The Basic Reliability Calculations
Definition
Reliability is a general quality of an object – an ability to
perform a desired function, sustaining the values of rated
operational indicators in given limits and time according to
given technical conditions.
Reliability is probability that an activity of an appliance in
given time and given operation conditions will be adequate
to its purpose.
EIA (Electronic Industry Association, USA)
Reliability Calculations
1. Reliability of single parts of networks in the time of
production of project documentation
2. Reliability of already operated networks
3. Reliability in the area of control of electric power
system operation
Numerical Representation of Reliability
(Classical - reliability of elements)
 failure rate  [ year-1]
 mean time of failure  [ h ]
 probability of failure-free run R [ - ]
 probability of failure Q [ - ]
 mean time between failures tS [ h ]
Restored x Not restored objects
Mean time between failures x Mean time to failure
Global Indices of Reliability
(Reliability of electric energy supply)
– Outage rate - SAIFI
average system outage rate
(number of outages/year/consumer)
– Total time of all outages - SAIDI
average system outage time
(min/year/consumer)
– Time of one outage - CAIDI
(min/outage)
average outage time at a customer
Bathtub curve

Early failure
period
I
Constant failure rate period
II
Wear-out failure
period
III
t
The relation between the function of reliability and
failure rate is:
For failure rate it is valid:
Division of Probability of Failure
Exponential division
Exponential rule of failure
Poisson’s division
If k = 0, there is probability of no failure, therefore probability of failure-free
running.
Weibull’s deal
Calculation of Reliability in Electricity
Industry
Obtaining of input values for reliability calculations
A priori reliability – determination of reliability quantities from data of a producer.
Empirical reliability – monitoring of failures in electricity industry.
The empirical method is mostly used for obtaining the input values for reliability
calculations, because an application of a priori reliability method requires different
attitude to every element of electricity system.
Analysis of Distribution Network Failure Databases
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Exclusive outage databases had been on the rise since 1975 in the
former Czechoslovakia.
Unfortunately, database building stopped in1990 because of political
and social changes.
Thanks to the expert group CIRED Czech distributors opted for
unified monitoring of global reliability indices and the reliability of
selected pieces of equipment in 1999 again.
Data for the reliability computation is centrally processed and
analyzed at the VSB - Technical University of Ostrava since the year
2000.
Collected data are often heterogeneous.
It is necessary to solve the storage, indexing, and also transformation
of such data.
We need to create a common relational scheme for the storage of the
data, a new relation makes the querying and analysis possible.
• Database range
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
Region1 Region2 Region3 Region4 Region5 Region6 Region7 Region8
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Heterogeneous Data
•
We developed a common relational schema.
Order
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
Attribute
Distribution company
Outage identification
Outage type
Distribution point
Distribution area
Network type
Network voltage
Equipment voltage
Original outage identification
Outage cause
Equipment type
Failed equipment
Type of equipment
Amount of failed equipment
Short circuit type
Producer
Production year
Outage start time
First manipulation
End of manipulations
End of outage
End of equipment failure
Time of dead earth
Unsupplied power at the outage start
Unsupplied power at the end of manipulations
Unsupplied distribution transformers at the outage start
Unsupplied distribution transformers at the end of manipulations
Unsupplied customers at the outage start
Unsupplied customers at the end of manipulations
Number of unsupplied customers multiplied by time of their outage
Failure type
Description
Unique number of a distributor
Unique number of an outage
Accidental, planned or forced
Type of substation: single, double busbar substation, . . .
Specification of the location
Insulated system, resonant grounded neutral system, . ..
0.4 kV, 22 kV, . . .
0.4 kV, 22 kV, . . .
Unique number of the outage cause
Foreign influences, causes before starting operation, . . .
Overhead line, underground line, . . .
Specific equipment: conductor, switch, pole, fuse, . . .
Further specification: wooden pole, steel pole, . . .
One-line-to-ground fault, ground fault, line-to-line grounded fault, . . .
Siemens, ABB, . . .
Production year of the equipment
Failure limitation start time
Failure limitation time
Supply renewal time for all consumers
Equipment reparation end time
With or without equipment fail
Results

Failure rate
N
λ 
Z  P



(year-1)
N = number of failures (-)
Z = number of elements of the given type
in the network (-)
P = the considered period (year)
Results

