Journal of Power and Energy Engineering, 2014, 2, 139-145
Published Online April 2014 in SciRes.
How to cite this paper: Buchta, J., Oziemski, A. and Pawlik, M. (2014) Probabilistic Issue of Reliability for Power Machinery
Operating in Coal Fired Power Plants. Journal of Power and Energy Engineering, 2, 139-145.
Probabilistic Issue of Reliability for Power
Machinery Operating in Coal Fired Power
Janusz Buchta, Andrzej Oziemski, Mac iej Pawlik
Lodz University of Technology, Institute of Electrical Power Engineering, Lodz, Poland
Received January 2014
The paper presents results of reliability analysis made fo r lignite fired 370 MW rated power units
installed in the Belch atow Power Plan t (Polan d). The concept of standardized power unit and the
method of a histogram with a set number of observations in each class wer e applied in a s tud y.
The study includes analysis of probability distributions of operation times and repair times for the
main power unit components. Empirical probability distribution functions have been identi fied
and their parameters estimated in the study. Th e final fo recast include s an estimation of suc h re-
liability measures like expected operation time, expected failure rate , average repair time and
expected annual failure duration.
Power Uni t; Reliability Indices; Probability Distribution Functions of Operation Time
and Repair Ti me
1. Introduction
The maintenance practice shows that failure frequency of power unit components is many times higher than
other components of the power system (overhead lines, transformers, switchgear, control systems, protections
etc.) [1,2]. Intense wear and tear of machine elements as well as the necessity of maintenance works effect rou-
tine repairs of power units. The economic impact of power unit shut-down depends mainly on failure frequency,
duration of rep air and the period of the year in which the shut-down happens. A very important issue is therefore
to optimize the values of these quantities.
The main reasons of power machinery failures can be classified as follows:
Errors in con struction and design;
Material defects;
Assembly errors;
Operational wear and tear of material (corrosion, erosion, fatigue, strain, ageing);
Influence of external conditions (e.g. lack of fuel or water, power system disturbances);
Mistakes in repairs;
J. Buchta et al.
Mistakes in machinery operation;
Insufficient care to keep installation in appropr ia te technical cond ition .
The first three groups of mentioned above reasons g ener ally reveal during the early lifetime of power machi-
nery and are removed au courant during the warranty repairs. Reasons belonging to the fourth rev ealed after a
longer period of oper atio n. However, particular attention should be paid to the last three groups of reasons which
depend directly on skills of operational staff worki ng in power p lant.
2. Mathematical Bas is of the Reliability Estimation of Power Units by the Method
of a Histogram
Power units are complicated thermal and mechanical installations consisting of several components with a dif-
ferent level of redundancy. Basic po wer unit components (boiler, turbine, generator) are singular. However, an
auxiliary devices (ID fans, FD fans, co al mills, feed pumps, condensate pumps, cooling water pumps, ash han-
dling etc.), in order to increase reliability of operation, have overt or latent reserve (structur al r edund ancy).
The characteristic feature of power unit reliability, directly resulting from the redundancy of an auxiliary sys-
tem is the possibility to appear failures of different kind, i.e. power unit continues operation with its rates, power
unit operates with power limitation or power unit must be shut-down. This means that power unit is a mul-
ti-stage object in sense of its reliability, on the contrary to the two-stage installations staying in ability or inabil-
ity to operate, which refers to many power tr ansmission and distribution devices. A multi-stage feature causes
that reliability analysis of power unit uses Markov or more complex semi-Mark ov processes, which form an
adequate probabilistic model.
An operational reliability analysis of complex installation is performed using a reliability diagram which
layout usually differs from the functional structure of an installation. Reliability diagram maps the impact of any
item on the reliability of the entir e system. The layout of reliability diagram of power plant is of a mixed type
with dominance of serial connections of elements.
Analytical methods for the assessment of power unit reliability based on Markov processes, in which reliabil-
ity measures for the entire unit are determined from reliability indices of single components, have one important
drawback—the lack of a sufficiently large populatio n of analyzed events for most components under considera-
Since the first of dozen 370 MW units was commissioned in the Bełchatów Power P lan t in 1982, a systematic
research on power units reliability was initiated by the Institute of Electrical Power Engineering at Lodz Univer-
sity of Technology. The principal target of analyses prepared yearly has been estimatio n of actual reliability
measures of main generating devices of pow e r u n its. Statistical data files of the successive years of the power
plant operation have been systematically complemented and v er ified. The v erificatio n of statistical da ta consist
in elimination of events which are not of random origin (i.e. actively influenced by operation and maintenance
staff) and these, which were not qualified as break-downs only because there was enough ready-reserve power
during failur e.
