2 hli analytical

8/12/2019 2 HLI Analytical

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HL-I Adequacy Evaluation(Analytical Approach)

B. Bagen

Section Outl ine

Objective and System Representation

Deterministic Assessment

Probabilistic Evaluation

Analytical Approach

Single Area Reliability Analysis

n erconnec e ys em e a y na ys s


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Objective and System Representation(A small representative system: RBTS)

1X40 MW4x20 MW2x5 MW

Bus 2

2

Capacity=

240 MW

Peak Load=

185 MW

2x40 MW

1x20 MW

1x10 MW

20 MW

1 6

3

2 7

4

Bus 1

Bus 4Bus 3

1

20 MW

20 MW

40 MW85 MW5 8

9Bus 5

Bus 6

Objective and System Representation

Main Ob ective: To determine the s stem

resource required to satisfy the system

demand

System representation:

Total

TotalSystem

Generation

System

Load


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Basic Evaluation Model

Generation Model Load Model

Risk Model

Analytical Approach

(Basic Models)

Generation System Model

Load Model

Risk Model (Margin Model)

Other Features can be included by: Modifying generation model: energy limited units, intermittent

generation, Unit derated states, external assistances

Adjusting load model: load forecast uncertainty, interruptible

load, contracts, demand side management


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System reserve margin/capacity isdetermined based on bein able to su lthe forecast peak load with a specifiednumber of units out of service.

Indices Percent Reserve (PR):

Reserve Margin: %100

PL

PLICRM

Reserve Capacity: RC=IC-PL=RMxPL

Loss of the Largest Unit (LOLU):

Reserve Capacity:

Combination of the Two

CLURC


(Pros and Cons)

Easy to apply and understand

PR: Does not incor orate an individual eneratin unitreliability data or load variation information

LOLU: Considers the size of the largest unit but does notrecognize the system risk due to an outage of one or moregenerating units. The system reserve increases with theaddition of larger units to the system.

In Summar deterministic method not reco nize andreflect uncertainties associated with power systems(Review example on Page 28 of Billinton and Allanbook)


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Probabilistic Evaluation

System reserve margin/capacity is set to keep

demand below a specified value considering

variabilities and uncertainties associated with

both system resource and demand.

Most commonly used indices:

Loss of Load Probability (LOLP)

Loss of Load Expectation (LOLE) Loss of Energy Expectation (LOEE)

Analytical Approach

(Generation Model: Individual Unit)

Repairable Forced Failure (Two State Representation)

d

r

Time (years)

d+r

Up

Down

r

1

rdf

1

d

1


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Analytical Approach



MTBF

Up

MTTF

MTTRTime (hours)

Down

Analytical Approach



11

MTTRr

8760

MTTFd Note: and f are two

completely differentquantities. In mostcases MTTR is ,however, a very small

8760

8760

11

MTTRMTTFrdf

MTTF and therefore and f are numericallyclose


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Analytical Approach


State Probabilities

AP0

FORUP

Up Down

0 1

MTTRMTTF

MTTR

MTTR

U

8760

Analytical Approach


is typically a function of time (A bath tub curve )

Infant Period Wear-out Period

Normal Operating Period

)(t

It is fairly common to assume constant transitionrates in reliability modeling.

Time


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Analytical Approach


Repairable Forced Failure (Multi-State Representation)

Operating

Partial

RepairDerated

Partial

Failure

Complete

FailureComplete

Repair

Capacity(MW)

Failed

Time Period

Analytical Approach


Three State Representation

Up

121

12

31

13

32 23

ya a1

DFORP 2

2 3 FORP 3


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Analytical Approach

(Generation Model: System)

System Capacity Outage Probability Tables (COPT)

Capacity Out

(MW)

Capacity In

(MW)

Probability Cumulative

Probability

0 150 0.970299 1

50 100 0.029403 0.029701

100 50 0.000297 0.000298

150 0 0.000001 0.000001

Can be developed using several different

methodologies Discuss: state enumeration and recursive methods

Analytical Approach

(Load Model) Daily Peak Load Variation Curve (DPLVC): The

cumulative load model formed by arranging theindividual dail eak loads in descendin order.

Chronological Daily Load

0.4

0.6

0.8

1

1.2

di

nP

ercentofPeak

Daily Peak Load Variation Curve

0.6

0.8

1

1.2

n

PercentofPeak

0

0.2

1 61 121 181 241 301 361

Day

Loa

0

0.2

.

