Reliability Analysis

NATIONAL ELECTRIFICATION ADMINISTRATIONU. P. NATIONAL ENGINEERING CENTER

Certificate in

Power System Modeling and Analysis

Competency Training and Certification Program in Electric Power Distribution System Engineering

U. P. NATIONAL ENGINEERING CENTERU. P. NATIONAL ENGINEERING CENTER

Power System Reliability Analysis

Training Course in

2

Competency Training & Certification Program in Electric Power Distribution System Engineering

U. P. National Engineering CenterNational Electrification Administration

Training Course in Power System Reliability Analysis

Course Outline

1. Reliability Models and Methods

2. Distribution System Reliability Evaluation

3. Economics of Power System Reliability

3




� Reliability Definition

� Probability Function

� The Reliability Function

� Availability

� System Reliability Networks

Reliability Models and Methods

4




Reliability Definition

A reliable piece of equipment or a System is understood to be basically sound and give trouble-free performance in a given environment.

Reliability is the probability that an equipment or system will perform satisfactorily for at least a given period of time when used under stated conditions.

5




Probability Function

� Subjective Definition (or Man-in-the-Street)� The probability P(A) is a measure of the degree of belief one

holds in a specified proposition A

Example: Out of 100 equipment that were upgraded by introducing a new design, 75 will perform better

P(improved performance) = 75/100 = 0.75

6





SET THEORY CONCEPTS

� SET� A finite or infinite collection of distinct objects or elements with

some common characteristics

Venn Diagram of a SET of Geometric Figures

7





SET THEORY CONCEPTS

� SUBSET� A partition of the SET by some further characteristics that

differentiate the members of the SUBSET from the rest of the SET

Venn Diagram of the SUBSET “Circles” from the SET of Geometric Figures

8





SET THEORY CONCEPTS

� Identity SET� SET that contains all the elements under consideration. Also

called Reference SET and denoted by letter I

� Zero SET� SET with no element denoted by letter Z

9




Probability and Statistics

SET THEORY CONCEPTS

� Size of a SET� The number of elements in the SET A is denoted by m(A) and is

referred to as the size of the SET A

Example: The NEC SET company employs ten non-professional workers. Three of these are Assemblers (the Set A), five are Machinists (the Set M), and two are Clerks(the Set C) A M C

m(A) = 3 m(M) = 5 m(C) = 2 m(I) = 10

10





SET THEORY CONCEPTS� The SET Q, made up of all workers who are both machinists

and assemblers, does not contain any element (mutually exclusive), i.e., Q = AM = Z. Hence,

� The SET F, consisting of all factory workers (assemblers and machinists), is the Union of SETs A & M, i.e., F = A + M

This SET contains eight distinct elements, three from A and five from M. Thus,

m(Q) = m(Z) = 0

m(F) = m(A+M) = 3 + 5 = 8

m(A+B) = m(A) + m(B) if AB = Z

11





SET THEORY CONCEPTS

Example: In addition to the 10 Non-professional workers, the NEC SET Company also employs eight full time Engineers (the Set E), three full time supervisors (the Set S), and two individuals whoare both engineers and supervisors (the Set ES).

The size of the Set of all professional employees (engineers andsupervisors) is 13 m(E+S) ≠ m(E) + m(S)

A M CE ES S

13 ≠ 10 + 5

12





SET THEORY CONCEPTSNote the Set ES is counted twice. Hence,

m(E+S) = m(E) + m(S) – m(ES)

= 10 + 5 - 2

= 13

m(A+B) = m(A) + m(B) – m(AB) if AB ≠ Z

“NOT MUTUALLY EXCLUSIVE”

13





PROBABILITY AND SET THEORY� The PROBABILITY of some Event A may be regarded as

equivalent to comparing the relative size of the SUBSET represented by the Event A to that of the Reference SET I

� Example: The probability of the employee of NEC SET Company being both Engineers and supervisor

m(A)M(I)

P(A) =

m(ES)M(I)

P(ES) =2

23=

14




� Random VariableA function defined on a sample space

•Tossing two Dice•Operating time (hours)•Distance covered (km)•Cycles or on/off operations•Number of revolutions•Throughput volume (tons of raw materials)

Discrete Random Variable - Countable and Finite

Continuous Random Variable - Measured and Infinite


15




21,1

31,2

41,3

51,4

61,5

71,6

82,6

72,5

62,4

52,3

42,2

32,1

Value of R.V.

Sample Point

43,1

53,2

63,3

73,4

83,5

93,6

104,6

94,5

84,4

74,3

64,2

54,1

Value of R.V.

Sample Point

65,1

75,2

85,3

95,4

105,5

115,6

126,6

116,5

106,4

96,3

86,2

76,1

Value of R.V.

Sample Point

Results of Tossing two dice� Random Variable


16




1

2

3

4

5

6

5

4

3

2

1

Occur.m(xi)

12

11

10

9

8

7

6

5

4

3

2

Value of R.V.

1/36

2/36

3/36

4/36

5/36

6/36

5/36

4/36

3/36

2/36

1/36

Probablityp(xi)

( )

⎪⎪⎩

⎪⎪⎨

⎧

−

−

=

36

13

36

1

i

i

i

x

x

xp

7 6, 5, 4, 3, 2 ,xi =

12 11,10,9, 8,=ix

� Probability Distribution

0

0.05

0.1

0.15

0.2

2 3 4 5 6 7 8 9 10 11

Random Variable y = x1 + x2

Pro

bab

ility

12


17




0<2

12

11

10

9

8

7

6

5

4

3

2

Value of R.V.

36/36 = 1.0

35/36

33/36

30/36

26/36

21/36

15/36

10/36

6/36

3/36

1/36

Cum. ProbablityF(xi)

( ) ( )∑≤

=ixx

ii xpxF

� Cumulative Distribution

0

0.2

0.4

0.6

0.8

1

122 3 4 5 6 7 8 9 10 11


Cu

m. P

rob

abili

ty


18




� For Continuous Random Variable� Probability Density Function

� Cumulative Probability Function

( )xf x – random variable

( ) ( ) τdxfxFx

∫ ∞−=


19




The Reliability Function

21,1

31,2

41,3

51,4

61,5

71,6

82,6

72,5

62,4

52,3

42,2

32,1

Value of R.V.

Sample Point

43,1

53,2

63,3

73,4

83,5

93,6

104,6

94,5

84,4

74,3

64,2

54,1

Value of R.V.

Sample Point

65,1

75,2

85,3

95,4

105,5

115,6

126,6

116,5

106,4

96,3

86,2

76,1

Value of R.V.

Sample Point

Results of Tossing two dice� Random Variable

20





1

2

3

4

5

6

5

4

3

2

1

Occur.m(xi)

12

11

10

9

8

7

6

5

4

3

2

Value of R.V.

1/36

2/36

3/36

4/36

5/36

6/36

5/36

4/36

3/36

2/36

1/36

Probablityp(xi)

( )

⎪⎪⎩

⎪⎪⎨

⎧

−

−

=

36

13

36

1

i

i

i

x

x

xp

7 6, 5, 4, 3, 2 ,xi =

12 11,10,9, 8,=ix

� Probability Distribution

0

0.05

0.1

0.15

0.2

2 3 4 5 6 7 8 9 10 11


Pro

bab

ility

12

21





0<2

12

11

10

9

8

7

6

5

4

3

2

Value of R.V.

36/36 = 1.0

35/36

33/36

30/36

26/36

21/36

15/36

10/36

6/36

3/36

1/36

Cum. ProbablityF(xi)

( ) ( )∑≤

=ixx

ii xpxF

� Cumulative Distribution

0

0.2

0.4

0.6

0.8

1

122 3 4 5 6 7 8 9 10 11


Cu

m. P

rob

abili

ty

22





� For Continuous Random Variable� Failure Density Function

� Cumulative Probability Function

( )tf t – random variable time-to-failure

( ) ( ) ττ dftFt

∫=0

23





The probability that a component will fail by the time t can be defined by cumulative distribution function of failure

where t is a random variable denoting time-to-failure.

Since success and failure are mutually exclusive, then the Reliability Function can be defined by

( ) ( )tFtTP =≤ 0≥t

( ) ( )tF1tR −= 0≥t

24




If the time to failure random variable t has a density function f(t), then

or

( ) ( )∫−=t

0df1tR ττ

( ) ( )∫∞

=t

dftR ττ

∫=t

dtftF0

)()( τ


( )tf

timet

λ f(t)

F(t)

R(t)

25




Example

What is the probability that an equipment will not fail in one year if its failure density function is found to be exponential ( ) where λλλλ = 0.01failure/yr

The reliability function is

( ) τλ λτ de1tRt

0∫−−=

t0|e1 λτ−+=

0t ee1 −− −+= λ

( ) tetR λ−=

( ) tetf λλ −=


( )( ) == − yryr/f.e 1010

26





� Hazard Function (Failure Rate)

The propones of failure of a system or a component is expressed by a the Hazard Function h(t).

