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Evaluación de la
confiabilidad del sistema de generación eléctrica utilizando elementos
algebraicos
Deysi Margoth Guanga <=
span
class=3DSpellE>Chunata , Chunata .
This article
presents the application of a Bayesian model that consists in the
implementation of an infinite state Markov chain for the evaluation of the
reliability of the electric generation system of three generators. The anal=
ysis
will allow obtaining a model of the current and future operating behavior of
three generator sets. The data to analyze are the operative registers elabo=
rated
by the operators of the generators, who in their logbook write down the
operation and failure times. The analysis sample comprises five (05) years,=
in
twelve-hour intervals. Therefore, with the development of the present model=
it
will be possible to predict parameters and indicators such as: availability,
maintainability, reliability, failure rate, repair rates, average time for =
repair
and average time between failures; with the purpose of achieving a better
management of the assets within the maintenance management, having as a cas=
e of
application that of the electric generation system.
Keywords: Reliability, Active Indicator, Markov, Failure, Vector.
Resumen.
Este artículo
presenta la aplicación de un modelo Bayesiano que consiste en la implementa=
ción
de cadena de Markov de estado finito para la evaluación de la
confiabilidad del sistema de generación eléctrica de tres grupos electrógen=
os. El
análisis permitirá obtener un modelo del comportamiento de operación actual=
y
futuro de tres grupos electrógenos. Los datos a analizar son los registros
operativos elaborados por los operadores de los grupos electrógenos, quiene=
s en
su bitácora anotan los tiempos de operación y falla. La muestra de análisis
comprende cinco (05) años, en intervalos de doce horas. Por lo tanto, con el
desarrollo de la presente modelación se logrará predecir parámetros e
indicadores relacionados a la confiabilidad, tasa de fallas, tasas de repar=
ación,
tiempo medio para reparar y tiempo medio entre fallas; con la finalidad de
conseguir una mejor gestión de los activos dentro de la gestión de
mantenimiento, teniendo como caso de aplicación el del sistema de generación
eléctrica.
Palabras claves: Confiabilidad, Indicador Activo, Markov ,
Falla, Vector.
Introducción.
Los conceptos estadísticos, necesarios para la aplicación del anális=
is
de confiabilidad son herramientas de análisis que comprenden distribuciones,
modelos e indicadores necesarios para interpretar el comportamiento de un
activo. La suposición de que la detección de fallos de un grupo electrógeno=
se
puede modelar a través de una cadena de Markov
aplicado al diseño de varios tipos de simuladores de baja complejidad, que
luego deben procesarse para determinar si se comete un error de decisión cu=
yo
resultado es la inoperatividad de los equipos.
Por más de medio siglo, los
investigadores han tratado de adaptarse al tiempo discreto de medición de f=
allos
del sistema basado en modelos que van desde los primeros canales
Gilbert-Elliott de dos estados a más complejos modelos como los basados
en cadenas ocultas de Markov (Garcia M=
arkov
del Estado Finito (FSMC) posee buen equilibrio entre la precisión y la
complejidad (Wang, H. 1997).. El modelamiento se basa en la estructura de un
número finito de estados logrando que el uso de un proceso de Markov de primer orden sea una buena aproximación usa=
ndo la
teoría de la información (Tan, C. 2000)..
Los problemas de Markov vistos como una =
matriz
asociada al comportamiento lineal de los grupos electrógenos conducen a
problemas de valores propios si se desea conocer el estado límite del proce=
so
en el cual el vector del estado X se reproduce bajo la multiplicación por la
matriz estocástica A que rige el proceso.
Metodología.
