Mathematical models of reliability parameters for complicated power systems
A method of investigation of reliability parameters for complicated power systems by means of generating functions is developed taking account of aging of the system’s output elements. Main time parameters for reliability evaluation are examined in this paper. Mathematical models in the form of anal...
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| Cite this: | Mathematical models of reliability parameters for complicated power systems / A.R. Sydor, V.M. Teslyuk // Технічна електродинаміка. — 2012. — № 3. — С. 37-38. — Бібліогр.: 5 назв. — англ. |
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Sydor, A.R. Teslyuk, V.M. 2014-05-17T10:25:14Z 2014-05-17T10:25:14Z 2012 Mathematical models of reliability parameters for complicated power systems / A.R. Sydor, V.M. Teslyuk // Технічна електродинаміка. — 2012. — № 3. — С. 37-38. — Бібліогр.: 5 назв. — англ. 1607-7970 https://nasplib.isofts.kiev.ua/handle/123456789/62174 519.873 A method of investigation of reliability parameters for complicated power systems by means of generating functions is developed taking account of aging of the system’s output elements. Main time parameters for reliability evaluation are examined in this paper. Mathematical models in the form of analytical expressions are worked out for the average duration of the system’s stay in each of its states and for the average duration of the system’s stay in the prescribed availability condition provided that the lifetime of ageing output elements is circumscribed by the Rayleigh distribution. en Інститут електродинаміки НАН України Технічна електродинаміка Електроенергетичні комплекси, системи та керування ними Mathematical models of reliability parameters for complicated power systems Article published earlier |
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Mathematical models of reliability parameters for complicated power systems |
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Mathematical models of reliability parameters for complicated power systems Sydor, A.R. Teslyuk, V.M. Електроенергетичні комплекси, системи та керування ними |
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Mathematical models of reliability parameters for complicated power systems |
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Mathematical models of reliability parameters for complicated power systems |
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Mathematical models of reliability parameters for complicated power systems |
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mathematical models of reliability parameters for complicated power systems |
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Sydor, A.R. Teslyuk, V.M. |
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Sydor, A.R. Teslyuk, V.M. |
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Електроенергетичні комплекси, системи та керування ними |
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Електроенергетичні комплекси, системи та керування ними |
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A method of investigation of reliability parameters for complicated power systems by means of generating functions is developed taking account of aging of the system’s output elements. Main time parameters for reliability evaluation are examined in this paper. Mathematical models in the form of analytical expressions are worked out for the average duration of the system’s stay in each of its states and for the average duration of the system’s stay in the prescribed availability condition provided that the lifetime of ageing output elements is circumscribed by the Rayleigh distribution.
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https://nasplib.isofts.kiev.ua/handle/123456789/62174 |
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Mathematical models of reliability parameters for complicated power systems / A.R. Sydor, V.M. Teslyuk // Технічна електродинаміка. — 2012. — № 3. — С. 37-38. — Бібліогр.: 5 назв. — англ. |
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AT sydorar mathematicalmodelsofreliabilityparametersforcomplicatedpowersystems AT teslyukvm mathematicalmodelsofreliabilityparametersforcomplicatedpowersystems |
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2025-11-27T01:12:46Z |
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2025-11-27T01:12:46Z |
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ISSN 1607-7970. . 2012. 3 37
519.873
MATHEMATICAL MODELS OF RELIABILITY PARAMETERS FOR COMPLICATED POWER SYSTEMS
A.R.Sydor, V.M.Teslyuk,
Lviv Polytechnic National University,
12 S.Bandery Str., Lviv, 79013, Ukraine.
A method of investigation of reliability parameters for complicated power systems by means of generating functions is developed
taking account of aging of the system’s output elements. Main time parameters for reliability evaluation are examined in this paper.
Mathematical models in the form of analytical expressions are worked out for the average duration of the system’s stay in each of its
states and for the average duration of the system’s stay in the prescribed availability condition provided that the lifetime of ageing
output elements is circumscribed by the Rayleigh distribution. References 5.
Key words: reliability parameters, power systems, Rayleigh distributed ageing elements.
ntroduction. Reliability prediction methods are widely used throughout the power engineering, and are often
used as a yardstick for comparing various equipment. But these models can be wildly inaccurate when compared with
rates of modern devices, and their use can lead to increased costs and complexity while deluding engineers into
following a flawed set of perceptions and leaving truly effective reliability improvement measures unrecognized.
In recent years there has been significant progress in the field of power electronic circuit topologies, typical for
power systems and, for this reason, one may expect innovations in high inverter systems [2].
There exist different methods of investigation for reliability parameters of complicated power systems [1,3,4]. But
existing traditional methods of reliability evaluation are not able to satisfy requirements of investigations of complicated systems
such as complicated power systems which provide control, management and monitoring that cover a broad range of tasks.
odels of reliability parameters. In power electronics most beneficial technological innovations have been
introduced into process and energy distribution; it being known that in numerous industrial and vehicular applications,
the assemblies of mechanically and electrically coupled devices are joined in electronic units of high complexity [5].
Let us consider a symmetric electropower system ramified to level 2 with ageing output elements, where a1
elements of level 1 are subordinate to an element of level 0, a2 elements of level 2 are subordinate to every element of
level 1. a1 is a coefficient of ramification to level 1, a2 is a coefficient of ramification to level 2.
We use T2R(x2) to denote the average duration of the system’s stay in a state of x2 operating output elements on
condition that lifetime of ageing output elements is circumscribed by the Rayleigh distribution.
