ee353 notes no. 2 - lole&loee

Department of Electrical and Electronics Engineering University of the Philippines - Diliman

EE 353 - Power System Reliability

Reliability of Generation SystemGenerating Capacity LOLE & LOEE

Prof. Rowaldo R. del Mundo

University of the Philippines

Department of Electrical and Electronics Engineering 2Prof. Rowaldo R. del Mundo

Conceptual Task & Generation System Representation

G LTotal System Generation Total System Load

Generation Model

Load Model

Risk Model



Generating UnitForced Outage Rate

If the hazard rate of a generating unit is constant

The failure density function is exponential

( ) =th

( ) ( ) ( )= t

0dh

ethtf

=

t

0d

e

( ) tetf =



Generating Unit Forced Outage Rate

The cumulative failure distribution function is

Then, the reliability function is

( ) ( ) == t0t

0dedftF

[ ]0tt0

eee

==

( ) = e1tF

( ) ( )tF1tR =[ ]te11 =

( ) tetR =




The mean-time-to-failure is

( )= 0 dttRMTTF

[ ]=0

te

1

m1MTTF == (mean up time)

=0

tdte

[ ]0ee1 =




Also, if the repair rate function is constant

the mean-time-to-repair is

The mean-time-between-failures

and the cycle frequency is

( ) =tr

rMTTR ==1 (mean repair time)

timerepairmeanuptimemeanMTBF +=MTTRMTTF +=

TrmMTBF =+=

T1f =




The Availability is the steady-state or long term probability that the generating unit is in operating condition (UP state)

time cycletime up meanA =

f

Tm

==A

+

=

+=

rm

mA

[ ][ ] [ ]

+

=

time uptime downtime up

A

time repair mean time up meantime up meanA

+=




The Unavailability of a generating unit is the probability of finding a unit on forced outage at some future time. This is technically termed as Forced Outage Rate or FOR.

f

TrU ==

+

=

+=

rm

rU

[ ][ ] [ ]

+=

time uptime downtime down

U

tyAvailabili1U =



Markov Modelsof Generation System

Up

Down

Single Generating Unit

State Generator Rate ofDepartureRate ofEntry

1 UP 2 DOWN



Markov Models of Generation System

Two Generating Units

2

4

3

1

State Gen 1 Gen 2 Rate ofDepartureRate ofEntry

1 UP UP 1 + 2 1 + 22 DOWN UP 1 + 2 1 + 23 UP DOWN 1 + 2 1 + 24 DOWN DOWN 1 + 2 1 + 2



Generating Capacity Model

(Capacity Outage Probability Table)

A. IDENTICAL UNITSFor identical units, binomial distribution is applicable. The probability of exactly r successes in n trials is

( ) ( )rnr

r p1p!rn!r!nP

=




EXAMPLE 1:2 10 MW units each with 0.03 forced outage rate

0.970.03-1p 2n ===

( ) ( ) ( ) 0009.097.0197.0!02!0!2P 0200 =

=

( ) ( ) ( ) 0582.097.0197.0!12!1!2P 1211 =

=

( ) ( ) ( ) 9409.097.0197.0!22!2!2P 2222 =

=

(probability that exactly two generating unit are available)

(probability that no generating unit is available)

(probability that exactly one generating unit is available)




The capacity outage table is summarized as follows:Units Out Capacity Out Capacity Available Probability

0 0 MW 20 MW 0.94091 10 MW 10 MW 0.05822 20 MW 0 MW 0.0009

1.0000

Note: The probability of all possible cases must sum to unity




EXAMPLE 2:3 5 MW units each with 0.03 forced outage rate

0.970.03-1p 3n ===

( ) ( ) ( ) 000027.097.0197.0!03!0!3P 0300 =

=

( ) ( ) ( ) 002619.097.0197.0!13!1!3P 1311 =

=

( ) ( ) ( ) 084681.097.0197.0!23!2!3P 2322 =

=

( ) ( ) ( ) 912673.097.0197.0!33!3!3P 3333 =

=




The capacity outage table is summarized as follows:Units Out Capacity Out Capacity Available Probability

0 0 MW 15 MW 0.9126731 5 MW 10 MW 0.0846812 10 MW 5 MW 0.0026193 15 MW 0 MW 0.000027

1.000000




For units with 0.03 forced outage rate313 - 4

Units Out Capacity Out Capacity Available Probability0 0.00 MW 13.33 MW 0.885292811 3.33 MW 10.00 MW 0.109520762 6.66 MW 6.66 MW 0.005080863 10.00 MW 3.33 MW 0.000104764 13.33 MW 0.00 MW 0.00000081

1.000000




EXAMPLE:A system consist of two 3 MW units and one 5 MWunit each with forced outage rate of 0.02.