Para el desarrollo del modelamiento fue necesaria la recopilación de
fallos en cada parte del sistema generador discretizados en el tiempo con un
registro histórico de cinco años. El análisis de los datos se lo realizó
mediante el programa estadístico SPSS. La asociación de variables para la
construcción de la matriz estocástica se la desarrolló empleando las medias=
de
los fallos contemplados por cada generador. A continuación, se detalla los
principales pasos seguidos para obtener los resultados:
Deducción del modelo matemático
Se seleccionó dos (03) grupos electrógenos para la implementación de
indicadores y registro de fallas, a quienes se los conocerá como MG1, MG2 y=
MG3
en adelante, como se muestra en la Tabla 1. La secuencia de fallas y de
reparaciones de un componente reparable define un proceso aleatorio que se
puede representar como una cadena de Markov de =
dos
estados discretos y tiempo continuo cuya representación se muestra a
continuación:
Figura 1 <=
/b>. Modelo reparable de=
dos
estados
Fuente: Elaboración propia
Las tasas de transición entre estados se definen: <=
/p>
Para un sistema de “n” componentes, cada uno con dos estados operati=
vos,
el número total de estados operativos del sistema viene dado por n
Si se considerase un intervalo de tiempo 𝑑𝑡 , muy pequeño, de tal manera que la probabilidad de que ocurran más =
de
una falla o más de una reparación sea muy pequeña y por lo tanto pueda
despreciarse la ocurrencia de estos eventos (Wang, H. 1997).. Así:
Probabilidad de una falla en t =3D Probabilidad de una falla en (𝑡 + 𝑑𝑡 ) =3D 𝜆 ( 𝑡 𝑑𝑡 .
Probabilidad de una reparación en t =3D Probabilidad de una reparaci=
ón en
( 𝑡 + 𝑑𝑡 ) =3D 𝜇 ( 𝑡 𝑑𝑡 .
La probabilidad de estar en el estado operativo después de un interv=
alo
de tiempo 𝑑𝑡 es igual a la
probabilidad de estar operativo en t y no haber fallado en 𝑑𝑡 + probabilidad de estar fallado en t y haber sido reparado en 𝑑𝑡 :
𝑃 1( 𝑡 𝑑𝑡 ) =3D 𝑃 𝑡 )[1 − 𝜆 𝑡 ) 𝑑𝑡 ] + 𝑃 2( 𝑡 𝜇 ( 𝑡 𝑑𝑡 ] (2.1)
La probabilidad de estar en=
el
estado de reparación (fallado) después de un intervalo de tiempo 𝑑𝑡 es igual a la probabilidad de estar fallado en t y no haber sido
reparado en 𝑑𝑡 + probabilidad =
de
estar no fallado en t y haber fallado en 𝑑𝑡 :
𝑃 2( 𝑡 𝑑𝑡 ) =3D 𝑃 𝑡 )[1 − 𝜇 𝑡 ) 𝑑𝑡 ] + 𝑃 1( 𝑡 𝜆 ( 𝑡 𝑑𝑡 ] (2.2)
Así reorganizando términos en ambas ecuaciones:
De (2.1):
𝑃 1( 𝑡 𝑑𝑡 ) =3D 𝑃 𝑡 )[1 − 𝜆 𝑡 ) 𝑑𝑡 ] + 𝑃 2( 𝑡 𝜇 ( 𝑡 𝑑𝑡 ]
𝑃 1( 𝑡 𝑑𝑡 ) =3D 𝑃 𝑡 ) − 𝜆 𝑡 ) 𝑑𝑡𝑃 1( 𝑡 ) + 𝜇 𝑡 ) 𝑑𝑡𝑃 2( 𝑡 )
𝑃 1( 𝑡 𝑑𝑡 ) − 𝑃 𝑡 ) =3D [− 𝜆 𝑡 ) 𝑃 𝑡 ) + 𝜇 𝑡 ) 𝑃 𝑡 )] 𝑑𝑡
𝑃 1( 𝑡 𝑑𝑡 ) − 𝑃 𝑡 )/ 𝑑𝑡 =3D − 𝜆 ( 𝑡 𝑃 1( 𝑡 𝜇 ( 𝑡 𝑃 2( 𝑡
𝜆 ( 𝑡 𝑃 1( 𝑡 𝜇 ( 𝑡 𝑃 2( 𝑡 Se obtiene:
De (2.2):
𝑃 2( 𝑡 𝑑𝑡 ) =3D 𝑃 𝑡 )[1 − 𝜇 𝑡 ) 𝑑𝑡 ] + 𝑃 1( 𝑡 𝜆 ( 𝑡 𝑑𝑡 ]
𝑃 2( 𝑡 𝑑𝑡 ) =3D 𝑃 𝑡 ) − 𝜇 𝑡 ) 𝑑𝑡𝑃 2( 𝑡 ) + 𝜆 𝑡 ) 𝑑𝑡𝑃 1( 𝑡 )
𝑃 2( 𝑡 𝑑𝑡 ) − 𝑃 𝑡 ) =3D [ 𝜆 𝑡 ) 𝑃 𝑡 ) − 𝜇 𝑡 ) 𝑃 𝑡 )] 𝑑𝑡
dt <=
/span>
Lo cual resulta en el siguiente sistema de ecuaciones diferenciales
ordinarias:
=3D()1() - ()2()
El cual se puede expresar en alguna de las siguientes formas:
=
p t p t =3D<=
m:sty
m:val=3D"p"/> =
=
=
=
=
=
P t H T =
P t
Donde H es la matriz estocástica de tasas de transición entre estado=
s.