In terms of the Hazard Function, the Failure Density Function is

( )( )∫=

−t

dhetR 0

ττ

( ) ( ) ( )∫−=

t dhethtf 0 ττ

the Reliability Function in terms of Hazard Function is

27




Example

What is the reliability of a component in one year if it has constant hazard function of λλλλ = 0.01failure/yr

The reliability function is


tdee

t

λτλ−

−

=∫= 0

( )( )∫=

−t

dhetR 0

ττ

( )( )yryr/f.e 1010−=

=

Note: the failure density function for a constant hazard is exponential

( ) ( ) ( )∫−=

t dhethtf 0 ττ

∫=t de 0 τλ

λte λλ −=

28





Component Failure Data

26610

1869

1418

1117

866

635

464

343

202

81

Time-to-Failure (hrs.)Item No. How do you determine the Failure Density and Hazard Functions?

29





Estimating Failure Density Function

Data Density Function (fd(t))

The data density function (also called empirical density function) defined over the time interval Δti is given by the ratio of the number of failures occurring in the interval to the size of the original population N, divided by the length of the interval.

where n(t) is the number of survivor at any time t.

( )( ) ( )[ ]

i

iiid t

Nttntntf

Δ

Δ+−= iii tttt Δ+≤<for

30




Failure Density Function f(t)

80186 – 266

45141 – 186

30111 – 141

2586 – 111

2363 – 86

1746 – 63

1234 – 46

1420 – 34

128 – 20

80 – 8

f(t)ΔtiTime

0013.080101

=

0022.045101

=

0033.030101

=

0084.012101

=

0074.014101

=

0084.012101

=

0125.08101

=


measure of the overall speedat which failures are occurring.

0059.017

10=

0043.023

101=

0040.025

101=

1

31




0.2

0.4

0.6

0.8

1.0

1.2

1.4

100 200 300

f(t)

frac

tiona

l fai

lure

s/hr

.x10

-2

00

Operating time, hr.

Failure Density Function from Component Failure Data


32




Data Hazard Rate or Failure Rate [hd(t)]

The data hazard rate or failure rate over the time interval Δti is defined by the ratio of the number of failures occurring in the time interval to the number of survivors at the beginning of the time interval, divided by the length of the time interval.

( )( ) ( )[ ] ( )

i

iiiid t

tnttntnth

Δ

Δ+−=

iii tttt Δ+≤<for


33




Failure Hazard Function h(t)

80186 – 266

45141 – 186

30111 – 141

2586 – 111

2363 – 86

1746 – 63

1234 – 46

1420 – 34

128 – 20

80 – 8

h(t)ΔtiTime


0125.08101

=

093.012

91=

0096.014

81=

0119.012

71=

0111.030

31=

0111.045

21=

0125.080

11=

measure of the instantaneous speed of failure

0098.017

61=

0087.023

51=

0100.025

41=

34




0.2

0.4

0.6

0.8

1.0

1.2

1.4

100 200 3000

0

h(t)

failu

res/

hr.x

10-2

Operating time, hr.


Hazard Function from Component Failure Data

35





Constant Hazard Model

( ) λ=th

( ) tddht

0

t

0λτλττ == ∫∫

( ) tetf λλ −=

( ) tetR λ−=

( ) te1tF λ−−=

( )th

λ

t

36




c. Rising exponentialdistribution function

t

( )tF

1e11−

λ1t =

d. Decaying exponentialreliability function

e1t

( )tR

1

λ1t =

( )th

λ

t t

( )tf

λ

eλ

λ1t =

a. Constant Hazard b. Exponential failuredensity function


37




( ) Ktth =

( ) 2t

0

t

0Kt

2

1dKdh == ∫∫ ττττ

( )2Kt

2

1

Ktetf−

=

( )2Kt

2

1

etR−

=

0t ≥


Linearly Increasing Hazard Model

t

( )th

Kt

38




a. Linearly increasinghazard

t

( )th

Kt

b. Rayleigh densityfunction

( )tf

Kslope

t

K

K1

eK

K1

0slopeInitial =

( )tR

t

121e

K1t

( )tF1

21e1 −

c. Rayleigh distributionfunction

d. Rayleigh reliabilityfunction


39




Linearly Decreasing Hazard( )th

t

0K

10 KK0t

( ) =th

( )0

10

ttK

0

tKK

−

−

+∞≤<

≤<

≤<

tt

ttKK

KKt0

0

010

10

( ) =∫ ττ dht

0

( )

( )

( ) ( ) ( )20

t

t 0

t

KK

KK

0 10

1

20

t

KK

KK

0 10

210

t

0 10

ttK2

1dtKd0dKK

K

K

2

1d0dKK

tK2

1tKdKK

010

10

10

10

−=−++−

=+−

−=−

∫∫∫

∫∫

∫

τττττ

τττ

ττ


40




( ) =tf

( )

( )( )

2ttK

2

1K

K

2

1

0

tK2

1tK

10

01

20

210

eettK

0

etKK

−−−

⎟⎠⎞

⎜⎝⎛

−−

−

−

( ) =tR

( )2

ttK2

1K

K

2

1

tK2

1tK

01

20

210

ee

0

e

−−−

⎟⎠⎞

⎜⎝⎛

−−

+∞≤<

≤<

≤<

tt

ttKK

KKt0

0

010

10

+∞≤<

≤<

≤<

tt

ttKK

KKt0

0

010

10

Linearly Decreasing Hazard


41




( ) mKtth =

( ) 1mt

0

mt

0Kt

1m

1dKtdh +

+== ∫∫ τττ

( )1mKt

1m

1meKttf

+

+−

=

( )1mKt

1m

1

etR+

+−

=

1m −>

Weibull Hazard Model

( ) Kth

1

2

3

4

5

1 2→t5.0m −=

0m =5.0m =

1m =

2m =3m =


42




a. Hazard function b. Density function

( ) ( )[ ] ( )1mm

K

1m

K

tf ++

1

2

3

4

5

1 2→τ

5.0m −=0m =

5.0m =1m =2m =

3m =

( ) Kth

1

2

3

4

5

1 2→t5.0m −=

0m =5.0m =

1m =

2m =3m =

Weibull Hazard Model


43




c. Distribution function d. Reliability function

( )

t1m

K1m1

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛

+=

+

τ

( )tF

1

2

3

4

5

1 2→τ

5.0m −=0m =

5.0m =1m =2m =3m =

( )tR

1

2

3

4

5

1 2→τ( )

t1m

K1m1

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛

+=

+

τ

5.0m −=0m =

5.0m =

1m =2m =3m =

The Reliability FunctionWeibull Hazard Model

44




a. Hazard Function

b. Failure Density Function

The Bathtub Curve


45




Hazard Model for Different System

a. Mechanical b. Electrical c. Software


46





( ) ( )∫==t

0dtttftE

( )( ) ( )[ ] ( )

dt

tdR

dt

tRd

dt

tdFtf −=

−==

1

tofvalueExpectedMTTF =

( )( ) ( )∫∫∫ =−=−=

∞∞ tdttRttdRdt

dt

ttdRMTTF

000

Mean-Time-To-Failure

but

∑=

=n

1iitn

1MTTFFor a population of n components

47




• 1997: 996 DT Failures• Average of three (3) DT Failures/day• Lost Revenue during Downtime• Additional Equipment Replacement Cost• Lost of Customer Confidence

Distribution Transformer Failures

RELIABILITY ASSESSMENTof MERALCO Distribution Transformers*

* R. R. del Mundo, et. al. (1999)

� Identify the Failure Mode of DTs� Develop strategies to reduce DT failures


48




•Gather Equipment History (Failure Data)• Classify DTs (Brand, Condition, KVA, Voltage)•Develop Reliability Model•Determine Failure Mode• Recommend Solutions to Improve Reliability

METHODOLOGY: Reliability Engineering(Weibull Analysis of Failure Data)


49




Parametric Model• Shape Factor Failure Mode• Characteristic Life

Shape Factor Hazard Function Failure Mode< 1 Decreasing Early= 1 Constant Random> 1 Increasing Wear-out


50




MERALCO DTs (1989–1997)

51,1292,3381,5881,11844,341TOTAL

69----H

79----G

168----F

192----E

2,344-901162,037D

6,5612131496,358C

6,5862691351185,986B

34,7122,0481,33383529,960A

TotalConvertRewindRecondNewBrand

Note: Total Include Acquired DTs


51




Reliability Analysis: All DTs


Interval Failures Survivors Hazard200 1444 57095 0.0269400 797 48852 0.0178600 638 39997 0.0174800 508 32802 0.0167

1000 475 27515 0.01891200 363 22129 0.01781400 295 18200 0.01781600 224 14690 0.01671800 159 11865 0.01512000 89 9010 0.01142200 98 6473 0.01772400 51 4479 0.0152600 19 2254 0.01222800 2 821 0.00423000 0 127 0

52




0

0.005

0.01

0.015

0.02

0.025

0.03

0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400 2600 2800 3000

Haz

ard

Time Interval

Weibull Shape = 0.84


Reliability Analysis: All DTs

Failure Mode: EARLY FAILURE

Is it Manufacturing Defect?