1 61 121 181 241 301 361

Day

Load

i


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Analytical Approach

(Load Model)

Load Duration Curve (LDC): The cumulative load modelformed by arranging the individual hourly load values in

escen ng or er. n s case, e area un er ecurve represents the energy required by the system in

a given period.Chronological Hourly Load

0.8

1

1.2

fPeak

Load Duration Curve

1

1.2

ak

0

0.2

0.4

0.6

1

801

1601

2401

3201

4001

4801

5601

6401

7201

8001

Hour

Loa

d

in

Percent

0

0.2

0.4

0.6

0.8

1

801

1601

2401

3201

4001

4801

5601

6401

7201

8001

Hour

Load

in

PercentofPe

Analytical Approach(Convolution of Generation wi th DPLVC)

Installed Capacity

W

kt

Peak Load

kO

DPLVC

LoadorCaacit(kW/M

Time Period (days)

valueactualis),/(

valueunitperis,

1 k

kn

k

kktifyeardaysinLOLE

tifLOLPtp


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Analytical Approach(Convolution of Generation with LDC)

kt

kE

Peak Load

Reserve

kO

LDC

orCapacity(kW/MW)

Time Period (hours)

Load

Analytical Approach(Convolution of Generation with LDC)

valueunitperis, kn tifLOLP

t

n

k

kk EpLOEE1

n

k

kk

E

EpEIR

1

1

va ueactuas,1 kk tifyearhoursinLOLE

610E

LOEEUPM60

PL

LOEESM

orma ze :


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Analytical Approach(A Simple Example)

1 Generation: 3 generators,

2 Load

3

identical FOR of 0.01 and

each generator fails

independently.

Develop: COPT0.2050

ProbabilityLoad (MW)

0.2050

ProbabilityLoad (MW)

a cu a e: oss o oa

probability (LOLP) andLOLE for the given load

distribution

0.05150

0.75100

0.05150

0.75100

Analytical Approach

(State Enumeration: System State Space)

11 11 11 11

2 oa

3

2 oa

3

2 oa

3

2 oa

3

2 oa

3

2 oa

3

2 oa

3

2 oa

3

Load

1Load

1Load

1Load

1Load

1Load

1Load

1Load

1

(1U,2U,3U) (1D,2U,3U) (1U,2D,3U) (1U,2U,3D)

33 33 33 33

(1D,2D,3U) (1D,2U,3D) (1U,2D,3D) (1D,2D,3D)


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Analytical Approach(Capacity Outage Probability Table)

Capacity Out Capacity In Individual Cumulative

(MW) (MW) Probability Probability

0 150 0.970299 1

50 100 0.029403 0.029701

100 50 0.000297 0.000298

100 0 0.000001 0.000001

Analytical Approach

(Load: 50 MW)

2 50

1

2 50

1

2 50

1

2 50

1

2 50

1

2 50

1

2 50

1

2 50

1

2 50

3

1

2 50

3

1

2 50

3

1

2 50

3

1

2 50

3

1

2 50

3

1

2 50

3

1

2 50

3

1

150 100 100 100(1U,2U,3U) (1D,2U,3U) (1U,2D,3U) (1U,2U,3D)

50 050 50(1D,2D,3U) (1D,2U,3D) (1U,2D,3D) (1D,2D,3D)

P{ (1D,2D,3D)} = 0.000001


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Analytical Approach(Load: 100 MW)

2 100

1

2 100

1

2 100

1

2 100

1

2 100

1

2 100

1

2 100

1

2 100

1

2 100

3

1

2 100

3

1

2 100

3

1

2 100

3

1

2 100

3

1

2 100

3

1

2 100

3

1

2 100

3

1

150 100 100 100(1U,2U,3U) (1D,2U,3U) (1U,2D,3U) (1U,2U,3D)

50 050 50(1D,2D,3U) (1D,2U,3D) (1U,2D,3D) (1D,2D,3D)

P{(1D,2D,3U),(1D,2U,3D),(1U,2D,3D), (1D,2D,3D)}

= 0.000297 + 0.000001 = 0.000298

Analytical Approach(Load: 150 MW)

2 150

1

2 150

1

2 150

1

2 150

1

2 150

1

2 150

1

2 150

1

2 150

1

33 33 33 33

2 150

1

2 150

1

2 150

1

2 150

1

2 150

1

2 150

1

2 150

1

2 150

1

150 100 100 100(1U,2U,3U) (1D,2U,3U) (1U,2D,3U) (1U,2U,3D)

50 050 50(1D,2D,3U) (1D,2U,3D) (1U,2D,3D) (1D,2D,3D)