Using reliability model and computer database worked out for the 370 MW power unit, the most defective
elements of the boiler, turbine, generator and auxiliary systems have been identified [3,4]. For selected elements,
empirical probability density functions of operation times, repa ir times have been determined with use of the
histogram with a set number of observations in each class.
The h istog ram is often used in statistical data analysis to illustrate the major features of the distribution of the
data in a convenient form. It divides up the range of possible valu es in a data set into classes. In a classical his-
togram, each data class spans the interval of the same width. The height of a rectangle drawn above each class is
proportional to the number of observations in the class. The shape of the histogram sometime s is particularly
sensitive to the number of classes. If th e classes are too wid e, important information might get omitted. On the
other hand, if the classes are too narrow, the meaningful information fades due to random variations and small
number of data in each class. The histogr am with a constant width of class interval dispossesses a random sam-
ple of its statistical attributes. The classical histogram is generally used when dealing with large d ata sets (more
then 100 observations). It’s difficult to get such a large data set in a reliability analysis of an engineered system.
The method of a histogram with a set number of observations in each class was applied in a presented study in-
stead a class ical histogra m. On the contrary to a classical histogram, the histogram with a set number of obser-
vations in each class preserves all statistical features of data set. This histogram is prepared according to the fol-
J. Buchta et al.
lowing steps:
sorting numbers in a data set of n observations in an ascending order t1, t2, t3, tn;
calculating number of classes r by the formula r = 2ln(n) and rounding down r to the nearest integ er;
calculating number of observations m in each class by the formula m = n / r and rounding down m to the
nearest integer;
assigning all observations to each class in such a way that equal numbers from a d ata set ar e included in the
same class; therefore the number of observations in the i-th class—ni can vary fro m an assumed value m;
determining the lower and upper bound of each class; for example the upper bound of the 4th class (lower
bound of the 5th class) is an arithmetic mean of the greatest number included in the 4th class and the lowest
number included in the 5th class;
calculating the i-th class interval width -
ti as the difference of its upper and lower bound; determining the
i-th class mid valu e —
as an arithmetic mean of its lower and upper bounds;
calculating th e value of an empirical probability density function by the formula
ft nt
. (1)
The histogram is completed by drawing a bar for each class.
The main advantage of this histogr am is the p ossib ility to apply for the amount of classes r 7 and less nu-
merous data sets. If take into account the minimum required amount of observations in a sing le class (ni 5), the
minimum statistical sample (n 35) is easy to achieve in practice.
Calculation unit was implemented in computer database to identify probabilistic models for empirical distri-
butions of operational times and shut-down times of s tandardiz ed 370 MW unit. Considering times of failures
recorded in a computer database, times to failure and times of failure duration have been calculated. An empiri-
cal probability density function fx(t) is compared w ith th e shape of the density function of different theoretical
distributions (expon en tial, Weib u ll, normal, log-normal [5]), and subjectively best distribution is selected as the
one representing the random variable under investigation. The hypothesis thus constructed is verified by means
of (Pearson’s and Kolmogorov) statistical tests of goodness of fit and only on this basis is the decision made to
accept or reject it. All calculations are performed for the standard significance level
= 0.05.
The Kolmogorov test compares the empirical distribution function with the cumulative distribution function
specified by the null hypothesis and tests goodness of fit with th is d istr ibution . The hypothesis regarding the dis-
tributional form is r ejected if the test s tatisti c,
, is greater than the critical value
obtained from a table (
0.05 =
1.358). Person’s chi-square test is an alternative to the Kolmogorov goodness-of-fit test. For the chi-square
goodness-of-fit computation, the data are divided into r classes and the t est statistic χ2 is calculated. The hypo-
thesis that the data are from a popu latio n with the specified distributio n is rejected if χ2 χ2(
, r-c ). The critical
value χ2(
,r-c) is the chi-squar e p ercent point function with r-c degrees of freedom and a sig n if icance level of
The variable c is the number of estimated parameters for the distribution increased by 1. For example, for a
3-parameter Weibull distribution, c = 4.
A Weibu ll distributio n is very popular statistical model in reliability engineering and failure analysis. Opera-
tion times are modeled in the study by a Weibull distribution with the probability density function defined as
( )exp
bt t
ft aa a
 
=⋅ ⋅−
 
 
where b is the shape parameter and a is the scale parameter.
