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EE5712 Power System Reliability:: Reliability Theory
Panida Jirutitijaroen
8/30/2010 1EE5712 Power System Reliability
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Announcement#1
Should we start the class at 6PM?
Do not follow lecture notes in 2009!!
Download reading materials from the given references.
Homework 2 uploaded & Homework 1 due today!
Ask your questions about homework earlier, dont wait until the last minute!
8/30/2010 EE5712 Power System Reliability 2
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Announcement#2
ECE NUS and IEEE PES Invited seminar next week.
Solar system integration by Dr. Thomas Reindl, SERIS on Tuesday August 31st, 4-5pm at E5-03-50.
Optimal Hydrothermal Generation Scheduling using Self-Organizing Hierarchical PSO by Dr. WeerakornOngsakul, AIT on Friday September 3rd,3-4pm at E5-03-21.
How reliabile electricity system in Singapore is?
http://www.channelnewsasia.com/stories/singaporelocalnews/view/1077279/1/.html
8/30/2010 EE5712 Power System Reliability 3
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Outline
Reliability Theory
Reliability Measure
Simple reliability evaluation methods
Example Quick introduction to MATLAB
8/30/2010 EE5712 Power System Reliability 4
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X: Time to Failure
Outcome: time to failure
Random Variable, X
GA generator start working at time x = 0
It can fail at any time, x 0
x 0
8/30/2010 5EE5712 Power System Reliability
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Failure Probability Density Function
Probability that a component fails at time x.
1 2 3 4 5
Exponential distribution function
3
2
5.05.032Pr dxeX x
0,5.0 5.0 xexf x0.5
x
xxXxxf
x
Prlim
0
8/30/2010 6EE5712 Power System Reliability
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Failure Probability Distribution Function
A probability that a component fails before or at time x.
Exponential distribution function
dttfxXxFx
Pr
1 2 3 4 5
8/30/2010 7EE5712 Power System Reliability
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Survival Function
Time to failure of a component is a random variable, X.
Commonly used in reliability theory
A function that gives probability of a component survival beyond time x.
1 2 3 4 5
xFxXxXxR 1Pr1Pr
4
5.05.04Pr4 dxeXR x0.5
8/30/2010 8EE5712 Power System Reliability
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Hazard Rate Function
Denoted by (x),
A function that gives a rate at time x, at which a component fails ( i.e. failure rate), given that it has survived for time x.
x
xXxxXxx
x
|Prlim
0
Probability of a component fails between time x and x+x given that it has survived for time x
8/30/2010 9EE5712 Power System Reliability
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Triangle Relationship
xf
xR x
x
dttf
xf
x
dttf
dx
xdR
dx
xdR
xR
1
x
dtt
ex 0
x
dtt
e 0
Probability density function
Survival function Hazard function
8/30/2010 10EE5712 Power System Reliability
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Important Note
Although , for simplicity, survival function and hazard rate function have been described with respect to a component failure, they apply to any random variable.
For example if the random variable is time to repair, then (x) represents the repair rate
8/30/2010 11EE5712 Power System Reliability
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RELIABILITY THEORY
Reliability evaluation
Failure distribution
Survival function
Hazard rate function
8/30/2010 12EE5712 Power System Reliability
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Reliability Evaluation
Concern with time of a component/system to fail.
At time = 0, probability of failure = 0.
As time , probability of failure 1.
t = 0
This looks like cumulative distribution function!
8/30/2010 13EE5712 Power System Reliability
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Cumulative Failure Distribution
Cumulative failure distribution
Measure of probability of failure as a function of time
Denote as Q(t).
t = 0
8/30/2010 14EE5712 Power System Reliability
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Survival Function
More interested to know what is the survival probability.
Denoted by R(t), or equivalently, S(t)
Probability of a component/system surviving
Complement of probability of failure
R(t) = 1 Q(t)
8/30/2010 15EE5712 Power System Reliability
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Failure Probability Density Function
Denoted by f(t).