State Gen. 1 Gen. 2 Gen. 3 Capacity In Capacity Out1 UP UP UP 3+3+5 = 11 MW 0 MW2 DOWN UP UP 0+3+5 = 8 MW 3 MW3 UP DOWN UP 3+0+5 = 8 MW 3 MW4 UP UP DOWN 3+3+0 = 6 MW 5 MW5 DOWN DOWN UP 0+0+5 = 5 MW 6 MW6 UP DOWN DOWN 3+0+0 = 3 MW 8 MW7 DOWN UP DOWN 0+3+0 = 3 MW 8 MW8 DOWN DOWN DOWN 0+0+0 = 0 MW 11 MW

Markov Transition States (23 = 8)`

B. NON-IDENTICAL UNITS Binomial distribution is not applicable Basic probability concepts can be applied




Note: The probability of exactly 3 MW is out of service is the sum of the probabilities of states 2 and 3. Similarly, the probability of exactly 8 MW is out of service is the sum of the probabilities of states 6and 7. Therefore states 2 and 3 can be combined and states 6 and 7 can be combined to come up with the reduced states capacity probability table.

New State Capacity In Capacity Out Probability1 11 MW 0 P12 8 MW 3 P2 + P33 6 MW 5 P44 5 MW 6 P55 3 MW 8 P6 + P76 0 MW 11 P8




Step 1: Combine the two identical units(binomial distribution may be applied)

Step 2: Add the 5 MW units considering that it can exist in two state (in service or out of service)

Capacity Out Capacity In Probability0 MW 6 MW 0.96043 MW 3 MW 0.03926 MW 0 MW 0.0004

1.0000

02.0P98.0 P

serviceof out

servicein

=

=




Step 2a: 5 MW unit in service

Step 2b: 5 MW unit out of service

0 + 0 = 0 MW 6 + 5 = 11 MW ( 0.9604 ) ( 0.98 ) = 0.9411923 + 0 = 3 MW 3 + 5 = 8 MW ( 0.0392 ) ( 0.98 ) = 0.0384166 + 0 = 6 MW 0 + 5 = 5 MW ( 0.0004 ) ( 0.98 ) = 0.000392

0.980000

Capacity Out Capacity In Probability

0 + 5 = 5 MW 6 + 0 = 6 MW ( 0.9604 ) ( 0.02 ) = 0.0192083 + 5 = 8 MW 3 + 0 = 3 MW ( 0.0392 ) ( 0.02 ) = 0.0007846 + 5 = 11 MW 0 + 0 = 0 MW ( 0.0004 ) ( 0.02 ) = 0.000008

0.020000

Capacity Out Capacity In Probability




Step 2c: Combine probability tables in steps 2a and 2bCapacity Out Capacity In Probability

0 MW 11 MW 0.9411923 MW 8 MW 0.0384165 MW 6 MW 0.0192086 MW 5 MW 0.0003928 MW 3 MW 0.00078411 MW 0 MW 0.000008

1.000000




Cumulative Probability TableAn additional column can be added to the capacity outage probability table which gives the cumulative probability. This is the probability of finding a quantity of capacity on outage equal or greater than the indicated amount.

Capacity Out of Service Individual Probability Cumulative Probability0 MW 0.941192 1.0000003 MW 0.038416 0.0588085 MW 0.019208 0.0203926 MW 0.000392 0.0011848 MW 0.000784 0.00079211 MW 0.000008 0.000008

1.000000( ) 118653 PPPPP3 out capP ++++=( ) 11865 PPPP5 out capP +++=




Truncating the Probability Table

Theoretically, the capacity outage probability table incorporates all the system capacity. The table, however, can be truncated by omitting all capacity outages for which the cumulative probability is less than a specified amount, e.g., 10-8.




Capacity Rounding Probability TableIn a system containing a large number of units of different capacities, the table will contain several hundred positive discrete capacity outage levels. This number can be greatly reduced by grouping the units into identical capacity groups prior to combining or by rounding the table to discrete levels after combining.

The rounding process is done by calculating

For all states i falling between the required rounding states j and k.

( ) ( )ijk

ikj CPCC

CCCP

= ( ) ( )ijk

jik CPCC

CCCP

=




EXAMPLE:Obtain the rounded probability table of the generation system in the previous example rounded in 5 MWincrements.

For Capacity Out of Service Ci Cj Ck P(Ci) P(Cj) P(Ck)

0 MW 0 MW 0.941192 0.9411920 0.00000003 MW 3 MW 0 MW 5 MW 0.038416 0.0153665 0.02304975 MW 5 MW 0.019208 0.0192080 0.00000006 MW 6 MW 5 MW 10 MW 0.000392 0.0003136 0.00007848 MW 8 MW 5 MW 10 MW 0.000784 0.0003136 0.0004704

(Exact)

(Exact)

Capacity on Outage (MW)

0 MW 0.941192 + 0.0153664 = 0.95655845 MW 0.0230496 + 0.019208 + 2(0.0003136) = 0.0428848

10 MW 0.0000784 + 0.0004704 + 0.0000064 = 0.000555215 MW 0.0000016 = 0.0000016

1.0000000

Individual Probability

The final Rounded Probability Table is:




Recursive Algorithm for Capacity Model Building Addition in Probability Table one generating unit at a

time Suitable for computerized computations Cases

No Derated States Derated States Included Unit Removal




Case 1: No Derated StatesThe cumulative probability of a particular capacity outage state of X MW after a unit of capacity C MW and forced outage rate FOR is added is given by:

