factors, such as their configuration,
intensity of use, maintenance and repair, and
environmental stress. Any user of such a system
is typically interested in the analysis of
performance of the components, and/or
subsystems so as to suggest methods of
improving system utilization with reduced risk
and maintenance cost.
The traditional approach for addressing this
objective is to monitor the operation of
components and subsystems through their
degradation states [Chinnam, 2002[1].
Degradation of a subsystem or a component
may be reduced by two types of actions, viz.
repair and major overhaul [Pulcini, 2000[2]. In
view of this observation, a repairable system
may have two kinds of states: (i) operating state
(or ‗up‘ condition), and (ii) maintenance state
(or ‗down‘ condition, either corrective or
preventive type) [Nieuwhof, 1983; Jack,
1997[3,4]. While a component or subsystem
runs its several states, the current state of the
system may not be same as its original in the
beginning. When such a system fails, the repair
work, carried out to restore the system back to
its state just before the occurrence of its failure,
is minimum [Ansell and Phillips, 1989[5]. As
the frequency of failure of subsystems and/or
components increases over time, a corrective
maintenance action is performed to improve the
conditions of subsystems and components,
thereby reducing the probability of failure in
subsequent time-interval. Such a maintenance
action is often referred to as major overhaul
[Sherwin, 1983; Hokstad, 1997[6,7].
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The use of such a reliability model would
help an analyst identify the problem causes and
suggest remedial measures so as to continually
improve its reliability. This would in effect
ensure a consistent performance of the system
as a whole.
The gas turbine based power plant is
characterized by its relatively low capital cost
compared with the steam power plant. The gas
turbine (GT) is also known to feature low
capital cost to power ratio, high flexibility, high
reliability without complexity [1],short delivery
time ,early commissioning and commercial
operation and fast starting–accelerating. The
gas turbine is further recognized for its better
environ- mental performance, manifested in
the curbing of air pollution and reducing
greenhouse gases. It has environmental
advantages and short construction lead-time.
However, conventional industrial engines have
lower e
ffi
ciencies, especially at part load
Gas turbine engines experience degradations
over time that cause great concern to gas
turbine users on engine reliability, availability
and operating costs In gas turbine applications,
maintenance costs, availability and Reliability
are some of the main concerns of gas turbine
users. With Conventional maintenance strategy
engine overhauls are normally carried out in a
pre-scheduled manner regardless of the
difference in the health of individual engines.
As a consequence of such maintenance
strategy, gas turbine engines may be overhauled
when they are still in a very good health
condition or may fail before a scheduled
overhaul. Therefore, engine availability may
drop and corresponding maintenance costs
may arise significantly .For gas turbine engines
,one of the effective ways to improve
engine availability and reduce maintenance
costs is to move from prescheduled
maintenance to failure analysis-based
maintenance by using gas turbine health
information provided by failure analysis.
The reliability analysis of repairable system
like gas turbine power plant is necessary in this
context to have a better performance in
operation, low maintenance cost, and improved
performance in all respect. Finding out the
reliability pattern of components and system
level will help to judge the performance of the
plant either decreasing or increasing.
Later it can be diagnosed what can be done or
what are the available procedures and what are
the supports available to improve the
reliability. The study of reliability will help the
manager for advance planning of technology up
gradation, Logistic support and maintenance
planning.
Here a Case study of Rukhia gas turbine
Power plant is selected and the Reliability
analysis of the component and system level is
carried out to estimate the performance level in
both the component level and system level. The
arrangement of different section is as follows
Section 2 describe the description of the
system, Section 3 described the Methodology
for carrying out the analysis Section 4
described the Statistical tests and Reliability
models available, Section 5 described the
Results and Section 6 described the Discussions
, Section 7 described the conclusion of the
paper. and finally section 8 had ended up with
references.
II.
S
YSTEMD
ESCRIPTIONInternational Journal of Emerging Technology and Advanced Engineering
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As with all cyclic heat engines, higher
combustion temperatures can allow for greater
efficiencies .However, temperatures are limited
by ability of the steel, nickel, ceramic, or other
materials that make up the engine to withstand
high temperatures and stresses. To combat this
many turbines feature complex blade cooling
systems. As a general rule, the smaller the
engine the higher the rotation rate of the
shaft(s) needs to be to maintain tip speed. Blade
tip speed determines the maximum pressure
ratios that can be obtained by the turbine and
the compressor. This in turn limits the
maximum power and efficiency that can be
obtained by the engine. In order for tip speed to
remain constant if the diameter of a rotor were
to half the rotational speed must double. Thrust
and journal bearings are a critical part of
design.
Traditionally,
they
have
been
hydrodynamic oil bearing or oil-cooled ball
bearing. These bearings are being surpassed by
foil bearing, which have been successfully used
in micro turbines and units. In this paper the
journal bearing is designated as no#2bearing
and supportive thrust bearing is no#1 bearing.
[image:3.595.52.289.628.728.2]Gas turbines are constructed to work with oil,
natural gas, coal gas, producer gas, blast
furnace gas and pulverized coal with varying
fractions of nitrogen and impurities such as
hydrogen sulfide
are used as Fuel
.
Each unit of
GTPPS consists of five main components, viz
turbine, compressor, combustion chamber,
Generator and electric system supporting the
whole unit. The various stages of operation are
shown in the Figure 1 as shown below.
Figure 1: Block Diagram of Single Shaft Gas Turbine Power Plant
The main components of the GTPPS plant is
described with following section.