53





Reliability Analysis: By Manufacturer

BRAND Size Shape Failure ModeA 34712 0.84 Early FailureB 6586 0.81 Early FailureC 6561 0.86 Early FailureD 2344 0.76 Early FailureE 192 0.85 Early FailureF 168 0.86 Early FailureG 79 0.76 Early FailureH 69 0.98 Early Failure

54




The Reliability FunctionReliability Analysis:

By Manufacturer & Condition

BRAND New Reconditioned Rewinded ConvertedA 1.11 1.23 1.12 1.4B 0.81 1.29 1.27 1.23C 0.81 1.13 0.77 0.94D 0.67 1.11 1.49 -

55




Reliability Analysis: By Voltage Rating


PRI SEC All DTs New DTs20 7.62 0.75 -20 120/240 0.79 0.9420 139/277 1.14 1.120 DUAL 0.72 1.03

13.2 120/240 0.88 1.5413.2 240/480 0.91 -7.62 120/240 0.99 1.467.62 DUAL 0.77 -4.8 120/240 0.87 1.613.6 120/240 0.78 1.172.4 120/240 1.15 -

56




Reliability Analysis: By KVA Rating (New DTs)


KVA Shape Failure Mode10 1.3 Wear-out15 1.25 Wear-out25 0.92 Early

37.5 0.83 Early50 0.73 Early75 1.05 Random100 1.04 Random167 1.16 Random250 1.11 Random333 1.46 Wear-out

57




MERALCO Distribution Transformer Reliability Analysis: Recommendations

• Review Replacement Policies

- New or Repair

- In-house or Remanufacture

• Improve Transformer Load Management Program

- Predict Demand Accurately (TLMS)

• Consider Higher KVA Ratings

• Consider Surge Protection for 20 kV DTs


58




RELIABILITY ASSESSMENTof MERALCO Power Circuit Breakers*

* R. R. del Mundo (UP) & J. Melendrez (Meralco), 2001

VOLTAGE OCB VCB GCB MOCB ACB

34.5 KV 149 160 4113.8 KV 7 28 2 36 126.24 KV 26 3 1224.8 KV 2 11TOTAL 156 216 43 39 145

Number of Feeder Power Circuit Breakers


59




Annual Failures of 34.5 kV OCBs

3 Circuit Breakers failing per year!

Preventive Maintenance Policy: Time-based (Periodic)

RELIABILITY ASSESSMENTof MERALCO Power Circuit Breakers


0.645----1155--Mechanism Failure

1.3171145314931551158Bushing Failure

1.152145114921552158Contact Wear

FailedInstalled FailedInstalledFailedInstalledFailedInstalled

Average Failures

(Units/yr)

2000199919981997Causes of Failure

60




All PCBs

34.5 kV OCBS OCBs 34.5 kV GCBs 13.8 kV MOCBs

6.24 kV MOCBs 6.24 kV ACBs

Reliability Assessment of MERALCO Power Circuit Breakers

TIME-BASED HAZARD FUNCTION

HAZARD FUNCTION CURVE FOR 34.5 KV OCBs

0

0.1

0.2

0.3

0.4

3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60

Time Interval (months)

Haz

ard

Rat

e

HAZARD FUNCTION CURVE FOR 34.5 KV GCBs

0

0.05

0.1

0.15

0.2

6 12 18 24 30 36 42 48 54 60


Haz

ard

Rat

e

HAZARD FUNCTION CURVE FOR 13.8 KV MOCBs

0

0.1

0.2

0.3

0.4

3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60


Haz

ard

Rat

e


0

0.1

0.2

0.3

0.4

3 6 9 12 15 18 24 30 36 42 48 54 60


Haz

ard

Rat

e

HAZARD FUNCTION CURVE FOR 6.24 KV ACBs

00.05

0.10.15

0.2

6 12 18 24 30 36 42 48 54 60Time Interval (months)

Ha

zard

Rat

eHAZARD FUNCTION CURVE FOR ALL PCBs CONSIDERED

00.1

0.20.30.4

3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60

Time Interval (m onths)

Haz

ard

Rat

e

.

61




TRIPPING OPERATIONS-BASED HAZARD FUNCTION

All PCBs

34.5 kV OCBS OCBs 34.5 kV GCBs 13.8 kV MOCBs

6.24 kV MOCBs 6.24 kV ACBs

Reliability Assessment of MERALCO Power Circuit Breakers


00.050.1

0.150.2

0.25

5 10 15 20 25 30 35

Tripping Interval

Haz

ard

Rat

e


0

0.01

0.02

0.03

0.04

0.05

25 50 75 100 125 150

Tripping Interval

Haz

ard

Rat

e


0

0.01

0.02

0.03

0.04

0.05

25 50 75 100 125 150

Tripping Interval

Haz

ard

Rat

e


0

0.1

0.2

0.3

5 10 15 20

Tripping Interval

Haz

ard

Rat

e


0

0.1

0.2

0.3

5 10 15 20

Tripping Interval

Haz

ard

Rat

e


00.050.1

0.150.2

0.25

5 10 15 20 25 30 35

Tripping Interval

Haz

ard

Rat

e

62




Schedule of Servicing for 41XV4

0

0.02

0.04

0.06

0.08

0 10 20 30 40 50 60 70

Number of Tripping Operations

Haz

ard

Rat

e

Reliability-BasedPreventive Maintenance Schedule

RELIABILITY ASSESSMENTof MERALCO Power Circuit Breakers


63




System Reliability Networks

Series Reliability Model

This arrangements represents a system whose subsystems or components form a series network. If any of the subsystem or component fails, the series system experiences an overall system failure.

R(x1) R(x2) R(x3) R(x4)

Series System

( )∏=

=n

iis xRR

1

64




Example:

Two non-identical cables in series are required to feed a load from the distribution system. If the two cables have constant failure rates λλλλ1 = 0.01failure/year and λλλλ2 = 0.02 failure/year. Calculate the reliability and the mean-time-to-failure for 1 year period.


65




Parallel Reliability Model

R(x1)

R(x2)

R(x3)

R(x4)

Parallel Network

This structure represents a system that will fail if and only if all the units in the system fail.

( )[ ]∏=

−−=n

iis xRR

1

11


66




Example

Supposing two identical machines are operating in a redundant configuration. If either of the machine fails, the remaining machine can still operate at the full system load. Assuming both machines to have constant failure rates and failures are statistically independent, calculate (a) the system reliability for λλλλ = 0.0005 failure/hour, t = 400 hours (operating time) and (b) the mean-time-to-failure (MTTF).


67




Standby Redundancy Model

This type of redundancy represents a distribution with one operating and n units as standbys. Unlike a parallel network where all units in the configuration are active, the standby units are not active.

R(x1)

R(x2)

R(x3)

R(x4)


68




The system reliability of the (n+1) units, in which one unit is operating and n units on the standby mission until the operating unit fails, is given by

The above equation is true if the following are true:1. The switch arrangement is perfect.2. The units are identical.3. The units failure rate are constant.4. The standby units are as good as new.5. The unit failures are statistically independent.

( )( )∑

=

−

−=n

i

ti

i

ettR

1 !1

λλ


69




In the case of (n+1) non-identical units whose failure time density functions are different, the standby redundant system failure density is given by

Consequently, the system reliability can be obtained by integrating fs(t) over the interval [t,∞∞∞∞] as follows:

( ) ( ) ( ) ( )∫ ∫ ∫− =

+ −−=t

y

y

y

y

y

nnn

n

n

n

dy...dydyytf...yyfyf...tf1

2

1 0

21112211

( ) ∫∞

=t

dt)t(ftR


70




K-Out-of-N Reliability Model

This is another form of redundancy. It is used where a specified number of units must be good for the system success.