Mean duration of the failure
N
τ
t
i 1
i
N

N = number of failures (-)

ti = the considered period (h)
(h)
Results
• The value tendency of reliability indices of the
22 kV cable
Mean time to repair (h)
7
0.1
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
6
5
4
3
2
1
0
200020012002 20032004 200520062007 20082009 Total
)
 (year-1)
-1
 (h)
Failure rate (year
Cable 22 kV
Results
• Comparison with methodology ČEZ 22/80
ČEZ
22/80
Equipment
2000 - 2009
22 kV cable
 (year-1)
 (h)
14.5
215
5.480
4.034
22 kV overhead line
 (year-1)
 (h)
14
3
3.018
4.163
110 kV overhead line
 (year-1)
 (h)
5.2
3.5
0.370
3.992
MV/LV transformer
 (year-1)
 (h)
0.03
2500
0.007
4.315
110 kV/MV transformer
 (year )
 (h)
0.04
1300
0.059
0.480
22 kV circuit breaker
 (year-1)
 (h)
0.015
30
0.016
64.179
110 kV circuit breaker
 (year-1)
 (h)
0.01
100
0.052
47.425
-1
Results
• Division of failures according to their causes
Četnosti příčin událostí
1
2
3
4
8
9
Causes before starting operation
Operation and maintenance causes
Foreign influences
Forced outage
Cause not explained
Other causes
The Main Calculation Methods of
Reliability
Department of electrical power engineering
Markov‘s processes
Method of reliabilty schemes
Simulative methods
Method of Reliability Schemes
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-
make-up of reliability diagram,
assignment of relevant reliability quantities to single elements,
simplification of reliability diagram towards one element,
Advantages:
-
considered systems do not have to really exist as yet,
procedure of solving is well-arranged and not exacting concerning
mathematics,
mathematical procedure does not require iterative calculation,
accuracy of results depends only on the accuracy of input parameters
of calculation.
Disadvantages:
-
-
it is impossible to pursue power balance of network,
„T“ type bay can be modelled only approximately.
Rule of Multiplication of Probabilities :
P(A)
probability of occurrence of A
P(B)
probability of occurrence of B
Series systems
A failure of one element leads to a failure of a system.
Probability of failure-free run:
Parallel Systems
A failure of a system occurs when all elements have a failure
Probability of a failure:
Probability of failure-free run:
Simplified
Probability of failure-free run:
Methodology of Calculation of Reliability
according to ČEZ 22/80 Regulation
Advantages:
-
takes into account maintenance outages,
enables to include manipulation into calculation as well, takes so-called cold
reserves into account.
The disadvantage is that this methodology does not include so-called coordination
of maintenance.
These operating states are considered with calculation:
operation,
failure outage,
maintenance outage.
Supposition and simplification:
-
the effect of weather on failure rate and repair rate is not taken into account,
exponential division of distributive function of time of failures
and repairs for all elements of electric network is taken into account,
average data are started from.
Series connection of elements
For this circuit with two elements it holds:
P ... Failure rate [year-1]
U ... Maintenance rate [year-1]
 ... Outage rate (maintenance + repair) [year-1]
Mean times of outages of a two-elements system:
Parallel connection of elements – hot reserve
Failure rate:
Mean time of failure:
Maintenance outage cannot occur at this connection, because at the failure of one
element maintenance of another element will not begin.
Parallel circuit of elements – cold reserve
…. Manipulation time [h]
Simulation Methods of Calculation of Reliability
It is necessary to know the intensity of outages and mean time of outages of all the
elements of a system.
Simulation - numerical method which resides in experimenting with mathematical
models of real systems on numerical computers.
Advantages:
- considered systems do not have to really exist as yet,
- considered systems can be too complicated for using analytical methods,
- simulation makes possible study of behaviour of systems in real, accelerated, or
retarded time. The second possibility is the most important in this case,
because the processes of outage of elements and their re-introduction into
operation are very slow. It would be very inefficient to study them in any other
time but accelerated.
- with simulation it is possible to verify results obtained by other independant
processes,
- possibility of modelling „T“ type bays
- simple power balance of a diagram is carried out, outage is always simulated at
overloaded elements.
Disadvantages of simulation methods:
- construction of a useful simulation model is very time-consuming. Mostly
several variants of a model are needed.
- simulation is a numerical method, so a solution of certain problem cannot be
generally transferred on analogous problems.
- the results obtained from stochastic simulation models are values of
accidental quantities, and it would be very computer time-consuming if their
accuracy should be increased
- precision of results depends on the number of iterations,
- the needed number of iterations depends on the extent of the solved
network and on the required precision.