(1) Compressor: The compressor in a GTPPS
power plant handle a large volume of air or
working media and delivering it at about 4 to
10 atmosphere pressure with highest possible
efficiencies The axial flow compressor is used
for this purpose. The kinetic energy is given to
the air as it passes through the rotor and part of
it is converted into pressure. The common types
of failures applicable in the compressor of
GTPPS system is as follows. (a) Exhaust
temperature high: (b) Air inlet differential
Trouble
(2) Combustion Chambers: The combustion
chamber perform the difficult task of burning
the large quantity of fuel, supplied through
the fuel burner with extensive volume of air
supplied by the compressor and releasing the
heat in such a manner that air is expanded and
accelerated to give a smooth stream of
uniformly heated gas at all conditions required
by the turbine. The common types of failures
applicable in the combustion chambers of
GTPPS system is as follows.(a) Loss of Flame.
(b) Servo Trouble:
(3) Gas Turbine: A gas turbine used in power
plant converts the heat and kinetic energy of the
gases into work The basic requirements of the
turbines are lightweight, high efficiency;
reliability in operation and long working life.
The common types of failures applicable in the
Gas Turbine component of GTPPS system is as
follows. (a) High Pressure (H.P) Turbine under
speed. (b) Low Pressure (L.P) Turbine Over
speed. (c) Wheel space differential temperature
high. (d) Mist eliminator Failure/Trouble. (e)
Turbine Lube Oil Header Temperature High. (f)
Low hydraulic pressure. (g) Bearing drain oil
temperature high:
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(5) Electrical systems: The A.C. power circuit
ignition system receives an alternating current
that is passed through a transformer and
rectifier to charge a capacitor.
When the voltage in the capacitor is equal to
the breakdown value of a sealed discharge gap,
the capacitor discharges the energy across the
face of the ignition plug. Safety and discharge
resistors are fitted in the circuit. Except this
various circuit breakers, Relay system, Bus
Bars, control panels, transformers are used in
electrical systems. The main function is linking
the produced generation to hungry consumers‘.
The common types of failures found in the
Electrical systems of GTPPS system is as
follows
(a) De synchronization with Grid. The overall
Diagram of GTPPS Plant is described in
GTPPS operation diagram.
III.
A
RRANGEMENTO
FU
NITSI
NP
OWERP
LANTBefore carrying out any research work understanding of the system is necessary. Block diagram of the system will help to understand the inner physics of any system. So a block diagram is necessary to understand the behavior of the system. Figure 2 represents the block diagram of Rukhia gas turbine Power plant. Proper planning is necessary to carry out any research work. For the repairable system analysis a flowchart of where to start, what are the things to do; a step by step working procedure is necessary. This Framework is presented in figure 3(Appendix B). In this framework a detailed working procedure and step by step model identification is presented.
(
Appendix A :
Figure 2 gives the GTPPS Operation Diagram of Rukhia Gas Turbine Power Plant.)Methodology: In any Reliability analysis the identification of Proper model is necessary. As per the flow diagram the methodology for reliability modeling of Turbine, Compressor, Generator and Combustion Chamber and Electrical System consists of following steps
Step I: Identification of Relevant Parameters and Variables, and Collection of Relevant Data. Here all the failure data are collected as per the data
collection procedure. After that flow chart for reliability analysis is carried out. The different parameters and variables are identified and presented in the Parameters and variable sections. This detail descriptions is divided into three section
(a) Data collection
(b) Reliability Logic Diagram (c) Parameters and variables.
Data collection: The collection of Data is necessary to carry out the analysis. The data are collected from the maintenance logbook available in the plant and asking questions to the operators, supervisors and managers of the plant. Data are required to be collected over a period of time for providing satisfactory representation of the true failure characterization of the machine. Data used in recent studies have been collected for a period of 5 and half years. Approximately 956 failure data is collected for all the seven units over the stated Period. These Data are segregated according to component wise and unit wise. The failure time repair time and time of breakdowns reasons for failures are also collected.
(a) Reliability Logic Diagram:
Before carrying out any research work Understanding of the system is necessary. Block diagram of the system will help to understand the inner physics of any system. So a block diagram is necessary to understand the behavior of the system. Figure 3 represents the block diagram of Rukhia gas turbine Power plant. Proper planning is necessary to carry out any research work. For the repairable system analysis a flowchart of where to start, what are the things to do; a step by step working procedure is necessary. This Framework is presented in figure 3,and it is attached in Appendix A. In this framework a detailed working procedure and step by step model identification is presented. Parameters and variables.
Step-II: Component- and System-level Analysis
Decisions regarding relevance of component- or system-level analysis are to be taken on the basis of the following considerations. These considerations are taken under the heading of Component-level Analysis and System-level Analysis. The details of the analysis are described in the next section.
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(normal or accelerated life tests) are to be conducted for such components for reliability analysis. Measurement and evaluation of reliability of such components would follow the analysis of test data.
(APPENDIX B : Figure 3 gives the Methodology for Reliability Analysis of turbine, compressor, combustion chamber and electrical system)
(c) Parameters and variables
The parameters of Turbine, Compressor, and Combustion chamber Generators are mentioned in Table 1 given in Appendix C.