R(x1)

R(x2)

R(x3)

The system reliability for k-out-of-n number of independent and identical units is given by

∑=

−−⎟⎟⎠

⎞⎜⎜⎝

⎛=

n

ki

inis )R(R

i

nR 1


71




Primary side

Secondary side

1

2

3

45

6

7

89

10

11

12

13

1415

1617

18

19

2021

22

23

2425

2627

2829

30

31

32

33

34

35

3637

38

39

40

4142

43

4445

46

47

4849

50

51

52

5354

55

56

57

58

Scheme 1: Single breaker-single bus(primary and secondary side)

Reliability Network Models for Typical Substation Configurations of MERALCO*


* Source: A. Gonzales (Meralco) & R. del Mundo (UP), 2005

72




15λλλλc 29λλλλct 2λλλλbus 4λλλλd1 2λλλλb1λλλλp 2λλλλb2 3λλλλd2

Summary of Substation Reliability Indices for Scheme 1

0.8287840.2471521.0Total

Opened 115kV bus tie breaker & opened 34.5kV bus tie breaker (normal condition)

0.8287840.2471521.0

Us (hr/yr)λs (failure/yr)ProbabilityEvent 1

where: λλλλs - substation failure rate or interruption frequencyUs – substation annual outage time or unavailability

Reliability Network Diagram of Single breaker-single bus scheme (Scheme 1)


73




Primary side

Secondary side

12

3

4

7

65

8

9

10 11

12

13

1415

1617

18

1920

2122

23

24

25

26

2728

2930

31

32

3334

35

36

37

38

39

40

41

42

43

44

45

4647

4849

50

51

5253

5455

61

5657

58

5960

62

6364

6566

67

68

69

7071

72

7374

7576

77

78

79

80

8182

8384

8586

8788

89

9091

92

93

94

95

96

97

98

99

100

101

102

103

104105

106107

108

109110

111112

113

114

115

116

117

118

119

120

121

122123

124125

126

127128

129

130131

132133

134

135136

137138

139

140141

L1 L2

Bank 2Bank 1Scheme 2: Single breaker-double bus (primary side) and two single breaker-single bus with bus tie breaker (secondary side)

Reliability Network Models for Typical Substation Configurations of MERALCO


74





20λλλλc 37λλλλct 3λλλλbus 5λλλλd1 2λλλλb1 λλλλp 2λλλλb2 3λλλλd2

20λλλλc 37λλλλct 2λλλλbus 3λλλλd1 2λλλλb1 λλλλp 3λλλλb2 5λλλλd2


Event 1: Opened 115kV and 34.5kV bus tie breakers; P1 = 0.997985

Event 2: Closed 115kV bus tie breaker & opened 34.5kV bus tie breaker; P2 = 0.000188

Event 3: Closed 115kV bus tie breaker & closed 34.5kV bus tie breaker; P3 = 0.000000344

Event 4: Opened 115kV bus tie breaker & closed 34.5kV bus tie breaker; P4 = 0.00182614

Substation Reliability ModelsSubstation Reliability Models

Reliability Network Diagram of Single breaker-double bus with normally opened 115kV bus tie breaker (Scheme 2)

75




0.8492750.2518661.0Total

1.0238400.3089360.0018264

1.0238400.3089360.0000003443

1.0083740.3029660.0001882

0.8489190.2517520.9979851

Us (hr/yr)λs (failure/yr)ProbabilityEvent

Summary of Substation Reliability Indices for Scheme 2

Event 1: With two primary lines energized & opened 34.5kV bus tie breaker; P1 = 0.997985


76




λλλλΒ4λλλλ54 λλλλΒ3

λλλλΒ1

λλλλΒ6λλλλΒ4

λλλλΒ3

λλλλΒ7

λλλλΒ2

λλλλΒ7


λλλλΒ4

λλλλΒ3

λλλλ29

λλλλΒ1

λλλλ17

λλλλΒ1


λλλλΒ3

λλλλ29

λλλλΒ2

λλλλΒ6


λλλλΒ5


Substation Reliability ModelsSubstation Reliability ModelsReliability Network Diagram of Single breaker-double bus with normally closed 115kV bus tie breaker (Modified Scheme 2)

Event 2: With one line, L2 interrupted & opened 34.5kV bus tie breaker; P2 = 0.000188

λλλλB1 λλλλB2 λλλλB3 λλλλB4λλλλB5λλλλ17

Event 3: With one line, L2 interrupted and closed 34.5kV bus tie breaker; P3 = 0.000000344

λλλλ29 λλλλB1 λλλλB2λλλλ17 λλλλB5λλλλB8 λλλλB9 λλλλB10 λλλλB11

77




Event 4: With two lines energized and closed 34.5kV bus tie breaker; P4 = 0.001826140

λλλλΒ1

λλλλΒ6

λλλλΒ10

λλλλΒ1

λλλλΒ7 λλλλΒ4

λλλλΒ3

λλλλ17

λλλλΒ6

λλλλΒ7



λλλλΒ4

λλλλΒ3

λλλλΒ6

λλλλΒ2

λλλλ111λλλλ29 λλλλΒ11

λλλλΒ6


λλλλΒ3

λλλλΒ8

λλλλΒ7

λλλλΒ9


λλλλΒ4

λλλλΒ3

λλλλ17

λλλλΒ7

Substation Reliability ModelsSubstation Reliability ModelsReliability Network Diagram of Single breaker-double bus with normally closed 115kV bus tie breaker (Modified Scheme 2)

0.5839230.1761941.0Total

0.7584720.2332610.0018264

1.2615490.3771200.0000003443

0.8476210.2511220.0001882

0.5835480.1760760.9979851

Us,(hr/yr)λs (failure/yr)ProbabilityEvent

Summary of Substation Reliability Indices for Modified Scheme 2

78




0.5839230.176194Modified (closed 115kV bus tie breaker)

0.8492750.251866Original (opened 115kV bus tie breaker)

Us (hr/yr)λs (failure/yr)Scheme 2

Comparison of Substation Reliability Indices for Scheme 2

Note: A remarkable 30% improvement in the performance of Scheme 2 by making the 115kV bus tie breaker normally closed.


79




Primary side

Secondary side

1

23

4

56

78

9

1011

1213

1415

16

17

18

1920

2122

2324

2526

27

2829

30

31

32

33

34

35

3637

38

3940

4142

43

44 4546

47

48

495051

5253

54

5556

5758

5960

6162

63

64

65

66

67

6869

7071

7273 74

75

7677

7879

8081

82

83

95

8485

8687

88

89 9091

92

9394

96

97

98

99

100101

102

103

104 105106

107

108109

110111

112

113114 115

116

117

118

119

120

121

122

123

124

125

126127

128

129

130

B2

B3

B5

B6

B7 B8

B10

B9

B1 B4

3 69

Bank 1 Bank 2Scheme 3: Ring bus (primary side) and two single breaker-single bus with bus tie breaker (secondary side)



80




λλλλ17

λλλλB1

λλλλB4

λλλλB1

λλλλB5

λλλλB6

λλλλB4

λλλλB2

λλλλB3

λλλλB2

λλλλB5

λλλλ51 λλλλB7 λλλλB10


Event 2: With two primary lines energized & closed 34.5kV bus tie breaker; P2 = 0.00182614

λλλλ31

λλλλB1

λλλλB4

λλλλB1

λλλλB5

λλλλB6

λλλλB4

λλλλB2

λλλλB3

λλλλB3

λλλλB6

λλλλB8 λλλλB9 λλλλ51 λλλλB10

Substation Reliability ModelsSubstation Reliability ModelsReliability Block Diagram of Ring Bus Scheme (Scheme 3)

81




Event 3: With one primary line (L2) interrupted and opened 34.5kV bus tie breaker; P3 = 0.000188056

λλλλ17

λλλλB2

λλλλB3

λλλλB2

λλλλB5

λλλλB7 λλλλ51 λλλλB10λλλλB1

λλλλB2

λλλλ31

λλλλB2

λλλλB6

Event 4: With one primary line (L2) interrupted and closed 34.5kV bus tie breaker; P4 = 0.000000344

λλλλ31

λλλλB2

λλλλB3

λλλλB3

λλλλB6

λλλλB8 λλλλB9 λλλλ51λλλλB1

λλλλB3

λλλλ17

λλλλB3

λλλλB5

λλλλB10

Substation Reliability ModelsSubstation Reliability ModelsReliability Block Diagram of Ring Bus Scheme (Scheme 3)

CONT.

