Subido por Pedro Jimenez

paper ALTAE 171

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1
Generalized Renewal Process
as an adaptive probabilistic model
Pedro R. Jiménez G. 1
Abstract—The model denominated Generalized Renewal
Process (GRP) is patented for the Maryland University. It is
based on a probabilistic function that depends of the following
variables: the operation time from the last event, the previous
times (historical data) and three parameters: shape, scale and
effectiveness. This model is fitted for the random data and is
considered the magnitude and the occurrence order; it is not
possible using the MLE traditional models. This property is
denominated Adaptability. Although the model was created
originally for to estimate the repairable systems reliability (it use
the successive corrective maintenances and it measures its
effectiveness), it is possible to apply it to any probabilistic process
as: eolic, lightning and switching impulse. For performing this, a
MAPLE language codified algorithm was designed. This paper
suggests a methodology for the model numerical solution, using
MLE and GOF test, applied to switching impulse studies and
others random processes.
Index Terms— Adaptive estimation, Probability, Reliability,
Repairable systems, Switching impulse.
Renewal Process” (GRP). Kaminskiy and Krivtsov [2] have
demonstrated that the ORP and the NHPP are specific cases of
GRP; and also denominate the “Weibull generalized
function”.
Ordinary
Renewal
Process (ORP)
Weibull
Normal
Log-normal
100%
as good
as new
worse
than old
as bad
as old
NonHomogeneous
Poisson Process
(NHPP)
Generalized Renewal Process (GRP)
I. NOMENCLATURE
GRP: Generalized Renewal Process.
ORP: Ordinary Renewal Process.
MLE: Maximum Likelihood Estimation.
LSE: Least Square Estimation.
GOF: Goodness-Of-Fit
T
II. INTRODUCTION
he historical data associated to a probabilistic process can
experience anyone of the five possible states (Fig. 1):
1. as good as new;
2. as bad as old;
3. better than old but worse than new;
4. better than new;
5. worse than old.
The MLE traditional probabilistic models used in system
analysis, such as the Ordinary Renewal Process (ORP) and the
NonHomogeneous Poisson Process (NHPP), account for the
first two states, respectively. However, no practical and
accurate approach exists to address the remaining after states.
The main reason as to why the last three states have not
received much attention appears to be the difficulty in
developing a mathematically robust and efficient approach to
represent them.
Recently, Kijima and Sumita [1] have proposed a new
probabilistic model to address all states called “Generalized
1
Petróleos de Venezuela, S.A., e-mail: jimenezps@pdvsa.com.
Possible states
better than
old but
worse
than new
better
than new
Repairable
System
Fig. 1. Possible states of any probabilistic process. Generalized Renewal
Process (GRP) vs Ordinary Renewal Process (ORP).
Krivtsov [3] recognized the complexities and the
difficulties of developing a mathematically tractable
probabilistic model to the GRP, and discussed an alternative
maximum likelihood estimation (MLE) approach to solve the
GRP model without offering any solution. In this paper a
comprehensive MLE solution to estimate the GRP parameters
has been developed.
In this study is demonstrated that the model GRP has the
property of "Adaptability": the parameters estimation depends
so much on the magnitude as on the occurrence order of the
historical data.
The GRP parameter estimation based on the ML approach
leads on a non-linear system of three equations that should be
solved simultaneously. For performing that, an algorithm was
developed in MAPLE language.
The results obtained from this study open the door for using
the GRP approach in many industries. The application of the
model GRP permits the estimate of the reliability and
maintainability of repairable systems, eolic models,
atmospheric phenomena and any probabilistic process.
This paper presents three practical applications: 1) PDVSA
115kV transmission system availability; 2) 400kV Switching
impulse; and 3) Eolic potential.
2
III. THE MODEL
A. Description
(6)
The Cumulative Density Function (CDF) or unreliability
function of the GRP model is as in (1):
β
⎡
i −1
⎞ ⎤
⎛
⎢
tj ⎟ ⎥
β ⎜ t i +q
⎢ ⎛ q i −1 ⎞ ⎜
⎟ ⎥
j=1
⎢⎜
⎟ ⎥
t j ⎟ −⎜
η
⎢ ⎜ η j=1 ⎟ ⎜
⎟ ⎥
⎠ ⎜
⎢⎝
⎟⎟ ⎥
⎜
⎢
⎠ ⎥⎦
⎝
⎣
∑
F(t i , β, q, η) = 1 − e
∑
,t≥0
(7)
(1)
The parameters are: η = scale , β = shape and
q = effectiveness.