IV. SYSTEM LEVEL ANALYSIS
For this kind of analysis, the configuration of the system as a whole is to be known. Both parametric as well as nonparametric approaches for reliability measurement and evaluation are a possibility. The conditions under which parametric or non-parametric approach is recommended are as follows:
Condition of Parametric approach: The number of data points related to a system is made available within the given time period and sufficient enough to verify the distributional assumption for the variables under consideration at an acceptable level of significance. A parametric approach may be applied in three situations: When the failure data follows a homogeneous Poisson process (HPP) (no trend and dependence in data), or (ii) nonhomogeneous Poisson process (NHPP) (trend in data), and (iii) Branching Poisson process (BPP) (no trend but dependence in data). The parameters of the three processes as mentioned may be estimated with a number of tools and techniques, such as ‗probability plot‘ and ‗maximum likelihood estimation‘
Conditions for Nonparametric Approach:
A nonparametric approach is recommended
when the following conditions are met. The
number of data points related to a system is
made available but insufficient enough to verify
the distributional assumptions for the variables
under consideration at an acceptable level of
significance. Special tools and techniques, such
as Kaplan-Meier estimator, simple and standard
actuarial methods, and regression analysis, are
used in this case.
The failure rate data of Turbine, Compressor, Combustion Chamber and Generatormay be analyzed with such techniques. In this case the trend test is done on the data and presented in the result section of trend test .
The electrical system follow the Renewal Process as dependency were not proved in the serial correlation tests. Hence Branching Poison Process is rejected.
Step III:
Development of Appropriate
Reliability Model. In this section the
appropriate reliability model is described
into following three sections.
Statistical tests used: The statistical tests used here are trend test, serial correlation test, Laplace test and Military handbook tests. The procedure for carrying out all these tests are described in the Statistical tests and Reliability models section.(a). Available reliability model: The different reliability models used here are Power law process and log linear process under the non homogenious poisson process and renewal process under the homogeneous poison process. The procedure for carrying out all these tests are described in the Statistical tests and Reliability models section. In addition the different reliability models and estimation of parameters and reliability estimation are described in the Statistical tests and Reliability models section. Selection of appropriate reliability model: The appropriate Reliability model is selected by different testing procedure as described in section (b), i.e. by Trend test , Serial correlation test, Military Handbook test and Laplace test, and renewal process. Based on the above test described in Result section the following conclusion is attained and described in table 2.
Model
Power law process.
Generators Power law process.
Combustion chamber Power law process.
Electrical system Renewal Process
[image:5.595.322.553.574.716.2]Turbine Log linear model
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Step IV: Verification and Validation of the
Proposed Approach.
The proposed models are to be tested for its
verification and validation in a number of
situations and conditions as discussed for the
testing Procedure. Modifications in the
proposed model are based on analysis of
difference between the actual and estimated
performance data over a period. The
methodology for reliability modeling of GTPPS
and its subsystems like turbine compressor,
combustion
chambers
and
Generators
considered to be a repairable system, is
explained in detail with the help of flow
diagram shown in Figure 3.This methodology is
applied for reliability analysis of GTPPS and its
subsystems
like
turbine
compressor,
combustion chambers and Generators and the
details of the application of methodology are
given below.
4. Statistical tests and Reliability models:
Various statistical tests are available for the
analysis of data. These are Trend tests, TTT
plot, Nelson Allen Plot, Serial correlation test,
Duane Growth model, Apart from for deciding
the appropriate reliability models the Military
handbook tests and Laplace stets are used. The
various Reliability models available are
homogeneous
poison
process,
non
homogenious poison process, branching
poisson process, renewal process and models
for non repairable items. The various statistical
tests and reliability models are described under
the following two sections described below (1)
statistical models (2) Reliability models
(1) Statistical Tests: The various Statistical
tests used in this paper are described below.
These are
Trend test- Cumulative failures verses
cumulative time between failures.
Serial Correlation test: to find whether Data is
independently and identically distributed.
Military Handbook test: To find the data pattern
follow Power law process
Laplace test: To find out the data pattern follow
Log linear process .All these aforesaid tests are
stated below.
Trend Test:-
The graphical trend test and
serial correlation test of TBF data of Turbine ,
compressor, Generator, combustion chamber
and electrical components and their subsystems
are necessary for validity of assumption that
failure data are independently distributed (iid)
in an analysis of failure distribution model. The
trend test is done by plotting the cumulative
time
between
failure
(CTBF)
against
cumulative frequency of occurrence. Serial
correlation test is done by means of plotting ith
TBF against (i-1) th TBF. Trend plot of
Turbine, compressor, Generator, combustion
chamber and electrical components and their
subsystems will be five curves presented in
figure.. In the test, weak or absolute trends were
found for Turbine, compressor, Generator,
combustion chamber and no trend in electrical
components and exhibited concave upward and
concave downward respectively in trend
test..The trend test were further elaborated by
Trend test 1 and Trend test 2 in Reliability
modeling to determine Homogeneous poison
process or Non Homogeneous poison process in
trend test 1 and Renewal process or Non
Renewal Process in Trend test 2 Finally the
Reliability of each system are calculated by
selecting the Proper model in analysis and the
models are followed in the manner as described
below.
Serial Correlation test:
Serial correlation test
is done by means of plotting ith TBF against
(i-1) th TBF. The condition is = Points are not
randomly scattered and in straight line . *Data
are dependent and Branching Poison Process
can be applied . Points are randomly scattered
and not in straight line ≠ Data have no
correlation and dependence = Data are
independent and identically Distributed.
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serial correlation test, the points are randomly
scattered not in a straight line. For example,
serial correlation test of Turbine, compressor,
Generator, combustion chamber and electrical
components are presented in different figures.