Where, P (X) - Cumulative Probability of X MWor more on outage of new table

P(X) - Probability of X MW or more on outage of old table

C - Capacity of unit to be addedInitial Setting: P(X) = 1.0 for X 0,

P(X) = 0 otherwise

( ) ( ) ( ) ( ) ( )CX'PFORX'PFOR1XP +=




EXAMPLE 1:

Using the Recursive Algorithm, the system capacity outage probability table is created sequentially as follows:

UNIT RATING (MW) FOR1 3 0.022 3 0.023 5 0.02

STEP 1: Add Unit No.1 C = 3 MW( ) ( ) ( ) ( ) ( )30'PFOR0'PFOR10P +=

( )( ) ( )( )0.102.00.102.01 += 0.1=( ) ( ) ( ) ( ) ( )31'P02.01'P02.011P +=

( )( ) ( )( )0.102.0098.0 += 02.0=( ) ( ) ( ) ( ) ( )32'P02.02'P02.012P +=

( )( ) ( )( )0.102.0098.0 += 02.0=( ) ( ) ( ) ( ) ( )33'P02.03'P02.013P +=

( )( ) ( )( )0.102.0098.0 += 02.0=




STEP 2: Add Unit No. 2 C = 3 MW( ) ( ) ( ) ( ) ( )30'P02.00'P02.010P +=

( )( ) ( )( )0.102.00.198.0 += 0.1=( ) ( ) ( ) ( ) ( )31'P02.01'P02.011P +=

( )( ) ( )( )0.102.002.098.0 += 0396.0=( ) ( ) ( ) ( ) ( )32'P02.02'P02.012P +=

( )( ) ( )( )0.102.002.098.0 += 0396.0=( ) ( ) ( ) ( ) ( )33'P02.03'P02.013P +=

( )( ) ( )( )0.102.002.098.0 += 0396.0=( ) ( ) ( ) ( ) ( )34'P02.04'P02.014P +=

( )( ) ( )( )02.002.0098.0 += 0004.0=( ) ( ) ( ) ( ) ( )35'P02.05'P02.015P +=

( )( ) ( )( )02.002.0098.0 += 0004.0=( ) ( ) ( ) ( ) ( )36'P02.06'P02.016P +=

( )( ) ( )( )02.002.0098.0 += 0004.0=




STEP 3: Add Unit No. 3 C = 5 MW( ) ( ) ( ) ( ) ( )50'P02.00'P02.010P +=

( )( ) ( )( )0.102.00.198.0 += 0.1=( ) ( ) ( ) ( ) ( )53'P02.03'P02.013P +=

( )( ) ( )( )0.102.00396.098.0 += 058808.0=( ) ( ) ( ) ( ) ( )55'P02.05'P02.015P +=

( )( ) ( )( )0.102.00004.098.0 += 020392.0=( ) ( ) ( ) ( ) ( )56'P02.06'P02.016P +=

( )( ) ( )( )0396.002.00004.098.0 += 001184.0=( ) ( ) ( ) ( ) ( )58'P02.08'P02.018P +=

( )( ) ( )( )0396.002.0098.0 += 000792.0=( ) ( ) ( ) ( ) ( )511'P02.011'P02.0111P +=

( )( ) ( )( )0004.002.0098.0 += 000008.0=




EXAMPLE 2:Using the Recursive Algorithm, prepare the generating capacity outage probability table of the system with three (3) generating units:

No. of states = 23

STEPS1. Initialize Table2. Enter Unit A3. Enter Unit B4. Enter Unit C

UNIT RATING (MW) FORA 100 0.01B 150 0.02C 200 0.03




STEP 1: Initialize the outage table

Probability when no unit yet is added intothe generation system

STATE X MW or more on Outage Cumulative Probability1 0 1.0000002 50 0.0000003 100 0.0000004 150 0.0000005 200 0.0000006 250 0.0000007 300 0.0000008 350 0.0000009 400 0.00000010 450 0.000000




STEP 2: Add unit A : C = 100, FOR = 0.01, 1-FOR = 0.99 STATE X MW P'(X) (X - C) MW P'(X-C) P(X)

1 0 1.000000 -100 1.000000 1.0000002 50 0.000000 -50 1.000000 0.0100003 100 0.000000 0 1.000000 0.0100004 150 0.000000 50 0.000000 0.0000005 200 0.000000 100 0.000000 0.0000006 250 0.000000 150 0.000000 0.0000007 300 0.000000 200 0.000000 0.0000008 350 0.000000 250 0.000000 0.0000009 400 0.000000 300 0.000000 0.00000010 450 0.000000 350 0.000000 0.000000




STEP 3: Add unit B: C = 150, FOR = 0.02, 1-FOR = 0.98 STATE X MW P'(X) (X - C) MW P'(X-C) P(X)

1 0 1.000000 -150 1.000000 1.0000002 50 0.010000 -100 1.000000 0.0298003 100 0.010000 -50 1.000000 0.0298004 150 0.000000 0 1.000000 0.0200005 200 0.000000 50 0.010000 0.0002006 250 0.000000 100 0.010000 0.0002007 300 0.000000 150 0.000000 0.0000008 350 0.000000 200 0.000000 0.0000009 400 0.000000 250 0.000000 0.000000