When b = 1, then the Weibull distribu tion reduces to the exponential distribution. Failure duration times are
modeled in the study by a log-normal distributio n with th e probab ility density fu n ction defined as follows:
( )
( )exp2
ft t
= ⋅−
⋅⋅ 
where m is the mean of the natural logarithms of the failure duration times and σ the standard deviation of the
natural logarithms of the failure duration times.
J. Buchta et al.
Let’s consider the histogram shown in Figure 1(a). The data set consists of n = 1520 observations. The num-
ber of class intervals equals r = 14. According to the table shown in the upper-right corner, each class includes
108 or 109 observations, so th at it’s a histogram with set number of observations. The width of the 1st class in-
terval is 78 hours, the width of the 2nd class interval—70 hours, the 3rd class—108 hours, etc. The width of each
class interval varies from another but the number of observations in each class is nearly the same. The hypothe-
sis of the Weibull distribution has been tested. The shape and scale parameters of the Weibull distribution have
been estimated for a given data set. The statistics χ2 and
have been calculated in Pearson’s and Kolmogorov
tests. The obtained values are less then critical statistics χ
2 and
respectively, so that both tests accept the
Weibull distribution with parameters a = 1391 .2 and b = 0.828.
Probability density functions f(t), cumulative distribution functions F(t) and means E(T) for considered dis-
tributions are presented in Table 1.
3. Histograms of Operation Times and Repair Times for the 370 MW Power Unit,
Its Main Components and Selected Elements of the BB-1150 Steam Boiler
Doz en of lignite fuelled 370 MW rated power units operate currently in Polish power system, located in Bel-
chatow Power Plant, which is the largest power plant of this type in Europe.
The idea of a standardized unit has been introduced into the reliability study of the 370 MW units. All units
are homogenous in several respects such as constructional uniformity and similar conditions in which units op-
erate. The standardized unit is the unit in its useful life with stabilized failure rate that substitutes a dozen of
power plant units. The assumption that a standardized unit is in its useful life requires excluding a period of an
early life from an operation time of respective power units. The concept of a standardized unit allows to obtain
an appropriately numerous population of failures not only for a pow e r unit but also f or its main generating de-
vices (boiler, turbine, generator).
While establishing the population, the cases of incidental failures that occurred in the initial period of the
power plant operation were neglected. It particularly refers to units 1 and 2 during the first three years of the
power plant operation. The cases of failures that occurred in first year of operation were also skipped for re-
maining units. One can state that a higher failure frequency of a 370 MW power units in the initial period of
their operation was caused mainly by a design, construction and assembly defects. All these factors are distinc-
tive for an adaptation of new generation power units. Therefore this period was excluded while population of
failures for a standardized 370 MW unit was selected.
An investigated probability distrib u tion functio n s were identified as Weibu ll d istr ibution s w ith parameter b <
1 (Figure 1). In emergency states, strong dependence of the time of damage liquidation on the cause of its oc-
currence was fo und . Average times of shut-downs, in the case of permanent defects of the installations clearly
differ from the times of shut-downs caused by the incorrect operation of the automatic control systems and
safety devices, the operational staff mistakes and the like. And so, the values of times are as follows: 41.2 h and
1.76 h for the boiler, 59.3 h and 1.27 h for the turbine, 78.9 h and 1.48 h for the generator, 40.1 h and 1.95 h for
Table 1. Probability density functions f(t), cumulative distribution functions F(t) and means E(T) for considered distribu-
Distribution f(t) F(t) E(T)
> 0
exp( )
⋅ −⋅
a > 0
b > 0
bt t
aa a
 
⋅ ⋅−
 
 
1 exp
⋅Γ +
m 0
> 0
( )
1exp 2
0.5 tm
m > 0
> 0
( )
lg exp 2
⋅⋅ 
0.5 tm
( )
exp lg 2 lg
Explanations: Γgamma function, Φ—Laplace’s function.