So the above failure data can be assumed to be
independently and identically distributed (iid)
Military Hand book test:
The Military Handbook test is a test for the
null hypothesis
H0 :HPPversus alternative
hypothesis
H
1:
NHPP
with increasing power
law process. The test statistic of this test for
more than one process is given by
ˆ
1 1
2 i ln( )
n m
i i
C
i j ij i
b a
M
t a
---
(22)
And it is chi-square distributed with 2q degrees
of freedom,
Where
m
i i
n q
1
ˆ
under the null hypothesis of
HPP [Kvaloy and Lindquist, 1998]. The Mc
value is calculated corresponding to failure
parameters of component and compared with
chi square value of 2N degree of freedom and
if chi square value is > than Mc value than
Null hypothesis is rejected otherwise Null
hypothesis is accepted. thus deciding Log linear
model or Power law process. In case the null
hypothesis is not accepted for both the tests,
then the conflict regarding NHPP with
log-linear model or power law process is resolved
by calculating the so-called probability or
P-values of the tests as mentioned. The P-value
for a normal distribution is defined to be the
probability value corresponding to the z-value
against Laplace‘s test statistic. For Military
Handbook test, P-value corresponds to the χ
2-value with one degree of freedom. A smaller
P-value is indicative a stronger evidences a null
hypothesis [Rigdon and Basu, 2000][32].
Laplace test:- Laplace‘s test is conducted for
the null hypothesis,
H0 :HPPversus the
alternative hypothesis,
H
1:
NHPP
with
log-linear model. The generalization of the
Laplace‘s test statistic for more than one
process may be given by
ˆ
1 1 1
2
1 1
ˆ ( )
2 1
ˆ ( )
12
i
n
m m
ij i i i
i j i
C m
i i i
i
t n b a
L
n b a
---
(23)
Where
n
ˆ = n or (n-1) if the process is time
itruncated or failure truncated, respectively
[Kvaloy and Lindquist, 1998].[33] This test
statistic is asymptotically standard normally
distributed under the null hypothesis. The
Laplace
statistics
value
is
calculated
corresponding to failure parameters of
component and compared with chi square
value of N degree of freedom and if chi square
value is < than the Laplace value than Null
hypothesis
is
rejected
otherwise
Null
hypothesis is accepted. thus deciding Log linear
model or Power law process. (2) Reliability
models: The various Reliability models used in
this paper are described below. These are
Homogeneous
Poisson
Process,
Non
homogeneous Poisson Process,
and Branching Poison Process,Renewal Process and Non repairable
items
Homogeneous Poisson Process
The homogeneous Poisson process is a Poisson
process with constant intensity function
[Rigdon and Basu, 2000][11]. Since the
systems are identical, the failure process of
each system is an HPP with intensity
1, and
is same for all systems. A counting process is a
homogenous Poisson process with parameter λ
>0 if :
N
(0)=0. The process has independent
increments
The number of failures in any interval of length
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F(t) = 1- e
-λt= The cumulative density
function of the waiting time to the next failure
(or interarrival time between failures)
N(T) = the cumulative number of failures from
time 0 to time T. P{N(t)=k} = (λT)
ke
-λt/ k│
M (T) =λt = the expected number of failures by
time T, λ = the rate of occurrence of failure
ROCOF, 1/λ = The Mean time between
failures (MTBF). So the reliability R(t) in the
case of Homogeneous Process =- e
-λt---(1)
Branching Poison Process:
Figure: 4 Example of Branching Process, Ref [12]
The Branching Poison Process: In some cases
failures tend to be bunched together . This may
indicate that the system is not correctly repaired
causing a number of subsequent failures. A
model that can account the phenomenon is the
branching Poisson Process (BPP). The BPP was
originally proposed by Bartlett (1963)[13] as a
model for Traffic flow., where a slow moving
vehicle may be followed immediately by a
number of other vehicles. This is analogous to
to a single failure causing a number of
subsequent failures. Lewis (1965)[14] applied
the BPP to failure times of computers. To be
more precise suppose the primary failures are
generated according to poison process with rate
λ . After each failure there is probability 1-r that
the repair will be done correctly; In this case
the next failure will occur when the next
primary failures occurs. With probability r, the
repair is not done correctly; In this case the
primary failure will spawn a finite renewal
process of subsidiary failures.
The number of of subsidiary failures that follow
from a primary failure is a discrete random
variable. Lewis (1964) and cox and Lewis
(1966)[15,16] describe the cases where the
random variable has a geometric, negative
binomial and poison distribution. It is assumed
that both kind of failures (primary and
secondary) are indistinguishable. Define G to
be the number of subsidiary failures
corresponding to a single primary failure
conditioned on there being at least one such
subsidiary failures and let S be the
(unconditional) number of subsidiary failures.
Note that S=0 when the repair is done correctly.
Let Z1, Z2 ---- denote the times between the
primary failures and let Y
i (1),Y
i(2),---Y
i(s),times between the subsidiary failures
that are triggered by ith primary failures.