La solución de la cadena de Markov son
las probabilidades de ocurrencia d=
e los
estados 1 y 2 como una función del tiempo (Wang, H. 1997).:
𝑃 1( 𝑡 ) =3D 𝐴 ( 𝑡 )
Es la disponibilidad
instantánea del componente.
Es la probabilidad de
encontrar el componente en el estado "bueno" en algún instante =
del
tiempo
𝑃 2( 𝑡 ) =3D 𝑈 ( 𝑡 )
Es la indisponibilidad
instantánea del componente
Es la probabilidad de
encontrar el componente en el estado "fallado" en algún instante
del tiempo
Si los procesos de fallas y de reparaciones del componente reparable=
son
estacionarios, independientes y ajustan a distribuciones de probabilidad, se
podrá entonces utilizar como
modelo de confiabilidad una cadena de Markov
homogénea (García, P. 1997)..
Como la tasa de falla y de reparación son constantes, éstas se pueden
escribir como:
𝜆 ( 𝑡 ) =3D 𝜆 =3D 1/ 𝐸 ( 𝑇𝑓 <=
span
lang=3DES-EC style=3D'font-size:12.0pt;line-height:115%;font-family:"Time=
s New Roman",serif;
mso-fareast-font-family:"MS Mincho";letter-spacing:-.05pt;mso-ansi-langua=
ge:
ES-EC'>)
𝜇 ( 𝑡 ) =3D 𝜇 =3D 1⁄ 𝐸 ( 𝑇𝑟 )
La cadena de Markov homogénea se define
entonces por:
𝐹𝑡𝑓=
; ( 𝑡 )
La función de
También se puede utilizar la 𝑓&#=
119905;𝑓 ( 𝑡 ).
𝐹𝑡𝑟=
; ( 𝑡 )
La función de
También se puede utilizar la 𝑓&#=
119905;𝑟 ( 𝑡 ).
SOLUCIÓN ESTADÍSTICA DE LA CADENA DE MARKOV
Si se tiene evidencia de que las muestras de tiempos para fallas y de
tiempos para reparación son estacionarias, también es posible calcular en f=
orma
estadística las disponibilidades e indisponibilidades de estado estable y l=
as
tasas de eventos del modelo de confiabilidad del componente reparable de dos
estados, de la siguiente forma (Wang, H. 1997).:
P A t r T
λ n f T <=
span
style=3D'font-size:12.0pt;mso-ansi-font-size:12.0pt;mso-bidi-font-size=
:12.0pt;
font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math";
mso-fareast-font-family:"MS Mincho";mso-hansi-font-family:"Cambria Mat=
h";
mso-bidi-font-family:"Times New Roman";letter-spacing:-.05pt;mso-ansi-=
language:
ES-EC'> t r n f =
t r
μ n r T <=
span
style=3D'font-size:12.0pt;mso-ansi-font-size:12.0pt;mso-bidi-font-size=
:12.0pt;
font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math";
mso-fareast-font-family:"MS Mincho";mso-hansi-font-family:"Cambria Mat=
h";
mso-bidi-font-family:"Times New Roman";letter-spacing:-.05pt;mso-ansi-=
language:
ES-EC'> t f r <=
m:naryPr>t f
RESULTADOS
Es preciso saber que la calidad de estos cálculos depende del tamaño=
de
la muestra. Con los datos analizados de las tasas de falla (λ) y
reparación (μ) de los grupos electrógenos, se obtiene la matriz de
transición y estados, según Markov :
Tabla 1 Datos iniciales de opera=
ción
y falla
UNIDAD
CAPACIDAD
FOR
λ
μ
GEN 1
850
0.58
0.00057
0.19
GEN 2
1175
9.05
0.00251
0.0892
GEN 3
1250
14.47
0.0024
0.0908
Fuente: Elaboración propia
FOR: Tasa de salidas forzadas (Forced Outage Rate )
El sistema tiene 2³ =3D 8 estados descritos así
(A: disponible; U: indisponible)
El modelo de Markov da probabilidades de=
los
estados posibles en un sistema en función del tiempo.