The “β” parameter have the cases: β < 1 (early life), β = 1
(userful life) and β > 1 (wearout life).
The “q” parameter have the cases: q = 0 (as good as new),
q = 1 (as bad as old), 0 < q < 1 (better than old but worse than
new), q < 0 (better than new) and q > 1 (worse than old).
(8)
The Probability Density Function (PDF) is as in (2):
(9)
β
⎡
i −1
⎛
⎞ ⎤
⎢
tj ⎟ ⎥
β ⎜ t +q
⎢ ⎛ q i −1 ⎞ ⎜ i
⎟ ⎥
j
1
=
⎢⎜
⎟ ⎥
t j ⎟ −⎜
⎢ ⎜ η j=1 ⎟ ⎜
η
⎟ ⎥
⎠ ⎜
β−1 ⎢ ⎝
⎟⎟ ⎥
⎜
⎢
⎝
⎠ ⎥⎦
⎣
∑
∑
i −1
∂F(t i ) ⎛ β ⎞⎛
⎞
= ⎜⎜ β ⎟⎟⎜ t i + q ∑ t j ⎟ e
f(t i , β, q, η) =
(2)
j=1 ⎠
∂t i
⎝ η ⎠⎝
The hazard function (failure rate fuction) is as in (3):
β −1
i −1
⎛ β ⎞⎛
f (t i )
⎞
λ(t i , β, q, η) =
= ⎜ ⎟⎜ t i + q ∑ t j ⎟ (3)
j=1
1 − F( t 1 ) ⎜⎝ ηβ ⎟⎠⎝
⎠
B. Algorithm
The MLE Weibull model F( t ) = 1 − e
solving the equations system as in (4)
∂ ln(L)
∂ ln(L)
(β, q) = 0,
(β, q) = 0, η = η(β, q) (11)
∂β
∂q
β
is obtained
1
⎛ n β ⎞β
x βi ln(x i ) ∑ ln(x i )
⎜ ∑ xi ⎟
∑
1
i =1
i =1
⎜ i =1 ⎟ (4)
0
,
−
−
=
η
=
n
⎜ n ⎟
n
β
β
xi
⎜
⎟
∑
i =1
⎝
⎠
n
n
The GRP MLE model is obtained solving the equations
system as in (7), (8) and (9) equaling to zero. Applying MLE:
n
L = f(t1 , β, q, η)∏ f(t i , β, q, η)
(10)
Substituting (10) en (7) y (8), is obtained a nonlinear
equation system, dependent of β y q, as in (11).
The algorithm is divided in three phases:
a) Weibull MLE model.
b) GRP MLE model.
c) LSE Weibull model is fitted to MLE GRP model.
⎛t⎞
− ⎜⎜ ⎟⎟
⎝ η⎠
From (9), equal to zero, is solved as in (10):
(5)
i=2
Applying natural logarithm (Ln) and expanding, it is obtained
(6).
The numeric calculus of (11) imply the solution from two
planes, due the asymptotic form for β = 1, as:
Solution #1. Plane 0 < β < 1 y - ∞ < q < ∞.
Recommended Initial Conditions: q = 0.0001 y β = 0.5.
b) Solution #2. Plane 1 < β < ∞ y - ∞ < q < ∞.
Recommended Initial Conditions: q = 0.2 y β = 3.
Note: the recommended initial conditions are the result of
rehearsals, although in general it is subjected to verification.
a)
To know the best solution is necessary apply the
Kolmogorov-Smirnov (KS) test (or any other GOF test):
(
KS = maximum F( t i ) − ni , F( t i ) −
( i −1)
n
), i = 1..n
The smallest KS test value is the best solution. In case that the
MLE GRP model doesn't have solution, the Weibull MLE
model is assumed as solution.
3
For simplify calculus and interpretation, is accepted to fit
the MLE GRP model to the Weibull model F(t) = 1 − e
applying LSE, according to the following steps:
1.
2.
3.
β
The dependent values are the times and the independent
values are the GRP function evaluated for each time.
With the data of the step 1, the LSE is applied for solving
the parameters β and η. To be initialized it assumes a
value of the parameter γ.
The step 2 is repeated, varying the parameter γ, until to
maximize the R2 statistical test:
⎛ n
⎞
R = 1 − ⎜ ∑ ( y i − ŷ i ) 2 ⎟
⎝ i =1
⎠
2
4.