Finally let T
1< T
2< T
3denotes the failure time
regardless of the type. In practice we observe
only the Ti‘s and not the type of failure. The
ROOCOF function for the BPP can be obtained
as follows. Let H(t) denote the expected
number of subsidiary failure from a single
primary failure to an interval interval of length
t and let F
k(t) and f
k(t) denote the cdf and pdf
respectively of the random variable Z
1+ Z
2+---+ Z
k(i.e. the time of the k th primary
failure) Given that Z
1=z the expected events in
the interval[0,t] from the first subsidiary
process, the first subsidiary process, then E[N
(1)
(t) the number of failures in (0,t) due to first
subsidiary process then
E[[N
(1)(t)]=
E{
N
(1)(t)│Z
1]}
=
∫
(t)ІZ
2=z] f
1(z) dz
=
∫
(z)dz
A similar analysis could be applied to the time
of the k th failure. The expected no of failure N
(1)
(t) in (0,t) due to the kith subsidiary process is
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The expected number of failure of any type in
[0,t] is thus Λ(t) =E[N,(t)]=E( Number of
Primary
failure
in
[0,t])
+
∑from
k th primary failure)
= Λz(t) +
∑
∫
= Λz (t) +
∫ ∑
(z)]dz
= Λz (t) +
∫
(z) dz----[3}
The ROCOF for the branching process is
therefore
=Λ(t). =Λ
z(t) + d/dt
∫
zdz. .If no subsidiary failure is there the
ROCOF
=λ--- (2)
The Reliability function is = e
-λtRenewal Process: RP is defined as a process in which the different times to failure of a component or system, Xi, are considered independently and identically distributed random variables. This is consistent with the primary underlying notion of this process that assumes that the system is restored to its original (like new) condition following a relatively instant repair action. Because it represents an ideal situation, this model has very limited applications in the analysis of repairable systems, unless the system consists of primarily non-repairable (replaceable) components in sockets. That is, when a part of the system fails, it will be taken out and replaced by a new one [17] . The expected number of failures in a time interval [0, t] is given by: Λ (t) = F(t) +∫
---(3)
Where F (t) is the cumulative distribution function (cdf) of the time between successive repairs or replacements of the system. By taking the derivative of both sides of Eq. (1) with respect to t:
λ(t) = f(t) + ∫
---(4) Where f(t ) is the probability density function (pdf) of the time between successive failures. In case of the two parameter Weibull distribution, representing the random variable t, the cdf and the pdf are of the form:
F (t) = 1- --- (5)
F (t) =
--- (6)
Where a and b are the scale and shape parameters, respectively. The ML estimators for parameters a and b are [18,19]:
=
[
∑
]
---
(7)
∑
[
]
∑
--
^
=
∑
--- (8)
Where ti is the observed time between successive failures and n is the total number of failures observed. A close form solution to estimate the expected number of failures in the case of Weibull distribution has not been developed yet. Instead, a group of numerical solutions can be obtained. Smith and Lead [20] better have proposed an iterative solution to the renewal equation for cases where the failure interarrival times follow a Weibull distribution.
The maximum likelihood estimation is done by and β¯ and Reliability is estimated by the formula
= --- (9)
Non repairable item
For this model illustrated in figure 3. there is neither preventive maintenance nor corrective nor preventive maintenance, and the failure intensity equals 0 after failure has occurred. The component history (step 2) is recorded by X(t), where
X(i) =1, if i<X, and 0 otherwise--- (10) The variable Keeps track of whether the failure has occurred or not, which is the only event relevant to IC(t)The intensity process IC(t) (step 3) is
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Thus the intensity process is the hazard rate of the inherent TTF, truncated at time of failure and I(t) –E[ IC(t) –E[λ(t),X(t)]= λ(t).E[X(t)]= λ(t).R(t) =f(t)
–---(12)
This is a special case of eqn(12) and again gives the result that the mean intensity, I(t), for a nonreplicable items equals the pdf, f(t). The interpretation is that at time [t,t+dt) equals f(t). dt
Nonhomogenious Poison Process:
Nonhomogeneous Poisson processes are also useful for modeling repairable system reliability [Bertkeats and Chambal, 2002; Ryan, 2003].[21,22] The failure process between two successive overhaul actions is described by a nonhomogeneous process [Ascher and Feingold, 1984; Engelhard and Bain, 1986][23,24]. The NHPP model also assumes ‗minimal repair‘, which means that after each failure and following repair, the system is in the same state as it was just prior to that failure [Heggland and Lindqvist, 2007][25]. Moreover, it is also assumed that the failed part is small one of the system. During and after repair or replacement of this part the other parts will not be affected. The assumption for modeling of more than one system is that all systems are identical with respect to their technology and mean time between failures. Also, all systems are time-truncated with same starting and finishing points. Among the NHPP‘s, large attention has been devoted to the power law process [Walls and Bendell, 1986; Pulcini, 2001] [26,27] and log-linear processes [Cox and Lewis, 1966; Baker, 2001].[28,29] In this context, it is mentioned that if the failure data of a system rejects the null hypothesis of Laplace‘s Test, the data is considered to follow the log-linear model. On the other hand, if the failure data of a system rejects the null hypothesis of Military Handbook test, the data follows the power law model. The following sections describe the procedures for estimation of parameters for the two models (Power Law and Log-linear Processes), as mentioned.
Power Law Process
The intensity function of power law process is
given as follows [Crow, 1990]:[30].
t
t
1
, λ where, , , t > 0, and t is theage of the system. Hence, the power law mean value function is given by,
( ( )) , t 0
E N t
t ---(13)The probability that a particular system will experience n failures over its age (0, T) is given by the Poisson expression [Rigdon and Basu, 1989][31],
(
)
( ( )
)
, n 0, 1, 2,...
!
n T
T
e
P N t
n
n
----14)
And the reliability may be given by, R(t) = exp{intensity function -λt}. Maximum likelihood estimation of the parameters may be described as follows: Let, tij denotes the jth failure of ith system.
Suppose, ni failures are observed for system i.