Recall: pdf= derivative (cumulative function)
dt
tdQtf
dt
tdR
dt
tRdtf
1
dttftQt
0
dttfdttftRt
t
0
1 t
f(t)
Q(t) R(t) = Survival Function
0 time
8/30/2010 16EE5712 Power System Reliability
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Hazard Rate Function
Equivalently
transition rate function, failure rate function, repair rate function, force of mortality
Definition:
Compute as per unit to the number of components
Measure of the rate at which failure occur
8/30/2010 17EE5712 Power System Reliability
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Hazard Rate Calculation
Denote by (t).
Depends on
number of failure in given time
number of components exposed to failure
(t) = Number of failures per unit time
Number of components exposed to failure
8/30/2010 18EE5712 Power System Reliability
This implies that we only consider those components that are survived up to this time t!
-
Example
N0 = number of component tested
Ns(t) = number of component survived at t
Nf(t) = number of failure at t
R(t) = Ns(t)
N0
Q(t) = Nf(t)
N0
(t) = Number of failures per unit time
Number of components exposed to failure
(t) = dNf(t)/dt
Ns(t)8/30/2010 19EE5712 Power System Reliability
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(t) VS. R(t)
From f(t) = dQ(t)/dt,
Hazard rate function is a conditional function of failure density function.
f(t) = dNf(t)/dt
N0
(t) = dNf(t)/dt
Ns(t)= dNf(t)/dt
Ns(t) N0N0 =
R(t)f(t)
8/30/2010 20EE5712 Power System Reliability
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(t) VS. f(t)
Failure density function Probability of failure in any period of time
Hazard rate function Probability of failure in next period of time, given
that it has survived up to time t
(t) equivalent to f(t) but covers only time up to point of interest.
Need to normalize back to unity for times up to t.
8/30/2010 21EE5712 Power System Reliability
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Example: Hazard Rate Calculation
An equipment initially contain 1000 identical components has the following history data in 20 hours.
Taken from Reliability Evaluation of Engineering Systems: concepts and techniques by Roy billinton and Ron Allan8/30/2010 22EE5712 Power System Reliability
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Hazard Rate Function
8/30/2010 23EE5712 Power System Reliability
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A Bath Tub Curve
Typical hazard function of a component.
It is fairly common to assume constant transition rates in reliability modeling.
Burn-in period Wear-out periodUseful life
t
8/30/2010 24EE5712 Power System Reliability
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A Two-State Component
Consider a two-state component
(t) is a hazard function of the up time, called failure rate.
(t) is a hazard function of the down time, called repair rate.
Generally, hazard function is called transition rate in reliability work.
UP DOWN
(t)
(t)
8/30/2010 25EE5712 Power System Reliability
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RELIABILITY MEASURE
Mean time to failure
Mean time to repair
Mean time between failure
System availability
Frequency of failure
8/30/2010 26EE5712 Power System Reliability
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Mean Time to Failure (MTTF)
If system spend Tsuccess hours in success states, the mean up time is given by
MTTF = Tsuccess / nsuccess-to-failurensuccess-to-failure = number of transitions from success states to failure states during T hours.
Average time that the system is in good working condition, also called mean up time.
8/30/2010 27EE5712 Power System Reliability
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A Two-State Example
Mean time to failure (MTTF) =
nUD
Tup
Tdown
UP
DOWN
Tup
nUD nUD nUD
8/30/2010 28EE5712 Power System Reliability
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Mean Time to Repair (MTTR)
If system spend Tfailure hours in failure states, the mean down time is given by
MTTR = Tfailure / nfailure-to-successnfailure-to-success = number of transitions from failure states to success states during T hours.
Average time that the system is in repair condition, also called mean down time.
8/30/2010 29EE5712 Power System Reliability
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A Two-State Example
Mean time to repair (MTTR) =
nUD
Tdown
Tdown
UP
DOWN
Tup
nUD nUD nUD
8/30/2010 30EE5712 Power System Reliability
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Mean Cycle Time
If system has mean up time of MTTF, and mean down time of MTTR, the mean cycle time is
MTBF = MTTF + MTTR
Average time between failures, also called Mean time Between Failure (MTBF)
8/30/2010 31EE5712 Power System Reliability
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A Two-State Example
Tdown
UP
DOWN
Tup
nUD nUD nUD
Mean cycle time (MTBF) =
nUD
Tup + Tdown
8/30/2010 32EE5712 Power System Reliability
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System Availability
If system has mean up time of MTTF, and mean down time of MTTR, the mean cycle time is MTTF + MTTR, system availability is found from
A = MTTF/ (MTTF + MTTR)
Probability of being found in the success states.