⎛ t −γ ⎞
−⎜⎜
⎟⎟
⎝ η ⎠
n
⎛ n
⎞
⎜ ∑ y i − ∑ y j n ⎟ (12)
⎜
⎟
i
1
j
1
=
=
⎝
⎠
Conservative criterions. If the data have reliability
philosophy (is less probable long times) then the
parameter γ should be ≤ 0 (if LSE is γ > 0, should be recalculated for γ=0). For maintainability philosophy (is
more probable long times) the parameter γ should be ≥ 0.
The MLE traditional models are:
For compare the MLE traditional models with the MLE GRP
model, is apply any GOF test as Kolmogorov - Smirnov (KS).
This test value should be smallest that the critical values for
1% statistical significance.
The GRP model prospective time “P”, is obtained as in
(18) and (19).
Be (18) the CDF for the next time (tn+1).
∑
F(t n +1 ) = 1 − e
∞
δ
−∞
−δ
P = ∫ f(t)dt = ∫
⎛t⎞
∂ F(t n +1 )
dt n +1
∂ t n +1
(19)
∫
δ
−δ
t 2n +1
∂ F(t n +1 )
dt n +1 − µ 2
∂ t n +1
(20)
(13)
Generally, it is expressed in percentage of the prospective
time.
Note: δ substitute at infinite for numeric calculation. Range:
50η to 1000η.
(14)
The algorithm developed is codified in the MAPLE
language. The nonlinear equations system is resolved using
the “Newton” library.
Normal:
1 ⎛ t −µ ⎞
⎟
σ ⎠
(18)
The deviation “D” is as in (20).
β
− ⎜
1
f (t) =
e 2⎝
2π ⋅ σ
,t ≥ 0
Applied to reliability study is called mean life or “Mean Time
To failure” (MTTF) and applied to maintainability is called
“Service Outage Mean Time” (SOMT).
D=
∂F( t ) ⎛ β ⎞ β−1 −⎜⎜⎝ η ⎟⎟⎠
,t ≥ 0
= ⎜⎜ β ⎟⎟(t ) e
∂t
⎝η ⎠
∑
Then, the prospective time “P” (or average time) is as in (19).
Weibull:
f (t) =
β
⎡
n
⎞ ⎤
⎛
⎢
+q t j ⎟ ⎥
β ⎜t
⎢ ⎛ q n ⎞ ⎜ n +1
⎥
⎟
j=1
⎢⎜
⎟ ⎥
t j ⎟ −⎜
η
⎢ ⎜ η j=1 ⎟ ⎜
⎥
⎟
⎠ ⎜
⎢⎝
⎟⎟ ⎥
⎜
⎢
⎠ ⎥⎦
⎝
⎣
2
,−∞ < t < ∞
C. Why is said that the GRP model is adaptive?
Be the data according to the occurrence order:
LogNormal:
f (t) =
1 ⎛ ln t −µ ⎞
⎟
σ ⎠
− ⎜
1
e 2⎝
2π ⋅ t ⋅ σ
2
,t ≥ 0
(15)
and the ascending order data:
Then, the results are shown in the table I.
TABLE I
ADAPTABILITY PROPERTY
Extreme Value (EV):
⎛ t −µ ⎞
⎟
σ ⎠
1 −⎜
f (t) = e ⎝
σ
2
e
e
2
⎛ t −µ ⎞
−⎜
⎟
⎝ σ ⎠
−∞ < t < ∞
(16)
This Extreme Value model is obtained solving the MLE
equations system:
(17)
η
β
q
P
D
occurrence order
794
0.697
7.91
9231
100%
Ascending order
95.7
0.48
3.26
13996
102%
Conclusion: the GRP model parameters estimation depends
on the magnitude and the occurrence order of the data. This
property is denominated Adaptability.
4
D. Availability study of the PDVSA 115kV electrical system.
Hours
This part is a repairable system application whose objective
is to estimate the reliability (probability that a system doesn't
fail for a specific time, only associated to non-programmed
events) and maintainability (service restoration probability for
a specific time, associated to non-programmed and
programmed events) of the PDVSA 115kV electric system,
with the purpose of to estimate the system availability
(probability that a system is in service, associated to nonprogrammed or programmed events, whose value represents
the mean for a long time) and to compare with the
international standard.
Maintainability (SOT)
16
14
12
10
8
6
4
2
0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45
occurrence order
Fig. 3. SOT historical Data. Source: PDVSA statistics.
Conclusion: The 115kV system reliability and
maintainability study report the mean values of
MTTF = 4477hours and SOMT = 5.12hours, respectively. The
PDVSA 115kV electrical system availability (A) is calculated
as in (21).
MTTF
4477
100 =
100 = 99.885% (21)
MTTF + SOMT
4477 + 5.12
Important: the MLE traditional models are not applicable
for repairable systems availability studies.