82




0.4368360.1380341.0Total

0.6501140.2044670.0000003444

0.4682330.1472830.0001883

0.6183790.1951120.0018262

0.4364990.1379280.9979851


Summary of Substation Reliability Indices of Ring Bus (Scheme 3)


83




Secondary side

Primary side

1

2

34

56

78

9

10

11

12

13

14

15

16

17

18 1

920 2

122 2

324 2

628

29

30

313

2333

4 353

6 373

8394

0 414

2

119

43

44

45

79

46

474

8 49

505

1 52

53 5

455 5

65758

59 6

061

62

63 6

465

666

7686

9 707

1 72

7374

75

767

7

78

80

81

82

83

84

85

86

87

88

89

90

91

92

93

94

959

6979

8 9910

010110

2 10310

4 105

106

10710

8 10911

0 111

112

11311

4 115

116 11

7118

120

121

122

123

124

125 12

612712

812913

0 131

132

13313

4 135

136 13

7138

139

140

141

142

143

144

145

146

147

148

149

150

151

152

153

154

3

29

80

105

B1

B2

B3

B4

B5

B8

B6

B7

B9 B10

B11

25

27

L1

L2

Bank 1

Bank 2

Scheme 4: Breaker-and-a-half bus (primary side) and two single breaker-single bus with bus tie breaker (secondary side)



84




λλλλΒ1

λλλλΒ5

λλλλΒ12

λλλλΒ3

λλλλΒ4

λλλλΒ2

λλλλΒ8

λλλλ6

λλλλ17

λλλλΒ1λλλλ6

λλλλ17

λλλλ7

λλλλΒ1

λλλλΒ7

λλλλ33

λλλλΒ2

λλλλΒ3

λλλλΒ7

λλλλΒ6

λλλλΒ1

λλλλ34

λλλλΒ3

λλλλ33

λλλλΒ5

λλλλΒ2

λλλλΒ3

λλλλΒ4

λλλλΒ3

λλλλ119

λλλλΒ3

λλλλΒ7


λλλλΒ9 λλλλ62

λλλλΒ4

λλλλΒ3

λλλλΒ8

λλλλΒ3

A

Event 1: With two primary lines energized and opened 34.5kV bus tie breaker; P1 = 0.997985

λλλλΒ2

λλλλΒ5

λλλλ6

λλλλ119


λλλλΒ8

λλλλ7

λλλλΒ3

λλλλΒ5

λλλλ33

λλλλΒ2

λλλλΒ3

λλλλ17

λλλλΒ5

λλλλΒ3

λλλλ34

λλλλΒ3

λλλλ33

λλλλΒ6

λλλλΒ3

λλλλΒ5

λλλλΒ2

λλλλΒ2

λλλλ6

λλλλΒ4


λλλλ119

λλλλ7

λλλλΒ1

λλλλΒ2

λλλλ33

λλλλΒ2

λλλλΒ3

λλλλΒ7

λλλλΒ2

λλλλΒ1

λλλλ34

λλλλΒ3

λλλλ33

λλλλΒ8

λλλλΒ1

λλλλΒ2

λλλλ34

λλλλΒ3

λλλλ33

λλλλ17

λλλλΒ1

λλλλΒ4A

λλλλ33

λλλλΒ2

λλλλΒ3

λλλλ119

λλλλ17

λλλλΒ1

Substation Reliability ModelsSubstation Reliability ModelsReliability Block Diagram of Breaker-and-a-half Scheme (Scheme 4)

85




λλλλΒ1

λλλλΒ5

λλλλΒ12

λλλλΒ7

λλλλΒ8

λλλλΒ2

λλλλΒ8

λλλλ6

λλλλ17


λλλλΒ7

λλλλ7

λλλλΒ1

λλλλΒ2

λλλλ33

λλλλΒ2

λλλλΒ3

λλλλΒ7

λλλλΒ6

λλλλΒ1

λλλλ34

λλλλΒ3

λλλλ33

λλλλΒ5

λλλλΒ2

λλλλΒ3

λλλλΒ4

λλλλΒ3

λλλλ119

λλλλΒ7

λλλλΒ7


λλλλΒ11 λλλλ62

λλλλΒ4

λλλλΒ3

λλλλΒ7

λλλλΒ4

Aλλλλ139λλλλΒ10

λλλλΒ2

λλλλΒ5

λλλλ6

λλλλΒ7

λλλλ119λλλλ6

λλλλ17

λλλλ7

λλλλΒ4

λλλλΒ5

λλλλ33

λλλλΒ2

λλλλΒ3

λλλλΒ8

λλλλΒ6

λλλλΒ5

λλλλ34

λλλλΒ3

λλλλ33

λλλλΒ6

λλλλΒ4

λλλλΒ5

λλλλΒ2

λλλλΒ4

λλλλ6

λλλλΒ5


λλλλ119

λλλλ7

λλλλΒ2

λλλλΒ5

λλλλ33

λλλλΒ2

λλλλΒ3

λλλλΒ6

λλλλΒ5

λλλλΒ3

λλλλ34

λλλλΒ3

λλλλ33

λλλλΒ8

λλλλΒ2

λλλλΒ5

λλλλ34

λλλλΒ3

λλλλ33

λλλλ17

λλλλΒ3

λλλλΒ5A

λλλλ33

λλλλΒ2

λλλλΒ3

λλλλ119

λλλλ17

λλλλΒ5

λλλλΒ2

λλλλΒ5

λλλλ6

λλλλΒ6

λλλλ119

Event 2: With two primary lines energized and closed 34.5kV bus tie breaker; P2 = 0.001826


86




λλλλΒ3


λλλλΒ3

λλλλΒ8

λλλλΒ6

λλλλΒ7


λλλλΒ4

λλλλΒ3

λλλλ119

λλλλΒ6

AλλλλΒ12λλλλΒ9 λλλλ62 λλλλΒ5

λλλλΒ4

λλλλΒ3

λλλλΒ4

λλλλΒ6

λλλλΒ4

λλλλΒ3

λλλλ17

λλλλΒ7

λλλλΒ8


λλλλΒ3

λλλλΒ4

λλλλ17

λλλλ119


λλλλΒ4

λλλλΒ3

λλλλΒ4

λλλλΒ2

λλλλΒ4

λλλλΒ3

λλλλΒ2

λλλλ119

λλλλΒ4

λλλλΒ3

λλλλΒ3

λλλλ119

λλλλΒ7


λλλλΒ3

λλλλ17

λλλλΒ8

λλλλΒ3


λλλλΒ4

λλλλΒ3

λλλλΒ2

λλλλΒ8

A

Event 3: With one primary line (L1) interrupted and opened 34.5kV bus tie breaker; P3 = 0.000188


87




λλλλΒ12λλλλΒ11 λλλλ62

λλλλΒ7


λλλλΒ3

λλλλ17

λλλλΒ7

λλλλΒ7


λλλλΒ4

λλλλΒ3

λλλλΒ2

λλλλΒ7

λλλλ139λλλλΒ10

λλλλΒ7


λλλλΒ3

λλλλΒ7

λλλλ119

λλλλΒ7


λλλλΒ5

Event 4: With one primary line (L1) interrupted and closed 34.5kV bus tie breaker; P4 = 0.000000344

Summary of Substation Reliability Indices of Breaker-&-a-half (Scheme 4)

0.4355450.1374131.0Total

0.6431650.2044730.0000003444

0.4669720.1466740.0001883

0.6114330.1951200.0018262

0.4352140.1373060.9979851


Substation Reliability ModelsSubstation Reliability Models� Reliability Block Diagram of Breaker-and-a-half Scheme (Scheme 4)

88




0.4355450.137413Scheme 4 (Breaker-and-a-half bus)

0.4368360.138034Scheme 3 (Ring bus)

0.8492750.583923

0.2518660.176194

Scheme 2 (Single breaker-double bus)- with normally opened 115kV tie bkr.- with normally closed 115kV tie bkr.

0.8287840.247152Scheme 1 (Single breaker-single bus)

Us (hrs/yr)λs (failures/yr)Configuration

Comparison of Substation Reliability Indices (Scheme 1 to 4)

Note: Scheme 3 & 4 - better than Scheme 1 & 2 by 44% & 45% respectively for substation failure rates.Scheme 3 & 4 - better than Scheme 1 & 2 by 47% & 49% respectively for substation interruption duration or unavailabilty.Scheme 3 & 4 - better than Modified Scheme 2 by 22% & 25% for substation failure rates & unavailability, respectively

Substation Reliability ModelsSubstation Reliability ModelsSystem Reliability Networks

89




� Distribution System Reliability Indices

� Historical Reliability Performance Assessment

� Predictive Reliability Performance Assessment

� Substation Reliability Evaluation

Distribution System Reliability Evaluation

90




Outages, Interruptions and Reliability Indices

� Outage (Component State)Component is not available to perform its intended

function due to the event directly associated with that component (IEEE-STD-346).

� Interruption (Customer State)Loss of service to one or more consumers as a result of

one or more component outages (IEEE-STD-346).

� Types of Interruptions(a) Momentary Interruption. Service restored by

switching operations (automatic or manual) within a specified time (5 minutes per IEEE-STD-346).