8/30/2010 33EE5712 Power System Reliability
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System Unavailability
If system has mean up time of MTTF, and mean down time of MTTR, the mean cycle time is MTTF + MTTR, system unavailability is found from
U = MTTR/ (MTTF + MTTR)
Probability of being found in the failure states.
8/30/2010 34EE5712 Power System Reliability
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Example
A two-state system has the following history over 10 years, 20 transitions from up to down state
9 years spent in up state
Find the followings Mean up time in days
Mean down time in days
Mean cycle time in days
System availability
System unavailability
8/30/2010 35EE5712 Power System Reliability
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Other Measures
Frequency of failure
Equivalent failure rate
8/30/2010 36EE5712 Power System Reliability
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SIMPLE RELIABILITY EVALUATION METHODS
Assumptions
Probability convolution
Failure mode and effects analysis
8/30/2010 37EE5712 Power System Reliability
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Assumptions
Use only probability rules
Known failure probability of each component
independent failure
Simple system configurations, series/parallel
Solution: Failure probability (equivalently, system unavailability)
8/30/2010 38EE5712 Power System Reliability
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Probability Convolution Method
Two independent random variables, X and Y with probability density function
Interest to find the distribution of a new random variable Z = X + Y
Continuous case:
Discrete case:
dtyxtyxtz
m
mnymxnyxnz
8/30/2010 39EE5712 Power System Reliability
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Excess Generation at Load Point
X: generating capacity distribution
Y: load distribution
Interest to find Z: Excess generation
Z = X Y
System loss is when Z < 0
System unavailability is Pr{Z < 0}
Need to find the distribution of Z
8/30/2010 40EE5712 Power System Reliability
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Probability Convolution for Z
Z = X Y = X + Y, where Y = -Y
Continuous case:
Discrete case:
dtyxdtyxtyxtz
m
nmymxnyxnz
8/30/2010 41EE5712 Power System Reliability
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Excess Generation Distribution Example
Capacity (MW) Probability
0 0.000001
50 0.000297
100 0.029403
150 0.970299
Interest to find excess capacity (E), given that the generation capacity (G) and load (L) has the following distribution
Load (MW) Probability
0 0.00
50 0.20
100 0.75
150 0.05
8/30/2010 42EE5712 Power System Reliability
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Convolution Example
From z*n+ = (all m) x[m]y[m-n] E[-150] = G[0]L[150] E[-100] = G[0]L[100] + G[50]L[150] E[-50] = G[0]L[50] + G[50]L[100] + G[100]L[150] E[0] = G[0]L[0] + G[50]L[50] + G[100]L[100] + G[150]L[150] E[50] = G[50]L[0] + G[100]L[150] + G[150]L[100] E[100] = G[100]L[0] + G[150]L[50] E[150] = G[150]L[0]
n G: Capacity (MW) Probability L: Load (MW) Probability
0 0 0.000001 0 0.00
1 50 0.000297 50 0.20
2 100 0.029403 100 0.75
3 150 0.970299 150 0.05
8/30/2010 43EE5712 Power System Reliability
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Failure Mode and Effects Analysis
Model the system
Categorized into subsystems
Define function of each system and its requirement
Block diagram
Examine all failure modes of a component
For example, A circuit breaker ground faults
failure to open
undesired tripping
Study the effects according to the failure
8/30/2010 44EE5712 Power System Reliability
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You are the weakest link!!
"A chain is only as strong as its weakest link
Does this mean that system reliability is determined by the least reliable component in the system?
8/30/2010 45EE5712 Power System Reliability
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SYSTEM RELIABILITY-NETWORK METHODS
Reliability block diagram
System structure
Series-Parallel network
Conditional Probability Approach
Cut-set or tie-set method
8/30/2010 EE5712 Power System Reliability 46
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Structure
System is called structure by definition when the followings apply.