10 450 0.000000 300 0.000000 0.000000




STEP 4: Add unit C : C = 200, FOR = 0.03, 1-FOR = 0.97 STATE X MW P'(X) (X - C) MW P'(X-C) P(X)

1 0 1.000000 -200 1.000000 1.0000002 50 0.029800 -150 1.000000 0.0589063 100 0.029800 -100 1.000000 0.0589064 150 0.020000 -50 1.000000 0.0494005 200 0.000200 0 1.000000 0.0301946 250 0.000200 50 0.029800 0.0010887 300 0.000000 100 0.029800 0.0008948 350 0.000000 150 0.020000 0.0006009 400 0.000000 200 0.000200 0.000006

10 450 0.000000 250 0.000200 0.000006




Case 2: Derated States IncludedTo include the multi-state unit representations, the recursive equation is modified as follows:

Where, n - Number of unit statesCi - Capacity outage of state i for

the unit being addedpi - Probability of existence of the

unit state i

( ) ( )=

=

n

1iii CX'PpXP




EXAMPLE 1:

5 MW unit 3-state representation

Replacing the 2-state 5 MW unit in the previous example by the new 3-state unit,

n = 1; C1 = 0 MW

n = 2; C2 = 2 MW

n = 3; C3 = 5 MW

State Capacity Out State Probability1 0 0.9602 2 0.0333 5 0.007




( ) ( ) ( ) ( )50'Pp20'Pp00'Pp0P 321 ++=( )( ) ( )( ) ( )( )0.1007.00.1033.00.1960.0 ++= 0.1=

( ) ( ) ( ) ( ) ( ) ( ) ( )52'P007.022'P033.002'P960.02P ++=( )( ) ( )( ) ( )( )0.1007.00.1033.00396.0960.0 ++= 078016.0=

( ) ( ) ( ) ( ) ( ) ( ) ( )53'P007.023'P033.003'P960.03P ++=( )( ) ( )( ) ( )( )0.1007.00396.0033.00396.0960.0 ++= 0463228.0=

( ) ( ) ( ) ( ) ( ) ( ) ( )55'P007.025'P033.005'P960.05P ++=( )( ) ( )( ) ( )( )0.1007.00396.0033.00004.0960.0 ++= 0086908.0=

( ) ( ) ( ) ( ) ( ) ( ) ( )56'P007.026'P033.006'P960.06P ++=( )( ) ( )( ) ( )( )0396.0007.00004.0033.00004.0960.0 ++= 0006744.0=

( ) ( ) ( ) ( ) ( ) ( ) ( )58'P007.028'P033.008'P960.08P ++=( )( ) ( )( ) ( )( )0396.0007.00001184.0033.00960.0 ++= 000283272.0=

( ) ( ) ( ) ( ) ( ) ( ) ( )511'P007.0211'P033.0011'P960.011P ++=( )( ) ( )( ) ( )( )0004.0007.00033.00960.0 ++= 000002.0=




Case 3: Unit RemovalFrom the recursive algorithm for capacity model building,

The modified capacity model after a unit removal can be obtained by working on the algorithm in reverse. Thus,

( ) ( ) ( ) ( )FOR1

CX'PFORXPX'P

=

( ) ( ) ( ) ( ) ( )CX'PFORX'PFOR1XP +=

Where: P(X-C) = 1.0 for X C




EXAMPLE:Using the obtained capacity outage probability table for the 100 MW system (2 - 25 MW and 1 - 50 MW). Remove the 50MW Unit with FOR = 0.02

Capacity Out of Service (MW) Cumulative Probability0 1.00000025 0.05880850 0.02039275 0.000792

100 0.000008




Removing the unit from the capacity outage probability tables that gives the following modified capacity model:

( ) ( ) ( ) ( )( )02.01500'P02.00P0P

=

( )( )98.0

0.102.00.1 = 0.1=

( ) ( ) ( ) ( )( )02.015025'P02.025P25P

=

( )( )98.0

0.102.0058808.0 = 0396.0=

( ) ( ) ( ) ( )( )02.015050'P02.050P50P

=

( )( )98.0

0.102.0020392.0 = 0004.0=



Comparison of Deterministic and Probabilistic Criteria

Consider the following four systems which are very similar but not identical

SYSTEM 1 - 24 x 10 MW Units each having a FOR of 0.01

SYSTEM 2 12 x 20 MW Units each having a FOR of 0.01






0.0238550.02212522020

0.0017300.00163921030

0.2143220.19046723010

0.0000040.000004190500.0000910.00008720040

1.0000000.7856782400

Cumulative Probability


Capacity In (MW)

Capacity Out (MW)

SYSTEM 1: Capacity Outage Probability Table




0.0061750.00596920040

0.0002060.00020118060

0.1136160.10744122020

0.0000050.00000516080

1.0000000.8863842400



Capacity In (MW)

Capacity Out (MW)





0.0003310.00031416080

0.0486500.04380320040

0.0048470.00451618060

0.3061590.25750922020

0.0000010.0000011201200.0000170.000016140100

1.0000000.6938412400



Capacity In (MW)

Capacity Out (MW)