Applying the joint pdf theorem, the MLE of parameters of power law process is given by and estimated as stated. Hence, the intensity function of power law process for GTPPS -unit is given .With the help of this intensity function the reliability of the GTPPS -unit may be calculated
1
1
ˆ
k i i k
in i
n
t
--- (15)
and 1
1 1 1
ˆ
ˆ log( ) i log( )
k i i
n
k k
in in ij
i i j
n
t t t
--(16)
If the systems are time-truncated and tin=T for all
systems, the above-mentioned estimates may be written as
1
ˆ
k i i
n
kT
--- (17)and
1
1 1
ˆ
log( )
i
k
i i n k
i j ij
n
T t
---(18) (18)
Reliability R(t) = e –intensity function Log Linear Model
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565
( )
t
e
t
--- (19)Where , , t>0, and t is the age of the system. Similar to power law process the maximum likelihood estimates of parameters of log-linear process for more than one system is calculated as follows
ˆ 1
ˆ
ˆ
(
1)
k
i i
T
n
e
k e
---(20) (20)
and
ˆ
1 1
ˆ 1 1
0
ˆ
(
1)
i
k k T i i n
k
i i ij T i j
n
Te
n
t
e
-(21)
Reliability R(t) = e –intensity function
V. RESULTS
Data used in recent studies have been
collected for a period of 5 and half years.
Approximately 956 failure data is collected for
all the seven units over the stated Period. These
Data are segregated according to component
wise and unitwise.The failure time repair time
and time of breakdowns reasons for failures are
also collected. These failure data of different
units were collected from the maintenance log
book. Failure behavior of these machines has an
influence on availability or failure pattern of the
machine as a whole. The basic methodology for
reliability modeling is presented in figure 3. It
shows a detailed flow chart for model
identification and is used here as a basis for the
analysis of failure Data So TBF are arranged in
a chronological order for using statistical
analysis to determine the trend in failure and
other aspects of Reliability. In this section the
detail analysis of the test carried out on
different components are discussed.
Compressor: The Cumulative TBF verses
cumulative failures datas are drawn in a plot
and the plot is shown below in figure 5.
Similarly the failure Data of ith failure verses
(i-1) th failure data are plotted in figure 6.
The result shows that Failure data line has
slight deviation from straight line. and
the points [image:11.595.318.558.167.329.2]are randomly scattered and not in straight line in the serial correlation test.
[image:11.595.316.557.362.520.2]Figure :-5 Cumulative Failure nos verses cumulative time between failure Data of compressor system
Figure: 6 Serial correlation test of Compressor units of GTPPS.
Conclusion: Data is modeled by NHPP.and data are independently and identically distributed. Now to select the model for Log linear or Power law process the data are further processed by carrying out military Handbook Test. The result of Military Handbook Test is described in table 2.
It is mentioned that if the failure data of a system rejects the null hypothesis of Laplace‘s Test, the data is considered to follow the log-linear model. On the other hand, if the failure data of a system rejects the null hypothesis of Military Handbook test, the data follows the power law model. Data Pattern follow the Power Law Process. By applying the Formula of intensity function and appropriate formula of Reliability the Reliability at different time intervals are calculated and it is shown graphically in the figure 7 mentioned below.
Plot of CTBF verses cumulative failures of compressor failures datas
0 10000 20000 30000 40000 50000 60000
1 4 7 10 13 16 19 22 25 28 31 34 cumulative nos of failures
cu
m
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at
iv
e
T
.B.
F
.
Series1 Series2
Serial Corelation test of compressor failure Datas
0 5000 10000
0 5000 10000 ( i -1)th T.B.F
I t
h
T
.B.
F
.
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Table: 2 Result of Military Handbook test of Compressor
[image:12.595.315.564.128.278.2]Figure 7 Reliability at different intervals of compressor units.
Table: 3 Result of Military Handbook test of Combustion Chamber
Combustion Chamber: The Cumulative TBF verses cumulative failures datas are drawn in a plot and the plot is shown below in figure 8. Similarly the failure Data of ith failure verses (i-1) th failure data are plotted in figure 9. The result shows that Failure data line have slight deviation from straight line. and the points are randomly scattered and not in straight line in the serial correlation test.
Figure 8:-Cumulative failure vs. Cumulative T.B.F of combustion chambers used in GTPPS system
Figure:--9 Serial correlation test of Combustion Chamber units of GTPPS.
Conclusion: Data is modeled by NHPP.and data are independently and identically distributed. Now to select the model for Log linear or Power law process the data are further processed by carrying out military Handbook Test. The result of Military Handbook Test is as follows. It is mentioned that if the failure data of a system rejects the null hypothesis of Laplace‘s Test, the data is considered to follow the log-linear model. On the other hand, if the failure data of a system rejects the null hypothesis of Military Handbook test, the data follows the power law model. Data Pattern follow the Power Law Process. By applying the Formula of intensity function and appropriate formula of Reliability the Reliability at different time intervals are calculated. and it is shown graphically in the figure 10 mentioned below.
Reliability at different intervals of Compressor
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 100 200- 300 400 500 600 700 800 900 1000
operating Time
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ty
Cumulative T.B.F. verses cumulative failures of Combustion chamber failure data
0 10000 20000 30000 40000 50000
1 10 19 28 37 46 55 64 73 82 91 100 109
cumulative failures
cu
m
ul
at
iv
e
T
.B.
F
.
Serial corelation of combustion chamber failure data
0 2000 4000 6000 8000 10000
0 2000 4000 6000 8000 10000 (i-1)th T.B.F.
i t
h
T
.B.
F
.
Compon ent
MC
value
Sam ple size
Χ2 N,,0.5val
ue
Result
Compre ssor
-30.354
35 Χ2
35,,0.5=4
5.5
Null hypothesis rejected
Compon ent
MC
valu e
Samp le size
Χ2
N,,0.5valu
e
Result
Combust ion Chamber
-104. 3
60 <Χ2 60,,0.5=
88.38
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Figure 10: Reliability at different intervals of combustion chamber system.