(b) Sustained Interruption. An interruption not classified as momentary

91




Distribution System Reliability Indices

CUSTOMER-ORIENTED RELIABILITY INDICES

System Average Interruption Frequency Index (SAIFI)*The average number of interruptions per customer served during a period

System Average Interruption Duration Index (SAIDI)The average interruption duration per customer served during a period

servedcustomersof numberTotal

onsinterrupti-customerof numberotalTSAIFI =

servedcustomers of number Total

duration oninterrupti customerof umSSAIDI =

Note: SAIFI for Sustained interruptions. MAIFI for Momentary Interruptions

92





Customer Average Interruption Frequency Index (CAIFI)

The average number of interruptions per customer interrupted during the period

Customer Average Interruption Duration Index (CAIDI)

The average interruption duration of customers interrupted during the period

dinterrupte customers of number Total

onsinterrupti customerof number otalTCAIFI =

dinterrupte customers of number Total

duration oninterrupti customerof umSCAIDI =


93





Average Service Availability Index (ASAI)

The ratio of the total number of customer hours that service was available during a year to the total customer hours demanded

Average Service Unavailability Index (ASUI)

The ratio of the total number of customer hours that service was not available during a year to the total customer hours demanded


demanded hours Customer

serviceavailableof hoursustomerCASAI =

demanded hours Customer

serviceeunavailablof hoursustomerCASUI =

94




LOAD- AND ENERGY-ORIENTED RELIABILITY INDICES

Average Load Interruption Index (ALII)

The average KW (KVA) of connected load interrupted per year per unit of connected load served.

Average System Curtailment Index (ASCI)

Also known as the average energy not supplied (AENS). It is the KWh of connected load interruption per customer served.


loaddconnecte Total

oninterrupti load TotalALII =

servedcustomers of number Total

tcurtailmenenergy TotalASCI =

95




LOAD- AND ENERGY-ORIENTED RELIABILITY INDICES

Average Customer Curtailment Index (ACCI)

The KWh of connected load interruption per affected customer per year.


affected customers of number Total

tcurtailmenenergy TotalACCI =

96




Historical and Predictive Assessment

INCIDENTS

HISTORICALASSESSMENT

COMPONENTPERFORMANCE

PREDICTIVEASSESSMENT

ORGANIZATION,CUSTOMER, kVA

COMPONENTPOPULATION

SYSTEMDEFINITION

HISTORICAL SYSTEMPERFORMANCE

MANAGEMENTOPERATIONSENGINEERINGCUSTOMER INQUIRIES

PREDICTED SYSTEMPERFORMANCE

COMPARATIVE EVALUATIONSAID TO DECISION-MAKINGPLANNING STUDIES

Conceptual Design of an Integrated Reliability Assessment Program

97




Historical Reliability Performance Assessment

Required Data:1. Exposure Data

N - total number of customers served

P - period of observation

2. Interruption Data

Nc - number of customers interrupted on interruption i

d - duration of ith interruption, hours

Number of customers interrupted

Time

1N

1d

2N2d

3N

3d

98




L1 L2 L3

A B CSource

S 1 2 3

Load PointNumber ofCustomers

Average LoadDemand (KW)

L1 200 1000L2 150 700L3 100 400

SYSTEM LOAD DATA

InterruptionEvent i

Load PointAffected

Number ofDisconnected

Customers

Average LoadCurtailed (KW)

Duration ofInterruption

1 L3 100 400 6 hours

INTERRUTION DATA


99




yr-customeroninterrupti 222222.0

100150200

100

N

NSAIFI C

=

++==

∑∑

( )( )

yr-customerhours 333333.1

100150200

6100

N

dNSAIDI C

=

++==

∑∑

( )( )

oninterrupti-custumerhours 6

100

6100

N

dNCAIDI

C

C

=

==∑∑


100




000152.0 8760

333333.1

8760

SAIDI

8760

NdNASUI C

=

===∑∑

999848.0

000152.01ASUI1ASAI

=

−=−=

( )( )

yrcustomerKWh 333333.5

100150200

6400

N

dL

N

ENSASCI a

−=

++===

∑∑

∑

Note: ENS - Energy Not Supplied


101




Outage & Interruption Reporting


102




1 01/08/04 3 1.5 Line Fault at C2* 02/06/04 All 4 Transmission3 02/14/04 5, 6 0.5 Line Fault at D4* 03/15/04 4, 5, 6 3 Pre-arranged5 04/01/04 6 1.5 Overload6* 05/20/04 3, 4 3.5 Pre-arranged7 05/30/04 1, 2, 3 0.5 Line Tripped8 06/12/04 1 2 Line fault9 07/04/04 5 1 Line Overload

10* 07/25/04 All 5 Transmission11 07/30/04 5 1 Line Fault12* 08/15/04 4 2 Pre-arranged13 09/08/04 2 1 Line Fault14* 09/30/04 1, 2, 3 2.5 Pre-arranged15 10/25/04 3 1.5 Line Tripped16 11/10/04 2, 3 1.5 Line Fault at A17* 11/27/04 3 2 Pre-arranged18* 12/14/04 3, 4, 5 3.5 Pre-arranged19* 12/27/04 2, 3 3 Pre-arranged20 12/28/04 1, 2, 3 0.075 Line Fault


*Not included in Distribution Reliability Performance Assessment


hoursAffectedDate

103




Month 1 2 3 4 5 6 TotalJanuary 900 800 600 850 500 300 3,950February 905 796 600 855 497 303 3,956March 904 801 604 854 496 308 3,967April 908 806 606 859 501 310 3,990May 912 804 608 862 509 315 4,010June 914 810 611 864 507 318 4,024July 917 815 614 866 512 324 4,048August 915 815 620 872 519 325 4,066September 924 821 622 876 521 328 4,092October 928 824 626 881 526 331 4,116November 930 826 630 886 530 334 4,136December 934 829 635 894 538 332 4,162Annual Average 916 812 615 868 513 319 4,043



Customer Count

104




Interruption Number

Load Points

Affected

Number of Customers Affected

Duration (Hrs.)

Customer Hours

Curtailed

Frequency (Inter/Cust.)

Duration (Hrs/Cust.)

Date

1 3 600 1.5 900 0.1519 0.2278 01/08/045 497 0.5 248.5 0.1256 0.06286 303 0.5 151.5 0.0766 0.0383

5 6 310 1.5 465 0.0777 0.1165 04/01/041 912 0.5 456 0.2274 0.11372 804 0.5 402 0.2005 0.10023 608 0.5 304 0.1516 0.0758

8 1 914 2 1,828.00 0.2271 0.4543 06/12/049 5 512 1 512 0.1265 0.1265 07/04/04

11 5 512 1 512 0.1265 0.1265 07/30/0413 2 821 1 821 0.2006 0.2006 09/08/0415 3 626 1.5 939 0.1521 0.2281 10/25/04

2 826 1.5 1,239.00 0.1997 0.29963 630 1.5 945 0.1523 0.2285

11/10/0416

3

7 05/30/04

02/14/04



105






Calculate the Annual Reliability Performance of the Distribution System (according to Phil. Distribution Code)

∑∑

=N

NSAIFI C

∑∑

=N

dNSAIDI C

∑∑

=N

NMAIFI C

106




Required Data:1. Component Reliability Data

λi - failure rate of component iri - mean repair time of component i

2. System Load Data

Ni - number of customers at point iLi - the demand at point i

DistributionSystemSource

A

B Loads

C

λB, rB, UBSource

A

B Loads

C

λA, rA, UA

λC, rC, UC

Predictive Reliability Performance Assessment

107




1 2 S

1

2

P

For series combinations:

n

λλλλs = Σ λλλλii=1

n

Σ λλλλirii=1

rs = _________

λλλλs

λλλλp = λλλλ1λλλλ2 (r1 + r2)

r1 r2rp = __________

r1 + r2

For parallel combinations:


Load Point Reliability Equivalents

108




L1 L2 L3

A B CSource

S 1 2 3

Feederλλλλ

(f/year)r

(hours)A 0.2 6B 0.1 5C 0.15 8

COMPONENT DATA

Load PointNumber ofCustomers

Average LoadDemand (KW)

L1 200 1000L2 150 700L3 100 400

SYSTEM LOAD DATA


109




Load Point Reliability Equivalents

� For L1

� For L2

� For L3

yrf 0.2 A1

=

= λλ

hrs 6

rr A1

=

=( )( )

yrhrs 1.2

62.0

rU 111

=

=

= λ

yrf 0.3

0.10.2 BA2

=

+=

+= λλλ

( )( ) ( )( )

hrs 676666.5 1.02.0

51.062.0

rrr

BA

BBAA2

=

+

+=

+

+=

λλ

λλ

( )( )yrhrs 1.7

5.6666673.0

rU 222

=

=

= λ

yrf 0.45

0.150.10.2 3

=

++=

++= CBA λλλλ

( )( ) ( )( ) ( )( )

hrs 4444446 1501020

8150510620

3

....