0.0202290.01889420020

0.1113350.00127219030

0.1983690.17814021010

0.0000020.000002170500.0000630.00006118040

1.0000000.8016312200



Capacity In (MW)

Capacity Out (MW)





20%

20%

20%

20%

% Reserve

0.004847200 MW240 MWSystem 3

0.000063183 1/3 MW220 MWSystem 4

0.000206200 MW240 MWSystem 2

0.000004200 MW240 MWSystem 1

Probabilistic RiskLoadTotal Capacity

System

Percentage Reserve Margin Criteria

Note that the risk in system 3 is more than 1000 times greater than that in system 1Analysis of the four systems shows that the variation in true risk depends upon FOR, number of units and load demand




22

12

12

24

Units

10 MW

20 MW

20 MW

10 MW

Reserve

0.048650220 MW20 MWSystem 30.020229210 MW10 MWSystem 4

0.006175220 MW20 MWSystem 20.023855230 MW10 MWSystem 1

Probabilistic RiskLoad

Cap/ Unit

System

Largest Unit Reserve Criteria

It is seen from the above comparison that deterministic approaches in power reserve margin planning are inconsistent, unreliable and subjective.



Loss of Load Expectation

Loss of Load Expectation (LOLE)The expected number of days in the specified period in which the daily peak load will exceed the available capacity.Notes:

LOLE is known in the power industry as the LOLP. LOLP is not a probability index but an expectation.

The expected risk of loss of load is determined by convolving the system capacity outage table and the system load characteristic.




Load Models

Daily Peak DemandJanuary 1

to

December 31Day

0 365D

a

i

l

y

P

e

a

k

,

M

W

Actual Load Curve




Load Models

The load is represented by its daily peak arranged in descendingorder to form a cumulative load model called load variation curve.

Load Variation Curve

Daily Peak DemandArrange in Descending Order

Linearized LVC




Where, Ok - Magnitude of the kth outage in the system capacity outage probability table

tk - Number of time units in the study interval that an outage magnitude Ok would result in a loss of load

pk - Individual probability of the capacity outage Ok.

=

=

n

1kkktpLOLP

( )=

=

n

1kk1kk PttLOLP

=

=

n

1kkk PT

Pk - Cumulative outage probability for capacity state Ok.




EXAMPLE 1:Consider a system containing 5 - 40 MWunits each with a FOR of 0.01. The capacity outage probability table is shown on the right.

The system load model is represented by the daily peak load variation curve in the figure on the right.

Capacity Out of

Service



0 0.950991 1.00000040 0.048029 0.04900980 0.000971 0.000980120 0.000009 0.000009

Note: Probability values less than10-8 have been deleted.

160

120

8064

0 100Time (%)

D

a

i

l

y

P

e

a

k

L

o

a

d

(

M

W

)

Installed Capacity = 200 MW

Time periods during which loss of load occurs

02 = 40MW 03 = 80MW 04 = 120 MW

T4 = 41.7%t3 = T3 = 41.7%

t4 = 83.4% 64




Capacity Out of Service (MW)

Capacity In Service (MW)

Individual Probability, pk

Total Time, tk (%)

LOLPpktk

0 200 0.950991 0.0 0.000000040 160 0.048029 0.0 0.000000080 120 0.000971 41.7 0.0404907

120 80 0.000009 83.4 0.00075060.0412413

Capacity Out of Service (MW)

Capacity In Service (MW)

Cumulative Probability, Pk

Total Time, Tk (%)

LOLPPkTk

0 200 1.000000 0.0 0.000000040 160 0.049009 0.0 0.000000080 120 0.000980 41.7 0.0408660

120 80 0.000009 41.7 0.00037530.0412413

LOLP using Individual Probabilities

LOLP using the Cumulative Probabilities

If the time considered is 365 days per year then,( ) yeardays LOLP 1505307.01000412413.0365 ==




It should be realized that there is difference between the terms capacity outage and loss of load. The term capacity outage indicates a loss of generation which may or may not result in a loss of load.

( ) ddays/perio LCPLOLP n1i

iii=

=

Where, Ci - Available capacity on day iLi - Forecast peak load on day iPi(Ci Li) - Probability of loss of load on day i. This

value is obtained directly from the capacity outage cumulative probability table.




EXAMPLE 2:What is the LOLP of the 450 MW system in a 5-day period?

Generator and Demand DataUnit Rating (MW) FOR Day MW

A 100 0.01 1 95B 150 0.02 2 120C 200 0.03 3 160

4 1105 90




Generation ModelSTATE G1 G2 G3 CAP IN CAP OUT IND CUMULATIVE

1 100 150 200 450 0 0.941094 1.0000002 0 150 200 350 100 0.009506 0.0589063 100 0 200 300 150 0.019206 0.0494004 100 150 0 250 200 0.029106 0.0301945 0 0 200 200 250 0.000194 0.0010886 0 150 0 150 300 0.000294 0.0008947 100 0 0 100 350 0.000594 0.0006008 0 0 0 0 450 0.000006 0.000006