A=
The data to estimate the reliability model are the times to
failure (TTF), which is defined as the lapsed operative time,
since a failure is restored until it is failure again.
E. 400kV Switching impulse study.
The data to estimate the maintainability model are the
service outage times (SOT), which is defined as the lapsed
time since a failure until it is restored.
Hours
Reliability (TTF)
30000
25000
20000
15000
GRP parameters
η = 2533 hours
β = 0.6368
q = 0.002218
P = 4477 hours
D = 143%
GRP parameters
q = 0.000311
β = 1.51
η = 5.75 hours
P = 5.12 hours
D = 68%
This part is a switching impulse application for 400kV
CIGMA-Furrial transmission line with length=310Km using
pre-insertion resistor and surge arrester at each terminal.
This study permit to obtain the line optimal CFO (Critical
Flash Over) to satisfy the Switching Forced Outage Rate
(SSFOR).
This study consists in the energization at the CIGMA terminal.
For the study, the basic equations are as in (22) to (25):
10000
5000
0
a
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56
occurrence order
E2 = a
Fig. 2. TTF historical Data. Source: PDVSA statistics.
is the undervoltage value more probable.
σ
σf ⎞ ,
⎛
CFOn = CFO⎜1 + z f
⎟ 0.03 < f < 0.07, typical = 0.05
CFO
CFO ⎠
⎝
is the equivalent CFO for n-tower.
(23)
TABLE II
GRP MODEL SOLUTIONS COMPARISON
Solution #1
2533hours
0.6368
0.002218
0.1488
Solution #2
836hours
0.6308
0.195705
0.3018
z f = inverse Normal = N −1 (1 − n _ tower 0.5 ), n_tower =
Comment: the GRP model solution that best fit is the 1 ,
due KS1 < KS2.
Simplifying to LSE Weibull:
E +D
2
1
SSFOR =
f ( t )dt ≤ 1 / 1000
2 CFOn=CFO∫(1+0.05zf )
(25)
Note: the MAPLE program has codified these equations.
TABLE III
RELIABILITY LSE WEIBULL MODEL
β
0.811
line length
0.4
(24)
The CFO estimation is obtained numerically solving (25):
st
η
3797 hours
(22)
0
Using the MAPLE codefied program:
η
β
q
KS test
∫ f (t )dt = 0.98, t ≥ 0
γ
0
Comment: young system, due that β = 0.811 < 1
Objective #1: to determine the CFOmin and CFOmax of each
function to satisfy the SSFOR criterion < 1/1000.
The MAPLE program include the estimation of:
CFOmin: is obtained when the 1st function, whose
KS test < critical value, reach SSFOR < 1/1000.
CFOmax: is obtained when all functions, whose
KS test < critical value, reach SSFOR < 1/1000.
5
TABLE VI
SWITCHING IMPULSE STUDY SUMMARY
Objective #2: to compare the MAPLE results between the
MLE traditional models and the GRP model.
GRP
Random Data:
The following random data (n = 149) are the phase-neutral
peak under-voltage per-unit values and they are obtained
through simulations applying a PSCAD switching impulse
model measured at 75%line.
Weibull
Normal
LogNormal
EV
E2
1.695
1.702
1.772
1.80588
P
1.489
1.496
1.492
1.492
1.503
D
0.1252
0.1258
0.1363
0.1424
0.1871
0.07386
0.0739
0.07694
0.07885
0.09409
1.82
1.828
1.908
1.948
2.175
0.1216
0.09736
D / E2
E2 + D
KS test
KS < CV
Yes
0.1303
Yes
Yes
1.988
0.1482
0.1728
No
No
Note: the KS critical value (CV) is 0.1335 for n = 149 and 1%
statistical significance.
TABLE VII
CFOMIN ESTIMATION
Fig. 4. MAPLE program input data (random values).
Note: the MAPLE language codified algorithm filter the
repeated data.
GRP
CFOmin
TABLE IV
GRP MODEL SOLUTIONS COMPARISON
η
β
q
Solution #1
1.557166
14.64
0.0000633
Solution #2
Not
Exist
2.0752
775
775
775
775
775
1.751
1.751
1.751
1.751
1.751
0.001358
0.013676
0.019799
0.046042
In the table VII is show the CFOmin estimation.
Comment: it demonstrate that the GRP model permit obtain
the optimum CFO value. It apply for the majority of the cases.