Only 2-state components: Up or Down.
System has only 2 states: Up or Down.
8/30/2010 EE5712 Power System Reliability 47
UP DOWN
(t)
(t)
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Monotonic Structures
A system is called monotonic structure when the system is structured and the followings apply. System operates if all components are up.
System fails if all components fail.
Failure of a component in an already-failed system cannot restore system to work, and the repair of a component in operation will not cause system failures.
Sometimes called Coherent.
8/30/2010 EE5712 Power System Reliability 48
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Reliability Block Diagram
Can only be used for monotonic structure system.
Also called Logic diagram.
Logical relationship between failure of the network and failure of the components.
A block represents working components.
Removal of a block represents failure of a component.
Usually consistent with system structure
Not necessarily refer to physical connections
8/30/2010 EE5712 Power System Reliability 49
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Series System
The success of all components causes the system to success.
Let PsA and PsB be the success probability of component A and B
System availability is
A = PsA PsB
A B
8/30/2010 50EE5712 Power System Reliability
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Parallel System
The failure of all components causes the system to fail.
Let PfA and PfB be the failure probability of component A and B
System unavailability is
U = PfA PfB
A
B
8/30/2010 51EE5712 Power System Reliability
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Series/Parallel Example
If all component has failure probability of 0.01, evaluate system reliability in terms of availability and unavailability
1
6
3
45
7
8
2
8/30/2010 52EE5712 Power System Reliability
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Applications
Probability Convolution
Develop excess generating capacity distribution
Single area analysis
Modeling injected power at each bus
Series/Parallel Network
Distribution systems
Simple substation configuration
Components Circuit breakers
Transformers
Transmission lines
8/30/2010 53EE5712 Power System Reliability
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Limitations
1
2
5
3
4
Series or Parallel??
2
8
5
3
9
1
7
4
10
6
8/30/2010 54EE5712 Power System Reliability
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Conditional Probability Approach
Decompose a complex system into simpler subsystems
Each subsystem is disjoint event
Use conditional probability rule to calculate system failure probability
Denote key component, X, the probability of system failure is calculated from.
Pf = Pr{system fails | X fails} Pr {X fails} +
Pr{system fails | X works} Pr {X works}
8/30/2010 55EE5712 Power System Reliability
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Conditional Probability Example
Assume that each component has failure probability of 0.01 and component 5 is key component, calculate system failure probability
1
2
5
3
4
1
2
3
4
1
2
3
4
5 fails5 is working
subsystem A subsystem B8/30/2010 56EE5712 Power System Reliability
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Failure Probability Calculation
Pf = Pr{system fails | 5 works} Pr {5 works} +
Pr{system fails | 5 fails} Pr {5 fails}
Pf = Pr{A fails} 0.99 + Pr{B fails} 0.01
8/30/2010 57EE5712 Power System Reliability
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Limitations
1
2
5
3
4
Series or Parallel??
2
8
5
3
9
1
7
4
10
6
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Cut-Set and Tie-Set Method
Evaluate reliability of a block diagram network
Assume independent failure
Definition Cut set
Minimal cut set
Tie set
Minimal tie set
Use probability rules to calculate system availability
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Cut-Set
A set of components whose failure causes system failure
1
2
5
3
4
Example (1,2), (1,2,3), (1,2,4), (1,2,5), (1,4,5) , (1,2,3,4) , (1,2,3,4,5) ,
8/30/2010 60EE5712 Power System Reliability
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Minimal Cut-Set
A smallest set of components whose failure causes system failure
1
2
5
3
4
Minimal cut-set = { (1,2), (3,4), (1,4,5) , (2,3,5) }
8/30/2010 61EE5712 Power System Reliability
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Cut-Set Method
Enumerate all minimal cut-set in the system. Failure of all components in a minimal cut-set
causes system failure. This implies parallel connections among these
components. Each minimal cut set causes system failure. This implies series connections among the
minimal cut sets. Draw equivalent system and use series/parallel
method to compute for system availability.