Generator:
The Cumulative TBF verses
cumulative failures data are drawn in a plot and
the plot is shown below in figure 11. Similarly
the failure Data of I th failure verses (i-1) th
failure data are plotted in figure 12. The result
shows that Failure data line has slight deviation
from straight line. and the points are randomly
scattered and not in straight line in the serial
correlation test.
[image:13.595.50.298.131.272.2]Figure 11:-Cumulative failure vs. Cumulative T.B.F of Generators used in GTPPS system
Figure 12: Serial correlation tests of Generator units of GTPPS.
[image:13.595.302.555.405.471.2]Conclusion: data is modeled by NHPP.and data are independently and identically distributed. Now to select the model for Log linear or Power law process the data are further processed by carrying out military Handbook Test. The result of Military Handbook Test is as follows
Table 4: Result of Military Handbook test of Generator
Conclusion: Data Pattern follow the Power Law Process. By applying the Formula of intensity function and appropriate formula of Reliability the Reliability at different time intervals are calculated. and it is shown graphically in the figure 13 mentioned below
Figure: 13 Reliability at different intervals of Generator system
Reliability at different intervals of combustion Chamber
0 0.2 0.4 0.6 0.8 1 1.2
0 100 200- 300 400 500 600 700 800 900 1000
Operating intervals
R
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ty
Plot of cumulative T.B.F. verses cumulative failures of Generator failure data
0 10000 20000 30000 40000 50000 60000
1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 cumulative failures
cu
m
ul
at
iv
e
T
.B
.F
.
Serial corelation test of Generator system failure data
0 1000 2000 3000 4000 5000 6000 7000
0 2000 4000 6000 8000
(i-1)th T.B.F.
I
th
T
.B
.F
.
Series1
Reliability at different operating intervals of Generator
0 0.2 0.4 0.6 0.8 1
100 200- 300 400 500 600 700 800 900 1000
Operating time
R
el
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bi
li
ty
Compon ent
MC
value Sam ple size
Χ2 N,,0.5val
ue
Result
Generat or
-46.85 60 Χ2 60,,0.5=8
8.38
Null hypothesis rejected
Compone nt
MC
value
Sampl e size
Χ2
N,,0.5valu
e
Result
Generator -46.85 60 Χ2
60,,0.5=88.
38
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Turbine: The Cumulative TBF verses cumulative failures data are drawn in a plot and the plot is shown below in figure 14. Similarly the failure Data of ith failure verses (i-1) th failure data are plotted in figure 15. The result shows that Failure data line has slight deviation from straight line. and the points are randomly scattered and not in straight line in the serial correlation test. [image:14.595.310.558.214.333.2]Figure 14:-Cumulative failure vs. Cumulative T.B.F of Turbines used in GTPPS system
Figure: 15 Serial correlation test of Turbine units of GTPPS.
Conclusion: data is modeled by NHPP.and data are independently and identically distributed. Now to select the model for Log linear or Power law process the data are further processed by carrying out military Handbook Test and Laplace test.. The result of Military Handbook Test is Null hypothesis is accepted It is mentioned that if the failure data of a system rejects the null hypothesis of Laplace‘s Test, the data is considered to follow the log-linear model.
On the other hand, if the failure data of a system rejects the null hypothesis of Military Handbook test, the data follows the power law model. Since Turbine failure Data rejects the Null hypothesis in Laplace test the failure Data follows Log Lineaer model. The result of Laplace Test is as follows
Com pone nt
Laplu s Statist ics
LC>0 or LC< 0
Laplace Statistics> <chi square statistics Χ2N,,0.5
Null Hypot hesis
Turb ine
-111.9 3
LC<0,
decreasing trend
> Χ2
N,,0.5 Reject
[image:14.595.319.573.450.598.2]ed
Table: 5 Result of Laplace test of Turbine unit system.
Conclusion: Data Pattern follow the Log linear model.By applying the Formula of intensity function and appropriate formula of Reliability the Reliability at different time intervals are calculated. and it is shown graphically shown in the figure 16 as mentioned below
Figure16: Reliability at different intervals of Turbine system
Electrical System: The Cumulative TBF verses cumulative failures datas are drawn in a plot and the plot is shown below in figure 17. Similarly the failure Data of ith failure verses (i-1) th failure data are plotted in figure 18. The condition is Deviations from straight line: trend is absent No deviations – trend present, Points are randomly scattered and not in straight line .
Data are independent and identically distributed. In straight line = Data have correlation and dependence.
Plot of cumulative failures verses cumulative T.B.F. of Turbine system failure
0 10000 20000 30000 40000 50000 60000 70000 80000
1 26 51 76 101 126 151 176 201 226 cumulative failures
C
um
ul
at
iv
e
T.
B
.F
.
Series1 Series2
Serial corelation test of Turbine system failure data
0 500 1000 1500 2000 2500 3000 3500
0 1000 2000 3000 4000 (i-1)th TBF
It
h
T
B
F
Series1
Reliability at different time intervals of Turbine system of GTPPS
0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1
0 100 200 300 400 500 600 700 1000
Operating time
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Figure 17: Cumulative failure vs. Cumulative T.B.F of electrical system used in GTPPS system
Figure18: Serial correlation tests of Electricals system units of GTPPS.
Conclusion: Trend is absent Hence it is Homogeneous Process Since correlation and dependence is absent Datas are independently and identically distributed. So next condition is after repair it is as good as new. Yes it is as good as new so this is a case of Renewal Process .By applying the Formula of intensity function and appropriate formula of Reliability the Reliability at different time intervals are calculated and it is shown graphically shown in the figure 19 as mentioned below
Figure19: Reliability at different intervals of Electrical system
Reliability analysis in system level:-
Estimation of Reliability of systems by using the Reliability of components.
