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EE5712 Power System Reliability:: Reliability Theory

Panida Jirutitijaroen

8/30/2010 1EE5712 Power System Reliability

Announcement#1

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Download reading materials from the given references.

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Ask your questions about homework earlier, dont wait until the last minute!

8/30/2010 EE5712 Power System Reliability 2

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


Outline

Reliability Theory

Reliability Measure

Simple reliability evaluation methods

Example Quick introduction to MATLAB


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


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


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


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


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


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


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


RELIABILITY THEORY

Reliability evaluation

Failure distribution

Survival function

Hazard rate function


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!


Cumulative Failure Distribution

Cumulative failure distribution

Measure of probability of failure as a function of time

Denote as Q(t).

t = 0


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)


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



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


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


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

(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)


(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.


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

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


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)


RELIABILITY MEASURE

Mean time to failure

Mean time to repair

Mean time between failure

System availability

Frequency of failure


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.


A Two-State Example

Mean time to failure (MTTF) =

nUD

Tup

Tdown

UP

DOWN

Tup

nUD nUD nUD


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.


A Two-State Example

Mean time to repair (MTTR) =

nUD

Tdown

Tdown

UP

DOWN

Tup

nUD nUD nUD


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)


A Two-State Example

Tdown

UP

DOWN

Tup

nUD nUD nUD

Mean cycle time (MTBF) =

nUD

Tup + Tdown


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.


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.


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


Other Measures

Frequency of failure

Equivalent failure rate


SIMPLE RELIABILITY EVALUATION METHODS

Assumptions

Probability convolution

Failure mode and effects analysis


Assumptions

Use only probability rules

Known failure probability of each component

independent failure

Simple system configurations, series/parallel

Solution: Failure probability (equivalently, system unavailability)


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


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


Probability Convolution for Z

Z = X Y = X + Y, where Y = -Y

Continuous case:

Discrete case:

dtyxdtyxtyxtz

m

nmymxnyxnz


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


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


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


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?


SYSTEM RELIABILITY-NETWORK METHODS

Reliability block diagram

System structure

Series-Parallel network

Conditional Probability Approach

Cut-set or tie-set method


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.


UP DOWN

(t)

(t)

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.


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


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


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


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


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


Limitations

1

2

5

3

4

Series or Parallel??

2

8

5

3

9

1

7

4

10

6


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}


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

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


Limitations

1

2

5

3

4


2

8

5

3

9

1

7

4

10

6


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


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) ,


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) }


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.


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


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


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


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) ,


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) }


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.


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

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


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


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


Limitations

1

2

5

3

4


2

8

5

3

9

1

7

4

10

6


INTRODUCTION TO MATLABExample of hazard function construction


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


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


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


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


Documents

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