J. Buchta et al.
Figure 1. Probability density of times between failures identified as Weibull’s distribution: (a) Boiler BB-1150, (b) Turbine
18K360, (c) Generator GTHW-370, (d) Power unit. Explanations: χ2Pearson's statistics, χ
2critical Pearson’s statistics
for the significance level
= 0.05,
Kolmogorov statistics,
critical value of Kolmogorov statistics (
= 0.05) ithe
class number, nthe sample size, rthe number of classes, sumthe total value of all the observations in the sample, nithe
number of observations in the i-th class, fithe value of empirical probability density function for the i-th class.
feed water pump, respectively. Thus, it shou ld be concluded th at the times of long -te rm failu re s and sh or t-ter m
defects belong to two statis tically different populations and that it is advisable to study their distributions sepa-
rately. Generally speaking, the distributions of failure duration times are log-normal distributions (Figure 2).
Empirical distribution functions of operation and failure times for each of dozen power units (Figure 3) have
been also examined.
4. The Assessment of Reliability Indices of 370 MW Power Units and Their
By means of estimation, parameters of identified probability distribution functions of operation times and repair
times have been calculated for power machinery of 370 MW rated power units under study. Such reliability in-
dices as expected failure rate, expected mean time of the shutdown, expected total time of the shutdowns and
expected mean time between failures have been estimated. Table 2 presents expected values of basic reliability
indices for pow e r machinery of the standardized 370 MW rated power unit. Similarly, Ta b le 3 presents basic re-
liability indices for all power units of Belchatow Power Plant.
The statistics shows that failure frequency of the steam boiler BB-1150 is crucial for availability factor
J. Buchta et al.
Figure 2. Probability density of failures times identified as log-normal distribution: (a) Generator GTHW-370 (times of
long-term failures), (b) Generator GTHW-370 (times of short-term failures).
Figure 3. Probability density of times between failures identified as Weibull’s distribution: (a) Unit 1, (b) Unit 12.
Table 2. Reliability indices for main power unit components.
Failure location Expected failure
rate [1/a] Mean time of a
shut-down [h] Total time of
shut-downs [h/a] Mean time between
failures [h]
B 3.90 37.8 147.3 1540
T 0.76 13.0 9.89 7880
G 0.69 25.0 17.3 8690
F 0.30 11.0 3.24 20350
W 0.16 9.3 1.52 36660
O 0.46 9.0 4.1 13160
Unit 5.36 31.0 166.1 1120
Explanations: Bsteam boiler and its auxiliaries, T—steam turbine and its auxiliaries, G—generator and its auxiliaries, F—system of feed water
pumps, Wsystem of cooling water and service water pumps, O—others (including failures of electrical devices).
J. Buchta et al.
Table 3. Reliability indices for dozen power units of Belchatow Power Plant.
Failure location Expected failure
rate, [1/a] Mean time of a
shut-down, [h] Total time of
shut-downs, [h/a] Mean time between
failures, [h]
Unit 1 7.05 30.1 212.4 850
Unit 2 7.27 29.9 217.2 830
Unit 3 4.27 29.7 126.6 1410
Unit 4 4.96 28.8 143.1 1210
Unit 5 5.92 33.5 198.3 1010
Unit 6 5.36 29.5 157.9 1120
Unit 7 6.08 36.0 219.0 990
Unit 8 5.78 29.7 171.4 1040
Unit 9 4.83 32.2 155.3 1240
Unit 10 4.86 25.7 124.8 1230
Unit 11 4.23 29.6 125.3 1420
Unit 12 4.67 35.3 164.5 1290
achieved by Belchatow Power Plant. On e can state that steam boiler marks out the potential to improve power
plant availability factor by reduction of emergency shutdowns with support of optimal s cheduling of power p lant
repairs. Such activities should improve operational durability of the weakest power unit components.
5. Conclusions
The security level of power system is determined by its weakest links represented by large power units which
failure rates are many times higher than other components of the power system. The lack of knowledge on the
reasons and frequency of failures of power machinery makes scheduling plant operation a difficult task as well
as optimal scheduling of po we r plant repairs and enab ling of pow e r supp ly c ontin u ity.
Statistical data on operational disturbances that appeared in power plant lifetime have been carried out syste-
matically for Polish power units rated at 370 MW and fuelled with lignite. Having this database, it’s possible to
analyze the changeability of reliability indices in a long-term perspective and define reasons or effects of defec-
tive operation of power machinery. Economically excused failure rates are reasonable measures for durability of
power machinery and their components in deter mination of modernization schedules and rationalization of
maintenance and materials management in entire pow er p lants under study.
The final decision on power machinery maintenance should be accepted on the ground of observation of the
element intended to repair with use of both diagnostic and prognostic methods. In global sense, power units
availability and reliability indices may be useful to determine operating and replacement reserve in power sys-
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