...

rrrr

BBA

CCBBAA

=

++

++=

++

++=

λλλ

λλλ

( )( )yrhrs 9.2

6.44444445.0

rU 333

=

=

= λ

110




�Reliability Indices

( )( ) ( )( ) ( )( )

yrcustomeroninterrupti 288889.0

100150200

10045.01503.02002.0

N

NSAIFI

i

ii

−=

++

++==

∑∑λ

( )( ) ( )( ) ( )( )


100150200

1009.21507.12002.1

N

NUSAIDI

i

ii

=

++

++==

∑∑

oninterrupti-customerhours 038462.6

288889.0

744444.1

SAIFI

SAIDI

N

NUCAIDI

ii

ii

=

===∑∑

λ

111




000199.0 8760

744444.1

8760

SAIDI

8760

NNUASUI iii

=

===∑∑

999801.0

000199.01ASUI1ASAI

=

−=−=

( ) ( )( ) ( )( ) ( )( )

yr-customerKWh 888889.7

100150200

9.24007.17002.11000

N

UL

N

ENSASCI

i

iia

i

=

++

++===

∑∑

∑

112




A

B

C

D

1 2 3 4Sourcea b c d

Typical radial distribution system


113




Length (km) λλλλ (f/yr) r (hrs)

1 2 0.2 42 1 0.1 43 3 0.3 44 2 0.2 4

a 1 0.2 2b 3 0.6 2c 2 0.4 2d 1 0.2 2

Lat

eral

Component

SYSTEM RELIABILITY DATA

Mai

n

Component No. of Customers Ave. Load Connected (KW)

A 1000 5000B 800 4000C 700 3000D 500 2000

SYSTEM LOAD DATA

114




λλλλ

(f/yr)r

(hrs)

U(hrs/yr)

λλλλ

(f/yr)r

(hrs)

U(hrs/yr)

λλλλ

(f/yr)r

(hrs)

U(hrs/yr)

λλλλ

(f/yr)r

(hrs)

U(hrs/yr)

1 0.2 4 0.8 0.2 4 0.8 0.2 4 0.8 0.2 4 0.8

2 0.1 4 0.4 0.1 4 0.4 0.1 4 0.4 0.1 4 0.4

3 0.3 4 1.2 0.3 4 1.2 0.3 4 1.2 0.3 4 1.2

4 0.2 4 0.8 0.2 4 0.8 0.2 4 0.8 0.2 4 0.8

a 0.2 2 0.4 0.2 2 0.4 0.2 2 0.4 0.2 2 0.4

b 0.6 2 1.2 0.6 2 1.2 0.6 2 1.2 0.6 2 1.2

c 0.4 2 0.8 0.4 2 0.8 0.4 2 0.8 0.4 2 0.8

d 0.2 2 0.4 0.2 2 0.4 0.2 2 0.4 0.2 2 0.4

2.2 2.73 6.0 2.2 2.73 6.0 2.2 2.73 6.0 2.2 2.73 6.0

RELIABILITY INDICES FOR THE SYSTEM

Total

Load pt. A Load pt. B Load pt. C Load pt. DM

ain

Lat

eral

Componentfailure

∑∑∑∑ === λλλ Ur ;UU ; :where totaltotaltotal

115




( )( ) ( )( ) ( )( ) ( )( )

yrcustomerint 2.2

5007008001000

5002.27002.28002.210002.2

N

NSAIFI

i

ii

−=

+++

+++==

∑∑λ

( )( ) ( )( ) ( )( ) ( )( )


5007008001000

5000.67000.68000.610000.6

N

NUSAIDI

i

ii

=

+++

+++==

∑∑


2.2

0.6

SAIFI

SAIDI

N

NUCAIDI

ii

ii

=

===∑∑

λ

116




000685.0 8760

0.6

8760

SAIDI

8760

NNUASUI iii

=

===∑∑

999315.0

000685.01ASUI1ASAI

=

−=−=

( )( ) ( )( ) ( )( ) ( )( )

yr-customerKWh 0.28 5007008001000

0.620000.630000.640000.65000

N

ULASCI

i

iai

=

+++

+++=

=∑∑

117




� Effect of lateral protection

A

B

C

D

1 2 3 4Source

a b c d

Typical radial distribution system with lateral protections


118




λλλλ

(f/yr)r

(hrs)

U(hrs/yr)

λλλλ

(f/yr)r

(hrs)

U(hrs/yr)

λλλλ

(f/yr)r

(hrs)

U(hrs/yr)

λλλλ

(f/yr)r

(hrs)

U(hrs/yr)

1 0.2 4 0.8 0.2 4 0.8 0.2 4 0.8 0.2 4 0.8

2 0.1 4 0.4 0.1 4 0.4 0.1 4 0.4 0.1 4 0.4

3 0.3 4 1.2 0.3 4 1.2 0.3 4 1.2 0.3 4 1.2

4 0.2 4 0.8 0.2 4 0.8 0.2 4 0.8 0.2 4 0.8

a 0.2 2 0.4

b 0.6 2 1.2

c 0.4 2 0.8

d 0.2 2 0.4

1.0 3.6 3.6 1.4 3.14 4.4 1.2 3.33 4.0 1.0 3.6 3.6

RELIABILITY INDICES WITH LATERAL PROTECTION

Total


ain

Lat

eral

Componentfailure


119




( )( ) ( )( ) ( )( ) ( )( )

yrcustomerint 153333.1

5007008001000

5000.17002.18004.110000.1

N

NSAIFI

i

ii

−=

+++

+++==

∑∑λ

( )( ) ( )( ) ( )( ) ( )( )


5007008001000

5006.37000.48004.410006.3

N

NUSAIDI

i

ii

=

+++

+++==

∑∑


153333.1

906667.3

SAIFI

SAIDI

N

NUCAIDI

ii

ii

=

===∑∑

λ

120




000446.0 8760

906667.3

8760

SAIDI

8760

NNUASUI iii

=

===∑∑

999554.0

000446.01ASUI1ASAI

=

−=−=

( )( ) ( )( ) ( )( ) ( )( )

yr-customerKWh 266667.18 5007008001000

6.320000.430004.440006.35000

N

ULASCI

i

iai

=

+++

+++=

=∑∑

121




� Effect of disconnects

A

B

C

D

1 2 3 4Source

a b c d

Typical radial distribution system reinforce withlateral protections and disconnects


122




λλλλ

(f/yr)r

(hrs)

U(hrs/yr)

λλλλ

(f/yr)r

(hrs)

U(hrs/yr)

λλλλ

(f/yr)r

(hrs)

U(hrs/yr)

λλλλ

(f/yr)r

(hrs)

U(hrs/yr)

1 0.2 4 0.8 0.2 4 0.8 0.2 4 0.8 0.2 4 0.8

2 0.1 0.5 0.05 0.1 4 0.4 0.1 4 0.4 0.1 4 0.4

3 0.3 0.5 0.15 0.3 0.5 0.15 0.3 4 1.2 0.3 4 1.2

4 0.2 0.5 0.1 0.2 0.5 0.1 0.2 0.5 0.1 0.2 4 0.8

a 0.2 2 0.4

b 0.6 2 1.2

c 0.4 2 0.8

d 0.2 2 0.4

1.0 1.5 1.5 1.4 1.89 2.65 1.2 2.75 3.3 1.0 3.6 3.6

RELIABILITY INDICES WITH LATERAL PROTECTION AND DISCONNECTS

Total


ain

Lat

eral

Componentfailure


123




( )( ) ( )( ) ( )( ) ( )( )

yrcustomerint 153333.1

5007008001000

5000.17002.18004.110000.1

N

NSAIFI

i

ii

−=

+++

+++==

∑∑λ

( )( ) ( )( ) ( )( ) ( )( )


5007008001000

5006.37003.380065.210005.1

N

NUSAIDI

i

ii

=

+++

+++==

∑∑


153333.1

576667.2

SAIFI

SAIDI

N

NUCAIDI

ii

ii

=

===∑∑

λ

124




000294.0 8760

576667.2

8760

SAIDI

8760

NNUASUI iii

=

===∑∑

999706.0

000294.01ASUI1ASAI

=

−=−=

( )( ) ( )( ) ( )( ) ( )( )

yr-customerKWh 733333.11 5007008001000

6.320003.3300065.240005.15000

N

ULASCI

i

iai

=

+++

+++=

=∑∑

125




� Effect of protection failures

λλλλ

(f/yr)r

(hrs)

U(hrs/yr)

λλλλ

(f/yr)r

(hrs)

U(hrs/yr)