8/30/2010 62EE5712 Power System Reliability
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Example
1
2
5
3
4
1
2
Minimal cut-set = { (1,2), (3,4), (1,4,5) , (2,3,5) }
3
4
1
5
4
2
5
3
8/30/2010 63EE5712 Power System Reliability
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System Unavailability Calculation
If the system has C1, , Cn minimal cut sets, system failure probability is found from
Pf = Pr,1 U 2 U U n},
is an event that the cut-set fails
These minimal cut sets are not disjoint,
Pf = Pr,1 U 2 U U n}
= i Pr{i} - ij Pr{i j- + ijk Pr{i j k}
- + (-1) Pr,1 n} 1
8/30/2010 64EE5712 Power System Reliability
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Probability Approximation
Total number of terms in previous equation is 2-1.
Use Booles inequality,
Pf = Pr,1 U 2 U U n- i Pr{i}
Upper bound on failure probability
Lower bound on success probability
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Tie-Set
A set of components whose function causes system success
1
2
5
3
4
Example (1,3), (1,3,4), (1,3,5), (1,4,5), (2,4) , (2,3,4) , (1,2,3,4,5) ,
8/30/2010 66EE5712 Power System Reliability
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Minimal Tie-Set
A smallest set of components whose function causes system success
1
2
5
3
4
Minimal tie-set = { (1,3), (2,4), (1,4,5) , (2,3,5) }
8/30/2010 67EE5712 Power System Reliability
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Tie-Set Method
Enumerate all minimal tie-set in the system. Success of all components in a minimal tie-set
causes system success. This implies series connections among these
components. Each minimal tie set causes system success. This implies parallel connections among the
minimal tie sets. Draw equivalent system and use series/parallel
method to compute for system availability.
8/30/2010 68EE5712 Power System Reliability
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Example
1
2
5
3
4
Minimal tie-set = { (1,3), (2,4), (1,4,5) , (2,3,5) }
1
2
3
4
1 54
2 538/30/2010 69EE5712 Power System Reliability
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System Availability Calculation
If the system has T1, , Tn minimal tie-sets, system success probability is found from
Ps = Pr{T1 U T2 U U Tn},
T is an event that the tie-set is success
These minimal tie sets are not disjoint,
Ps = Pr{T1 U T2 U U Tn}
= i Pr{Ti} - ij Pr{Ti Tj- + ijk Pr{Ti Tj Tk}
- + (-1) Pr,T1 Tn} 1
8/30/2010 70EE5712 Power System Reliability
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Probability Approximation
Total number of terms in previous equation is 2-1.
Use Booles inequality,
Ps = Pr{T1 U T2 U U Tn- i Pr{Ti}
Upper bound on success probability
Lower bound on failure probability
8/30/2010 71EE5712 Power System Reliability
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Bounds on Probability Approximation
Cut-set Method
Compute for failure probability
Upper bound on failure probability
Lower bound on success probability
Tie-set Method
Compute for success probability
Upper bound on success probability
Lower bound on failure probability
8/30/2010 72EE5712 Power System Reliability
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Limitations
1
2
5
3
4
Series or Parallel??
2
8
5
3
9
1
7
4
10
6
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INTRODUCTION TO MATLABExample of hazard function construction
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Goal
To review logical concepts of programming
To familiarize yourself with MATLAB environment
Feel free to work with program language of your choice!-same logic still applies
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Simple MATLAB Command
Initialize a vector, matrix
Manipulate a vector, matrix
Program control statement
Resources
http://www.mathworks.com/access/helpdesk/help/techdoc/learn_matlab/bqr_2pl.html
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Summary
Reliability theory Failure distribution Survival function Hazard rate function
Reliability Measure Simple reliability analysis
Probability convolution method Series/parallel system Conditional probability approach Cut-set/Tie-set method
Input: failure probability of each component Output: failure probability of a system
8/30/2010 77EE5712 Power System Reliability
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About Next Lecture
Interest to know how often the system fails
Frequency of failure is another reliability measure.
Need to study stochastic process
Model stochastic behavior of a system
Transition rate from one state to others
From success to failure state called Failure rate
From failure to success state called Repair rate
8/30/2010 78EE5712 Power System Reliability