Unit Level Reliability: Assuming the total System has identical units and all the 5 units are parallally connected keeping two standby units as shown in the operational Diagram of GTPPS. The unit and System level Reliability is calculated. Unit level Reliability: Since all the components are in series the series system reliability is Rs= R1x R2x R3x R4x
R5 [34] the system level reliability is calculated by
[image:15.595.49.301.132.253.2]applying the series level formula. The Calculated Reliability is shown in the Graph of figure 20.
Figure 20: Reliability at different intervals of identical units in series system.
Determination of Parallel system Reliability: Parallel Using the Formula Rp=1-Qp 1-[1-e(-λt))]n
[36]of Parallel system Reliability we get the Reliability of Parallel system at different time intervals. The Reliability at different time intervals and is shown in Figure 21.
Figure 21: Reliability Pattern of Parallel system unit failure Dataˆˇ Cumulative T.B.F. verses cumulative failure data of electrical
system failures
0 100000 200000 300000 400000 500000 600000
1 14 27 40 53 66 79 92 105 118 131 144 157 170
cumulative failures
cu
m
ula
tiv
e
T.
B.
F.
Serial corelation test of electrical system failure data
0 1000 2000 3000 4000 5000 6000
0 1000 2000 3000 4000 5000 6000
(i-1) th T.B.F.
ith
T
.B
.F
.
Reliability of Parallel unit from unit 1,4,5,6,8 units in GTPPS
0 0.2 0.4 0.6 0.8 1
0 100 200- 300 400 500 600 700 800 900 1000 Operating Time
R
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ty
Reliability Pattern of the GTPP system having all units in parallel
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
100 200- 300 400 500 600 700 800 900 1000 Operating Time
R
el
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bi
lit
y
Reliability at different intervals of electrical system
0 0.2 0.4 0.6 0.8 1
0 100 200 300 400 500 600 700 800 900 1000
Operating Time
R
e
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b
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Determination of standby system Reliability
The formula used here is
,Rs(t) = λ1t +
λ1t - λ2t)+ λ 1 λ2
X[
+
+
]
[35]---[27] Where
λ1 = Failure rate of the Parallel units
λ2 = Failure rate of the standby unit
λ3 = Failure rate of the second standby unit
[image:16.595.47.286.132.322.2]Using the appropriate formula we got the Parallel system reliability with two unit standby. The Reliability at different time interval is shown in figure 22
Figure 22: Reliability Pattern of whole plant unit if two stand by unit is kept
6.0 Discussion:-The selection of model for calculation of Reliability shows that the turbine unit follows Log Linear model and compressor unit, Generator and combustion chamber units follows Power law model and electrical system follow Renewal Process. The Reliability of different components of Rukhia Gas Power Plant is analyzed by taking the failure data of different operating units and consultation with Plant operating Personnel and management People about the arrangement of units in winter and summer season, It is known that in winter less Power is supplied to Grid and two units are kept as standby system. In Summer the demand is more and only one unit is kept as standby system. In both cases Reliability is determined by taking one and two units as stand by. Initially the component level Reliability is discussed one by one.
Compressor: The trend test shows that there is little deviations from straight line. So it can be concluded that Data Pattern of TBF data have trend.
In serial correlation test it is proved that Data pattern are independently and identically distributed. So Data are modeled by Non Homogeneous Poison Process Now for deciding whether Data Pattern follow the Power l;aw process or Log Linear model Data are again tested by Military Handbook test. It is Proved that the Data Pattern follow the Power law Process as Null hypothesis is rejected in Military Handbook test. By setting the Parameters of Power law Process the Reliability at different time interval are shown in figure 7. The Reliability from 0th hour to 100 th hour is drastically changing in the Diagram.
Combustion Chamber: The trend test shows that there is little deviations from straight line. So it can be concluded that Data Pattern of TBF data have trend. In serial correlation test it is proved that Data pattern are independently and identically distributed. So Data are modeled by Non Homogeneous Poison Process Now for deciding whether Data Pattern follow the Power l;aw process or Log Linear model Data are again tested by Military Handbook test. It is Proved that the Data Pattern follow the Power law Process as Null hypothesis is rejected in Military Handbook test. By setting the Parameters of Power law Process the Reliability at different time interval are shown in figure 10. Here the Reliability is almost constant fro 0 th hour to 200 hour and from 700 hour to 900 hour which requires further investigation as Reliability pattern does not show a constant Path.
Generator: The trend test shows that there is little deviations from straight line. So it can be concluded that Data Pattern of TBF data have trend. In serial correlation test it is proved that Data pattern are independently and identically distributed. So Data are modeled by Non Homogeneous Poison Process Now for deciding whether Data Pattern follow the Power l;aw process or Log Linear model Data are again tested by Military Handbook test. It is Proved that the Data Pattern follow the Power law Process as Null hypothesis is rejected in Military Handbook test. By setting the Parameters of Power law Process the Reliability at different time interval are shown in figure 13. Here the Reliability Pattern shows a decreasing trend with constant rate. It is almost acceptable..
Turbine: The trend test shows that there is little deviations from straight line. So it can be concluded that Data Pattern of TBF data have trend.
System Reliability at different time interval with 2 unit standby of Rukhia Gasthermal Power Plant
0 0.2 0.4 0.6 0.8 1
0 100 200 300 400 500 600 700 800 900 1000
Operating Time
R
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