P = 1 - P{ (1U,2U,3U) } = 0.029701


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Analytical Approach( Calculation of LOLP and LOLE)

LOLP = 0.0000010.20 + 0.0002980.75 +

0.0297010.05= 0.00170875

LOLE = 0.00170875*365=0.6237 days/year

Analytical Approach(Approximations)

Grouping

Rounding

Truncating

Continuous Approximations


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Analytical Approach

(Recursive Algorithm)

Capacity Out Individual Probability Cumulative Probability

X1 p1 1

X2 p2 P2 = pn +pn-1 ++p2

Xi pi Pi = pn +pn-1 ++pi

Xn-1 pn-1 Pn-1 = pn +pn-1

Xn pn Pn =pn

Analytical Approach(Recursive Algorithm)

X MW can be calculated as follows:

n

ii CXPpXP )(')(

i 1


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Analytical Approach(Recursive Algorithm)

n: number of unit state

i

pi: Probability of existence of the unit state i

P(X): the cumulative probabilities of a capacity outage level of

X MW before the unit of capacity C is added. For initialization:

P(X)=1 if X 0 and otherwise P(X)=0.

P(X): the cumulative probabilities of a capacity outage level of XMW afterthe unit of capacity C is added.

Analytical Approach

(Recursive Algori thm)

Step 1: Add the first unit

Capacity Out

(MW)

Capacity In

(MW)

Individual

Probability

Cumulative

Probability

= . - . - = . . =

P(50)=0.99*P(50-0)+0.01*P(50-50)=0.99*0+0.01*1=0.01

.50 0 0.01 0.01


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Analytical Approach


Step 2: Add the second unitP 0 =0.99*P 0-0 +0.01*P 0-50 =0.99*1+0.01*1=1

Capacity Out

(MW)

Capacity In

(MW)

Individual

Probability

Cumulative

Probability

P(50)=0.99*P(50-0)+0.01*P(50-50)=0.99*0.01+0.01*1=0.0199

P(100)=0.99*P(100-0)+0.01*P(100-50)=0.99*0+0.01*0.01=0.0001

.

50 50 0.0198 0.0199100 0 0.0001 0.0001

Analytical Approach


Step 3: Add the third unit

P(0)=0.99*P(0-0)+0.01*P(0-50)=0.99*1+0.01*1=1

P(50)=0.99*P(50-0)+0.01*P(50-50)=0.99*0.0199+0.01*1=0.029701

P(100)=0.99*P(100-0)+0.01*P(100-50)=0.99*0.0001+0.01*0.0199=0.000298

P(150)=0.99*P(150-0)+0.01*P(150-50)=0.99*0+0.01*0.0001=0.000001


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Analytical Approach


Ste 3 becomes:

P(0)=0.96*P(0-0)+0.033*P(0-20)+ 0.007*P(0-50)

P(20)= 0.96*P(20-0)+0.033*P(20-20)+ 0.007*P(20-50)

P(50)= 0.96*P(50-0)+0.033*P(50-20)+ 0.007*P(50-50)

P(70)= 0.96*P(70-0)+0.033*P(70-20)+ 0.007*P(70-50)

= . - + . - + . -

P(120)= 0.96*P(120-0)+0.033*P(120-20)+ 0.007*P(120-50)

P(150)= 0.96*P(150-0)+0.033*P(150-20)+ 0.007*P(150-50)

Analytical Approach(Recursive Algor ithm)

The cumulative probability of each sate is:

P(0)=0.96*1+0.033*1+ 0.007*1=1

P(20)= 0.96*0.0199+0.033*1+ 0.007*1=0.0591040

P(50)= 0.96*0.0199+0.033*0.0199+ 0.007*1=0.0267607

P(70)= 0.96*0.0001+0.033*0.0199+ 0.007*0.0199=0.000892

P(100)= 0.96*0.0001+0.033*0.0001+ 0.007*0.0199=0.0002386

P(120)= 0.96*0+0.033*0.0001+ 0.007*0.0001=0.000004

P(150)= 0.96*0+0.033*0+ 0.007*0.0001=0.0000007


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Analytical Approach


Step 3:Add the third unit

Capacity Out

(MW)

Capacity In

(MW)

Individual

Probability

Cumulative

Probability

0 150 0.9408960 1

20 130 0.0323433 0.0591040

50 100 0.0258687 0.0267607

. .

100 50 0.0002346 0.0002386

120 30 0.0000033 0.0000040

150 0 0.0000007 0.0000007

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