DAY MW1 952 1203 1604 1105 90

DAY MW1 1602 1203 1104 955 90

Load Model




DAY SYSTEM CAPACITY (MW)PEAK LOAD

(MW)RESERVE

(MW)PROB. OF LOSS-OF-

LOAD1 450 160 290 0.0008942 450 120 330 0.0006003 450 110 340 0.0006004 450 95 355 0.0000065 450 90 360 0.000006

LOLP 0.002106

System Reliability

LOLP = 0.002106 Days/5-Days

Annual LOLP 365 days period




EXAMPLE 3:100 MW system with annual peak load of 57 MW

Unit No. Capacity (MW) FOR1 25 0.022 25 0.023 50 0.02

System Capacity Data

Daily Peak Load 57 52 46 41 34No. of Occurences 12 83 107 116 47 = 365 days

System Load Data




Using the recursive algorithm, the capacity outage cumulative probability table is

Capacity Out of Service (MW) Cumulative Probability0 1.000000

25 0.05880850 0.02039275 0.000792

100 0.000008

( ) ( ) ( )( ) ( )34100P4741100P116

46100P10752100P8357100P12LOLP++

++=

( ) ( ) ( ) ( ) ( )66P4759P11654P10748P8343P12 ++++=( ) ( ) ( )

( ) ( )000792.047000792.0116 000792.0107020392.083020392.012

++

++=

yeardays 15108.2LOLP =




LOLP Vs. Peak Load




Scheduled OutagesThe system capacity evaluation examples previously considered assumed that the load model applied to the entire period and that the capacity model was also applicable for the entire period. This will not be the case if the units are removed from services for periodic inspection and maintenance in accordance with the planned program. During this period, the capacity available is not constant and therefore a single capacity outage probability table is not applicable.

Typical Annual Load& Capacity Model

Total Installed Capacity

January 1 December 31

S

y

s

t

e

m

D

a

i

l

y

P

e

a

k

L

o

a

d

Reserve

Units onmaintenance



Loss of Load ExpectationConsidering scheduled maintenance, the annual LOLPa can be obtained by dividing the year into periods and calculating the period LOLPP values using the modified capacity model and the appropriate period load model.The annual risk index is given by

Subdivision of Period

=

=

n

1ppa LOLPLOLP

Total Installed Capacity

P1 P2 P3

S

y

s

t

e

m

D

a

i

l

y

P

e

a

k

L

o

a

d

R1

R2R3




Approximate methods may also be used to take into account scheduled maintenance while using the original capacity outage probability table such as shown in the following figures:

Installed Capacity

Time load exceeded the indicated value

L

o

a

d

0

Reserve CapacityPeak Load

Capacity onMaintenance

Modified LoadCharacteristicOriginal LoadCharacteristic

Installed Capacity

Time load exceeded the indicated value

L

o

a

d

0

Reserve CapacityCapacity on Maintenance Peak Load



LOLE with Load Forecast Uncertainty

The calculation of reliability indices presented so far assumes that the actual peak load will differ from the forecast value with zero probability. But some uncertainties exists in load forecasting arising from the historical data considered, the methodology used and the assumptions made. There is, therefore, a need to improve the risks calculation by including these uncertainties in building the system load model.

Studies made in the past had shown that the uncertainty in the load forecasts can be reasonably described by a normal distribution with the peak load value as the distribution mean parameter. The distribution can be divided into a discrete number of class intervals. The load representing the class interval midpoint is assigned the probability for that class interval. A seven-step distribution is considered reasonable in the subdivision.



Load Forecast Uncertainty

-3 -2 -1 0 +1 +2 +3

0.006

0.061 0.2420.382

0.006

0.0610.242

Probability given by indicated area

No. of standard deviations from the mean

Mean = forecast load (MW)



0.00020562150.0000046420

0.0061745410

0.0000000725

0.1136151351.000000000


Capacity Outage (MW)

(probability values less than 10-B are neglected)

Example:The capacity model of a system consisting of twelve 5 MW units,

each with forced outage rate of 0.01 is shown below.




0.0095630500.156771320.06152+2

0.0025418510.010503520.24249-10.0048877280.012795100.382500

0.22468462

0.08619473

0.008116450.00562781

LOLP (days/year) (4)

0.0208591250.24251+1

0.0013481070.00653+3

0.0004951030.06148-20.0000337660.00647-3

(3) x (4)Probability of the Load (3)

Load (MW)

(2)

Number of S. D. (1)

Using a seven-step normal distribution approximation of the load forecast with peak load of 50MW, 2% S.D., 70% load factor and straightline load variation curve, calculate the annual LOLP of the system in days/year.Standard Deviation = (0.02)(50) = 1 MW Mean = 50 MW

Annual LOLP = 0.03972873 days/year




Peak megawattdemand

Alternative 2

Alternative 1

0 2 4 6 8 Years

MW0

M

e

g

a

w

a

t

t

s

Note: MW0 = capacity of existing generating plant

Application in Generation System Expansion Analysis



Application in Generation System Expansion Analysis

Percentage of days the daily peak load

exceeded the indicated value

Daily peak load variation curve Year Number Forecast Peak Load (MW)1 160.02 176.03 193.64 213.05 234.36 257.57 283.18 311.4

Load Growth at 10% p.a.



Consider a system containing five 40 MW units each with a forced outage rate of 0.01.