Tabla 2 Estados de disponibilida=
d e
indisponibilidad de los grupos electrógenos
Estado
GEN1
GEN2
GEN3
KW
KW
1
U
U
U
0
3275
2
A
U
U
850
2425
3
U
A
U
1175
2100
4
U
U
A
1250
2025
5
A
A
U
2025
1250
6
A
U
A
2100
1175
7
U
A
A
2425
850
8
A
A
A
3275
0
Fuente: Elaboración propia
Las tasas de transición entre estados (ℎ 𝑖𝑗 ) forman la matriz h:
Cada celda de la matriz contiene la probabilidad de transición Pij del estado i al estado j.
-0.37
0.19
0.0892
0.0908
0
0
0
0
0.00057
-0.18057
0
0
0.0892
0.0908
0
0
0.00251
0
-0.28331
0
0.19
0
0.0908
0
0.0024
0
0
-0.2816
0
0.19
0.0892
0
0
0.00251
0.00057
0
-0.094
0
0
0.0908
0
0.0024
0
0.00057
0
-0.092
0
0.0892
0
0
0.0024
0.00251
0
0
-0.195
0.19
0
0
0
0
0.0024
0.0025
0.0006
-0.00548
En estos sistemas, componentes o elementos poseen en general salidas=
que
afectan su disponibilidad (Wang, H. 1997).. Éstas pueden ser:
Tabla 3 Tipos de salidas en un sistema reparable <=
span
lang=3DES-EC style=3D'font-size:12.0pt;line-height:115%;font-family:"Times =
New Roman",serif;
mso-fareast-font-family:"MS Mincho";letter-spacing:-.05pt;mso-ansi-language:
ES-EC'>
Tipo de salida
Descripción
Ejemplo
No planeadas
Son de naturaleza
Fallas propias de
Planeadas
Se conoce de antemano
Mantenimiento
=
Fuente: Elaboración propia
Las salidas planeadas pueden tratarse de la siguiente manera:
1. Determinar la programación de salidas planeadas del componente: en
este caso al estudiar o generar la secuencia operativa del componente se de=
be
mirar en qué períodos de tiempo estará indisponible según el programa de
mantenimiento preventivo.
2. Incorporarlas al modelo probabilístico: implica asumir que se con=
oce
que el componente tendrá salidas planeadas, pero no se conoce su programaci=
ón.
Válido para estudios de largo plazo o para componentes donde el mantenimien=
to
preventivo no obedece a un programa riguroso.
Otro asunto, es que las salidas no planeadas y las salidas planeadas
deben incorporarse en los índices de confiabilidad pues para algunos
componentes del sistema eléctrico éstas salidas pueden tener una participac=
ión
muy alta
CONCLUSIONES
Con base en los registros opera=
tivos
de un sistema se puede considerar las siguientes conclusiones:
· =
Se
pueden aproximar los índices de confiabilidad que son estadísticas
descriptivas, por ejemplo, el tiempo promedio para falla, las horas de
indisponibilidad por año, etc.
· =
Con los índices de confiabilidad se pueden hacer l=
os
siguientes análisis:
ü
Evaluar el desempeño del componente con respecto a valores de refere=
ncia
u objetivo.
ü
Plantear medidas correctivas para mejorar la confiabilidad del
componente.
=
·&nb=
sp;
Con los modelos probabilísticos se pueden hacer los siguientes análi=
sis:
ü
Simular la operación pasada y presente del componente.