TABLE VIII
CFOMAX ESTIMATION
Model
Extreme Value
Normal
LogNormal
Weibull
µ
1.419
1.492
0.395
-
σ
0.146
0.136
0.095
-
η
1.550
GRP
CFOmax
MLE TRADITIONAL MODELS
Weibull
2.2152
n tower
β
14.56
EV
2.0752
0.000939
Comment: only exist one solution for the GRP model.
TABLE V
LogNormal
2.0752
CFOn
SSFOR
Normal
2.0752
n tower
To next be showing the MAPLE codified program reports.
Weibull
2.0752
CFOn
Normal
2.2152
LogNormal
2.2152
EV
2.2152
2.2152
775
775
775
775
775
1.869
1.869
1.869
1.869
1.869
0
0
0.000835
0.002833
0.019557
SSFOR
In the table VIII is show the CFOmax estimation. It is a
conservative value for design (107%CFOmin ).
F. Eolic potential study of the Venezuela Oriental region.
GRP parameters
η = 7.89 m/s
β = 6.56
q = 0.0046
P = 6.20 m/s
D = 21%
Eolic measurement. Venezuela Oriental Region
m/s
12
10
8
6
4
2
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
occurrence order
Fig. 6. Eolic measurement.Venezuela Oriental region.
TABLE IX
LSE WEIBULL MODEL
η
7.936 m/s
green: Extreme Value yellow: LogNormal
red: GRP blue: Weibull gold: Normal
Fig. 5. Probabilistic Density Function and histogram. From MAPLE program
β
6.6
γ
-1.2
Note: for this case, the β parameter not has interpretation. The
GRP model is only applied for to fit the data.
6
Comment: the eolic potential of the Venezuela Oriental region
is bigger than 5 m/s, therefore it is feasible the electric
generation.
IV. CONCLUSIONS
a)
b)
c)
d)
e)
f)
g)
h)
PGR is a mathematical model that has the property of
"Adaptability”: the parameters depend of the magnitude
and the occurrence order of the historical data.
It is not recommended to use the censored or suspended
data for the PGR model, due the nonexistence of a
defined occurrence order.
The GRP model, applied to a probabilistic switching
impulse study, permits the optimum CFO value
estimation.
Applying the GRP model for the PDVSA 115kV
electrical system availability study is obtain 99.885% ≈
99,9%, whose value is satisfactory according to
recommended practice.
The result β < 1 imply that the 115kV PDVSA electric
system is in infantile mortality period.
The “q” predominant value of the PDVSA 115kV system
is approximately “as good as new” (q ≈ 0).
For many data (usually for maintainability data) is
accepted the MLE Weibull model.
The PGR model opens a world of opportunities for the
investigation of any probabilistic model as: maintenance,
eolic, lightning and switching impulse.
V. RECOMMENDATIONS
a)
It is invited to apply this PGR to more complex systems
study as: refineries, compressors, large bombs, ovens,
cryogenic and any repairable industrial process.
b) The PGR model opens the doors to continue investigating
its application with censored data.
c) To verify with certainty the null hypothesis (that the
model is adjusted to the data) for the function PGR, is
necessary a complementary investigation for the
development of an adapted goodness-of-fit test.
VI. REFERENCES
[1]
[2]
[3]
[4]
[5]
Kijima M. y Sumita N. (1986). A useful generalization of renewal
theory: counting process governed by non-negative Markovian
increments. J Appl Prob 1986; 23:71–88.
Kaminskiy M. y Krivtsov V. (1998). A Monte Carlo approach to
repairable system relaibility analysis. Probabilistic safety assessment and
management, New York: Springer; p. 1063–8.
Krivtsov V. (2000). A Monte Carlo approach to modeling and
estimation of the generalized renewal process in repairable system
reliability analysis. Dissertation for The Degree of Doctor of Philosophy,
University of Maryland.
Xie, Yang y Gaudoin (2001). National University of Singapore,
Singapore. Institut National Polytechnique de Grenoble, France.
Regression Goodness-Of-Fit Test for Software Reliability Model
Validation.
Joglar F., Modarres M. y Yanez M. (2002). Reliability Engineering &
System Safety. Generalized renewal process (GRP) for analysis of
repairable systems with limited failure experience. Approved for
University of Maryland at April 29, 2002.
VII. BIOGRAPHIES
Pedro Jiménez was born in Puerto La Cruz city,
Venezuela country, on February 1, 1973. He
graduated electrical engineer with honors from the
UDO University on 1996 and Maintenance
Management MSc on 2006.
He has 11 years of experiences with Petróleos de
Venezuela, S.A., working electrical planning,
engineering and electrical simulations. He is
professor of the UDO University since 1996. He has
published in some national and international congresses related to power
systems analysis.
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