Capacity outof service

IndividualProbability

CumulativeProbability

0 MW 0.950991 1.00000040 MW 0.048029 0.04900980 MW 0.000971 0.000980

120 MW 0.000009 0.0000091.000000

Sytem installed capacity = 200 MW

Generation model for thefive-unit system.

160

120

8064

0 100Time (%)

D

a

i

l

y

P

e

a

k

L

o

a

d

(

M

W

)

Installed Capacity = 200 MW

Time periods during which loss of load occurs

02 = 40MW 03 = 40MW 04 = 40 MW

T4 = 41.7%t3 = T3 = 41.7%

t4 = 83.4% 64




Capacity Outof Service (MW)

Capacity InService (MW)

IndividualProbability

Total Timetk (%)

LOLE(%)

0 200 0.950991 0.0 -40 160 0.048029 0.0 -80 120 0.000971 41.7 0.0404907120 80 0.000009 83.4 0.0007506

1.000000 0.0412413

Capacity Outof Service (MW)

Capacity InService (MW)

CumulativeProbability

Total Timetk (%)

LOLE(%)

0 200 1.000000 0.0 -40 160 0.049009 0.0 -80 120 0.000980 41.7 0.040866120 80 0.000009 41.7 0.0003753

0.0412413

LOLE using individual probabilities

LOLE using cumulative probabilities




LOLE (days/year)System Peak

Load (MW)200 MWCapacity

100 0.001210120 0.002005140 0.086860160 0.150600180 3.447000200 6.083000220 -240 -250 -260 -280 -300 -320 -340 -350 -




System PeakLoad (MW)

200 MWCapacity

250 MWCapacity

100 0.001210 -120 0.002005 -140 0.086860 0.001301160 0.150600 0.002625180 3.447000 0.068650200 6.083000 0.150500220 - 2.058000240 - 4.853000250 - 6.083000260 - -280 - -300 - -320 - -340 - -350 - -

LOLE (days/year)




System PeakLoad (MW)

200 MWCapacity

250 MWCapacity

300 MWCapacity

100 0.001210 - -120 0.002005 - -140 0.086860 0.001301 -160 0.150600 0.002625 -180 3.447000 0.068650 -200 6.083000 0.150500 0.002996220 - 2.058000 0.036100240 - 4.853000 0.180000250 - 6.083000 0.661000260 - - 3.566000280 - - 6.082000300 - - -320 - - -340 - - -350 - - -

LOLE (days/year)




200 MWCapacity

250 MWCapacity

300 MWCapacity

350 MWCapacity

100 0.001210 - - -120 0.002005 - - -140 0.086860 0.001301 - -160 0.150600 0.002625 - -180 3.447000 0.068650 - -200 6.083000 0.150500 0.002996 -220 - 2.058000 0.036100 -240 - 4.853000 0.180000 0.002980250 - 6.083000 0.661000 0.004034260 - - 3.566000 0.011750280 - - 6.082000 0.107500300 - - - 0.290400320 - - - 2.248000340 - - - 4.880000350 - - - 6.083000

LOLE (days/year)System Peak

Load (MW)




Loss of Energy Indices

Load Model: Load Duration Curve (LDC) Individual hourly load values arranged in

decreasing orders The area under the curve represents the energy

required in the given period

Let, Ok - Magnitude of the capacity outage

Pk - Probability of the capacity outage equal to Ok

Ek - Energy curtailed by capacity outage equal to Ok.



Loss of Energy Expectation

Load Models

Hourly Load Curve(Luzon Grid)

HOUR LOAD P.U. LOAD1 1452 0.712 1422 0.693 1446 0.714 1405 0.695 1440 0.706 1464 0.717 1417 0.698 1455 0.719 1547 0.7510 1498 0.7311 1574 0.7712 1535 0.7513 1501 0.7314 1499 0.7315 1504 0.7316 1487 0.7317 1504 0.7318 1659 0.8119 2051 1.0020 1960 0.9621 1830 0.8922 1718 0.8423 1598 0.7824 1661 0.81



Loss of Energy Expectation

Load Models

Load Duration Curve(Hourly Peak arrange in descending order)

HOUR P.U. LOAD1 1.002 0.963 0.894 0.845 0.816 0.817 0.788 0.779 0.7510 0.7511 0.7312 0.7313 0.7314 0.7315 0.7316 0.7317 0.7118 0.7119 0.7120 0.7121 0.7022 0.6923 0.6924 0.69




The probability energy curtailed by a capacity outage equal to Ok is Ekpk.Loss of Energy Expectation is the total energy curtailment because of capacity outages.

The per unit LOEE value represents the ratio between the probable load energy curtailed due to deficiencies in available generating capacity and the total energy required to served the system demand.