Under condition 0<x2 N2 we obtain the following expression:
22 21 1 1 2 1 2 2
0 1 1 11 2 1 1 2 2 2
1 2 1 1 1 2 1 2
1 22
1
2
( ( )) 2
2 2
0 0 0
( ) ( 1) ( 1) .
x ja a x a x x t
x j tx x j j j j
R a a x a x a x x
j jxx ceil
a
T x C C C C e e dt
Notice that T2R(0)= .
The system availability condition is that there are not less than k operating output elements of the system
(0<k N2). The sum of average durations of the system’s stay in states over count of output elements from k N2 is
equal to the average duration of the system’s stay in the prescribed availability condition k.
Let T 2R(k) be the average duration of the system’s stay in the availability condition k provided that lifetime of
ageing output elements is circumscribed by the Rayleigh distribution. We obtain:
22 2
2 1 1 1 2 1 2 2
0 1 1 11 2 1 1 2 2 2
1 2 1 1 1 2 1 2
2 1 22
1
2
( ( )) 2
2
0 0 0
( ) ( 1) ( 1) .
x jN a a x a x x t
x j tx x j j j j
R a a x a x a x x
x k j jx
x ceil
a
T k C C C C e e dt
Let us consider an unsymmetrical electropower system with two nonequivalent branches on level 1, ramified to
level 2, with Rayleigh distributed output elements, where 2 elements of level 1 are subordinate to the element of level 0,
the first element of level 1 subordinates )1(
2a elements of level 2, the second element of level 1 subordinates )2(
2a
elements of level 2. Without loss of generality assume that )2(
2
)1(
2 aa .
We use T2R(x2) to denote the average duration of the system’s stay in a state of x2 operating output elements on
condition that lifetime of ageing output elements is circumscribed by the Rayleigh distribution. Under condition
0<x2
)2(
2
)1(
2 aa we obtain the following expression:
(1) (1) ( 2 )
2 , 2 1 1(1) ( 2 )
22 2 1
(1) (1) ( 2 ) ( 2) (1) ( 2 )
2 1 2 1 1 1(1) ( 2 ) (1) (1)
122 2 (1) ( 2 ) 22 2
1 1(1) ( 2)
2 2
min{ } 2 ( )1 1
2 2 2 ( )
0max{0, }
( )
x a x x
x x x j
R a x a x x x
jx x a x x x
x ceil x ceil
a a
T x C C C
(1) (1) ( 2) ( 2) 22 222 1 2 1 (1) ( 2 ) 20 1 11 11 2 2 2
(1) (1) ( 2) ( 2)
22 1 2 1
2
(
0 0
( 1) ( 1) .
x ja x a x x tx x j tj j j
a x a x x
j
C e e dt
38 ISSN 1607-7970. . 2012. 3
Let T 2R(k) be the average duration of the system’s stay in the availability condition k provided that lifetime of
ageing output elements is circumscribed by the Rayleigh distribution. We obtain:
(1) (1) (2)
2, 22 1 1(1) (2)
22 2 1
(1) (1) (2) (2) (1) (2)
2 1 2 1 1 1(1) (2) (1) (1)
2 122 2 2(1) (2)2 2
1 1(1) (2)
2 2
1
(1)
2
min{ } 2 ( )1 1
2 2 ( )
0max{0, }
( )
( 1)
x a x xN
x x x j
R a x a x x x
x k jx x a x x x
x ceil x ceil
a a
j
a x
T k C C C
C
(1) (1) (2) (2) 22 222 1 2 1 (1) (2) 20 1 11 12 2 2
(1) (2) (2)
21 2 1
2
(
0 0
( 1) .
x ja x a x x tx x j tj j
a x x
j
e e dt
In case of failure of an element of level 0 the system will fail completely, therefore probability of failure-free
operation Iof this element is maximal. Operation of all elements on other system’s levels depends on operation of this
element. The system’s operation depends on elements of the lowest level the least, therefore probability of failure-free
operation of them may be the least.
onclusions. The paper deals with mathematical models of main time reliability characteristics for unresto-
rable complicated electropower systems.
Without use of reliability characteristics it is impossible to settle a number of problems of systems’ design and
operation, for example: selection of structure and rational redundancy, organization of inspection monitoring and
preventive maintenance. It is necessary to work out methods of reliability prediction with regard for systems’ specific
features such as possibility of structure rearrangement, preservation of serviceability in case of partial failures at the
expense structural redundancy.
Thus, expressions are worked out for evaluation of two main time reliability parameters of complicated
electropower systems: – the average duration of the system’s stay in a state of x2 operating output elements; – the
average duration of the system’s stay in the prescribed availability condition.
1. Chen G., Yuang X., Tang X. Reliability analysis of hierarchical systems // Proceedings of Annual Reliability and
Maintainability Symposium RAMS-2005, Alexandria, Virginia USA. – 2005. – P . 146–151.
2. Dawidziuk T. Resonant pole inverters // Tekhnichna elektrodynamika. – 2006. – 1. – P . 89–94.
3. Linquist T., Bertling L., Eriksson R. A method of age modeling of power system components based on experiences from
the design process with the purpose of maintenance optimization // Proceedings of Annual Reliability and Maintainability
Symposium RAMS-2005, Alexandria, Virginia USA. – 2005. – P . 82–88.
4. Robinson D.G. Reliability analysis of bulk power systems using swarm intelligence // Proceedings of Annual Reliability
and Maintainability Symposium RAMS-2005, Alexandria, Virginia USA. – 2005. – P . 96–102.
5. Vodolazov V., Vinnikov D., Laugis J., Leita T. Interinstitutional activity in professional training in power electronics //
Tekhnichna elektrodynamika. – 2008. – 7. – P . 58–61.
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20.12.2011
Received 20.12.2011
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