λλλλ

(f/yr)r

(hrs)

U(hrs/yr)

λλλλ

(f/yr)r

(hrs)

U(hrs/yr)

1 0.2 4 0.8 0.2 4 0.8 0.2 4 0.8 0.2 4 0.8

2 0.1 0.5 0.05 0.1 4 0.4 0.1 4 0.4 0.1 4 0.4

3 0.3 0.5 0.15 0.3 0.5 0.15 0.3 4 1.2 0.3 4 1.2

4 0.2 0.5 0.1 0.2 0.5 0.1 0.2 0.5 0.1 0.2 4 0.8

a 0.2 2 0.4 0.02 0.5 0.01 0.02 0.5 0.01 0.02 0.5 0.01

b 0.06 0.5 0.03 0.6 2 1.2 0.06 0.5 0.03 0.06 0.5 0.03

c 0.04 0.5 0.02 0.04 0.5 0.02 0.4 2 0.8 0.04 0.5 0.02

d 0.02 0.5 0.01 0.02 0.5 0.01 0.02 0.5 0.01 0.2 2 0.4

1.12 1.39 1.56 1.48 1.82 2.69 1.3 2.58 3.35 1.12 3.27 3.66

RELIABILITY INDICES IF THE FUSES OPERATE WITH PROBABILITY OF 0.9

Total


ain

Lat

eral

Componentfailure


126




� Effect of load transfer to alternative supply

λλλλ

(f/yr)r

(hrs)

U(hrs/yr)

λλλλ

(f/yr)r

(hrs)

U(hrs/yr)

λλλλ

(f/yr)r

(hrs)

U(hrs/yr)

λλλλ

(f/yr)r

(hrs)

U(hrs/yr)

1 0.2 4 0.8 0.2 0.5 0.1 0.2 0.5 0.1 0.2 0.5 0.1

2 0.1 0.5 0.05 0.1 4 0.4 0.1 0.5 0.05 0.1 0.5 0.05

3 0.3 0.5 0.15 0.3 0.5 0.15 0.3 4 1.2 0.3 0.5 0.15

4 0.2 0.5 0.1 0.2 0.5 0.1 0.2 0.5 0.1 0.2 4 0.8

a 0.2 2 0.4

b 0.6 2 1.2

c 0.4 2 0.8

d 0.2 2 0.4

1.0 1.5 1.5 1.4 1.39 1.95 1.2 1.88 2.25 1.0 1.5 1.5

RELIABILITY INDICES WITH UNRESTRICTED LOAD TRANSFERS

Total

Load pt. A Load pt. B Load pt. C Load pt. D

Mai

nL

ater

al

Componentfailure


127




λλλλ

(f/yr)r

(hrs)

U(hrs/yr)

λλλλ

(f/yr)r

(hrs)

U(hrs/yr)

λλλλ

(f/yr)r

(hrs)

U(hrs/yr)

λλλλ

(f/yr)r

(hrs)

U(hrs/yr)

1 0.2 4 0.8 0.2 1.9 0.38 0.2 1.9 0.38 0.2 1.9 0.38

2 0.1 0.5 0.05 0.1 4 0.4 0.1 1.9 0.19 0.1 1.9 0.19

3 0.3 0.5 0.15 0.3 0.5 0.15 0.3 4 1.2 0.3 1.9 0.57

4 0.2 0.5 0.1 0.2 0.5 0.1 0.2 0.5 0.1 0.2 4 0.8

a 0.2 2 0.4

b 0.6 2 1.2

c 0.4 2 0.8

d 0.2 2 0.4

1.0 1.5 1.5 1.4 1.59 2.23 1.2 2.23 2.67 1.0 2.3 2.3

RELIABILITY INDICES WITH RESTRICTED LOAD TRANSFERS

Total

Load pt. A Load pt. B Load pt. C Load pt. D

Sec

tion

Dis

trib

uto

r

Componentfailure


� Effect of load transfer to alternative supply

128




Case 1 Case 2 Case 3 Case 4 Case 5 Case 6

λ (f/yr) 2.2 1.0 1.0 1.12 1.0 1.0r (hrs) 2.73 3.6 1.5 1.39 1.5 1.5U (hrs/yr) 6.0 3.6 1.5 1.56 1.5 1.5

λ (f/yr) 2.2 1.4 1.4 1.48 1.4 1.4r (hrs) 2.73 3.14 1.89 1.82 1.39 1.59U (hrs/yr) 6.0 4.4 2.65 2.69 1.95 2.23

λ (f/yr) 2.2 1.2 1.2 1.3 1.2 1.2r (hrs) 2.73 3.33 2.75 2.58 1.88 2.23U (hrs/yr) 6.0 4 3.3 3.35 2.25 2.67

λ (f/yr) 2.2 1.0 1.0 1.12 1.0 1.0r (hrs) 2.73 3.6 3.6 3.27 1.5 2.34U (hrs/yr) 6.0 3.6 3.6 3.66 1.5 2.34

SUMMARY OF INDICES

Load Point A

Load Point B

Load Point C

Load Point D

129




Case 1 Case 2 Case 3 Case 4 Case 5 Case 6

SAIFI 2.2 1.15 1.15 1.26 1.15 1.15SAIDI 6.0 3.91 2.58 2.63 1.80 2.11CAIDI 2.73 3.39 2.23 2.09 1.56 1.83ASAI 0.999315 0.999554 0.999706 0.999700 0.999795 0.999759ASUI 0.000685 0.000446 0.000294 0.003000 0.000205 0.000241ENS 84.0 54.8 35.2 35.9 25.1 29.1ASCI 28.0 18.3 11.7 12.0 8.4 9.7

Case 3. As in Case 2, but with disconnects on the main feeders.Case 4. As in Case 3, probability of successful lateral distributor fault clearing of 0.9.Case 5. As in Case 3, but with an alternative supply.Case 6. As in Case 5, probability of conditional load transfer of 0.6.

Sytem Indices

SUMMARY OF INDICES (cont.)

Case 1. Base case.Case 2. As in Case 1, but with perfect fusing in the lateral distributors.

130




� Impact of Power Interruptions

� Reliability Worth

� Optimal Power System Reliability

Economics of Power System Reliability

131




� To Electric Utility• Loss of revenues• Additional work• Loss of confidence

� To Customers• Dissatisfaction• Interruption of productivity• Additional investment for alternative

power supply

� To National Economy• Loss value added/income• Loss of investors• Unemployment

Impact of Power Interruptions

132




Impact to National Economy:�NEDA Study (1974)

� P 342,380 per day – losses due to brownout in Cebu-Mandauearea

�Business Survey (1980)� P1.4 Billion – losses due to brownouts in 1980

�CRC Memo No. 27 (1988)� P 3.4 Billion – loss of the manufacturing sector in 1987 due to

power outages

�Viray & del Mundo Study (1988)� P 25 – losses in Value Added per kWh curtailment

�Sinay Report (1989)� 45% – loss in Value Added in the manufacturing sector in

Cebu due to power outages


133




Impact to Customers:A. Short-Run Direct Cost

• Opportunity losses during outages• Opportunity losses during restart period• Raw materials spoilage• Finish products spoilage• Idle workers• Overtime• Equipment damage• Special operation and maintenance during restart period

B. Long-Run Adaptive Response Cost• Standby generators• Power plant• Alternative fuels• Transfer location• Inventory


134




(0.0086 + 0.0023D)F + 0.1730 Pesos/kWh

Source: del Mundo (1991)

Outage Cost to Industrial Sector in Luzon

Where, F – Frequency of Interruptions

D – Average Duration of Interruptions

Reliability Worth

Losses of MERALCO Industrial Customers in 1989

Energy Sales: 3.781 billion kWhOutage Cost: Php 0.3544/kWhTotal Losses: Php 1.34 billion

135




Reliability Worth

136




Reliability Worth

0.181.310.730.08

0.181.300.310.04

0.201.3820.21

0.221.5040.45

0.261.6170.94

0.341.73131.88

0.682.00386.25

1.122.117012.26

Outage Cost(Php/kWh)

Duration(Hours)

Frequency(per year)

LOLP(days/yr)

Luzon Grid Outage Cost*


137




Reliability Worth

Luzon Grid Outage Cost


138




Reliability Worth

ATC = ASC + AOC

1.291.311.110.08

1.321.301.140.04

1.291.381.090.21

1.281.501.060.45

1.291.611.030.94

1.351.731.011.88

1.622.000.946.25

2.022.110.9012.26

Total Cost(Php/kWh)

Outage Cost(Php/kWh)

Supply Cost(Php/kWh)

LOLP(days/yr)

Luzon Grid Total Cost


139




Optimal Power System Reliability


Luzon Grid Total Cost

140




Reliability Analysis

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