ü
Predecir el comportamiento futuro del componente. <=
/p>
· =
Para
una mejor cognición se deben tratar los vectores como un elemento concreto y
representable visualmente, como en el plano usando flechas desde el origen,
considerando que a medida que el estudiante aprende más sobre álgebra linea=
l,
es conveniente tener en cuenta que estos conocimientos son aplicables a un
nivel mucho más general y que éste es el motivo por el cual los textos estu=
dian
el tema de un modo más abstracto
=
·&nb=
sp;
El presente trabajo constituye una
herramienta para el aprendizaje cognitivo de los espacios vectoriales
abstractos. El resultado de la estrategia didáctica cognitiva permite a los
alumnos su construcción al menos como proceso de aprendizaje dejando de lado la memorización.=
Referencias Bibliográficas .
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<=
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PARA CITAR EL ARTÍC=
ULO
INDEXADO.
Guanga Chunata ,
D., Sarmiento Torres, L., & Inca Chunata , M=
. de
J. (2019). Evaluación de la confiabilidad del sistema de generación eléctri=
ca
utilizando elementos algebraicos. Ciencia Digital , 3 (3.2.1),
275-286. h=
ttps://doi.org/10.33262/cienciadigital.v3i3.2.1.818
El
artículo que se publica es de exclusiva responsabilidad de los autores y no
necesariamente reflejan el pensamiento de la Revista Ciencia Digital.
El artículo queda en propiedad de la revista y, por
tanto, su publicación parcial y/o total en otro medio tiene que ser autoriz=
ado
por el director de la Revista Cien=
cia
Digital.
<=
o:p>
Petroamazonas EP, Sucumbíos, Ecuador.
luis_sarmiento@petroamazonas.ec
Secretaría Nacional de Educación Superior, Pichin=
cha,
Ecuador. minca@senescyt.gob.ec
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Content-Location: file:///C:/E64510EF/19articulo_publicacion01_DeisiGuangaPagado_archivos/item0030.xml
Content-Transfer-Encoding: quoted-printable
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2 http://www.petroamazonas.gob.ec/quienes-somos/2015 1 ISO06Book {16D998BE-DF6B-47F=
0-8029-BF238F9E260E} Petroleum and natual gas industries -=
Collection and exchange of reliability and Maintenance data equipment 2006 ISO 14224 Switzerland ISO2 SAE99Book {1DD8990F-ED4C-4167-9E9D-CD23E=
1BC28FA} SAE JA1011 <=
/b:Author>SAE JA1011 Evaluation Criteria for Reliabilit=
y - Centered Maintenance (RCM) Process 1999 3 Now78 =
Report {166330C1-3989-436F-A35F-9A134231E834}Reliability-centered Maintenance, Departament of Defense, Report =
Number AD-A066579, Unclassified 1978 San F=
ranciso, California NowlanF. Stanley Heap Howard F. U.S. Depa=
rtment of Commerce. Springfield, VA 22161 http://relia=
bilityweb.com/ee-assets/my-uploads/docs/2010/Reliability_Centered_Maintenan=
ce_by_Nowlan_and_Heap.pdf 4 SAE02 Book {ACB93086=
-504E-467F-BC53-AC671031C01E} SAE =
JA1012 SAE JA1012 A guide to th=
e Reliability-centered Maintenance (RCM) standard. Society of Automotive En=
gineering 2011 5 Mos04 JournalArticle =
{DA29CB0A-1BDD-4D8E-8D3A-9BD6A1D461E7} =
Mostafa S. I. “Im=
plementation of Proactive Maintenance in the Egyptian Glass Company."2004 Journal Quality in Maintenance Engi=
neering 10 no. 2 107–122 6 New98 Book{8C30C2AB-4892-4265-9784-C10C9E2BD9E3} A=
dministración de mantenimiento industrial. 1998 <=
b:Author>Newbrough E.T. Mexico=
Diana 7 Gar10 Book {=
C4BF542D-DFA5-4B20-8B01-D709F72FBCA4} La contratación del=
Mantenimiento Industrial 2010 GarcÃa Santiago <=
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