=

=

n

kkk pELOEE

1




STATE CAP IN CAP OUT INDIVIDUAL CUMULATIVE1 450 0 0.941094 1.0000002 350 100 0.009506 0.0589063 300 150 0.019206 0.0494004 250 200 0.029106 0.0301945 200 250 0.000194 0.0010886 150 300 0.000294 0.0008947 100 350 0.000594 0.0006008 0 450 0.000006 0.000006


Example:




Hourly Load

1 160.002 152.903 142.764 134.025 129.586 129.427 124.668 122.799 120.6810 119.7511 117.3312 117.33

13 117.0914 116.9415 116.8616 116.0017 114.2118 113.5119 113.2720 112.8021 112.3422 110.9323 110.5424 109.61




Loss of Energy Expectation (LOEE)CAP IN = 150 PROB = 0.000294

HOUR LOAD CURTAILED EXPECTATION1 160.00 10.00 0.0029402 152.90 2.90 0.0008533 142.76 0.00 0.0000004 134.02 0.00 0.0000005 129.58 0.00 0.0000006 129.42 0.00 0.0000007 124.66 0.00 0.0000008 122.79 0.00 0.0000009 120.68 0.00 0.00000010 119.75 0.00 0.00000011 117.33 0.00 0.00000012 117.33 0.00 0.00000013 117.09 0.00 0.00000014 116.94 0.00 0.00000015 116.86 0.00 0.00000016 116.00 0.00 0.00000017 114.21 0.00 0.00000018 113.51 0.00 0.00000019 113.27 0.00 0.00000020 112.80 0.00 0.00000021 112.34 0.00 0.00000022 110.93 0.00 0.00000023 110.54 0.00 0.00000024 109.61 0.00 0.000000

12.901024 0.003793SUBTOTAL




CAP IN = 100 PROB = 0.000594HOUR LOAD CURTAILED EXPECTATION

1 160.00 60.00 0.0356402 152.90 52.90 0.0314233 142.76 42.76 0.0253994 134.02 34.02 0.0202095 129.58 29.58 0.0175686 129.42 29.42 0.0174757 124.66 24.66 0.0146498 122.79 22.79 0.0135379 120.68 20.68 0.01228510 119.75 19.75 0.01172911 117.33 17.33 0.01029312 117.33 17.33 0.01029313 117.09 17.09 0.01015414 116.94 16.94 0.01006115 116.86 16.86 0.01001516 116.00 16.00 0.00950517 114.21 14.21 0.00843918 113.51 13.51 0.00802219 113.27 13.27 0.00788320 112.80 12.80 0.00760521 112.34 12.34 0.00732722 110.93 10.93 0.00649323 110.54 10.54 0.00626124 109.61 9.61 0.005705

535.309605 0.317974SUBTOTAL

Loss of Energy Expectation (LOEE)




CAP IN = 0 PROB = 0.000006HOUR LOAD CURTAILED EXPECTATION

1 160.00 160.00 0.0009602 152.90 152.90 0.0009173 142.76 142.76 0.0008574 134.02 134.02 0.0008045 129.58 129.58 0.0007776 129.42 129.42 0.0007777 124.66 124.66 0.0007488 122.79 122.79 0.0007379 120.68 120.68 0.00072410 119.75 119.75 0.00071811 117.33 117.33 0.00070412 117.33 117.33 0.00070413 117.09 117.09 0.00070314 116.94 116.94 0.00070215 116.86 116.86 0.00070116 116.00 116.00 0.00069617 114.21 114.21 0.00068518 113.51 113.51 0.00068119 113.27 113.27 0.00068020 112.80 112.80 0.00067721 112.34 112.34 0.00067422 110.93 110.93 0.00066623 110.54 110.54 0.00066324 109.61 109.61 0.000658

2935.309605 0.017612SUBTOTAL





STATE CAP IN CAP OUT PROB CURTAILED EXPECTATION1 450 0 0.941094 0.00 0.000002 350 100 0.009506 0.00 0.000003 300 150 0.019206 0.00 0.000004 250 200 0.029106 0.00 0.000005 200 250 0.000194 0.00 0.000006 150 300 0.000294 12.90 0.003797 100 350 0.000594 535.31 0.317978 0 450 0.000006 2935.31 0.01761

0.33938LOEE (MWH)





Applications in Probabilistic Production Simulations

Consider the LDC below

75.0

52.5

30.0

0 20 100

Duration (hours)

L

o

a

d

(

M

W

)




and the generating units capacity data

Assume that the economic loading order is units 1, 2 & 3.

If there were no units in the system, the expected energy not supplied would be

MWh4575.0 Energy Required Total = (area under the curve)

MWh0.4575EENS0 =

Unit No. Capacity (MW) Probability1 0 0.05

15 0.3025 0.65

2 0 0.0330 0.97

3 0 0.0420 0.96




If the system contained only Unit 1, the EENS can be calculated as follows:

The expected energy produced by Unit 1= EENS0 - EENS1= 4575.0 - 2500.0

= 2075.0 MWh

Capacity Out ofService (MW)

Capacity inService (MW) Probability

Energy Curtailed(MWh)

Expectation(MWh)

0 25 0.65 2075.0 1348.7510 15 0.30 3075.0 922.5025 0 0.05 4575.0 228.75

EENS1 2500.00




EENS with Units 1 and 2

The expected energy produced by Unit 2= EENS1 - EENS2= 2500.0 - 401.7

= 2098.3 MWh




Expectation(MWh)

0 55 0.6305 177.8 112.0910 45 0.2910 475.0 138.2325 30 0.0485 1575.0 76.3930 25 0.0195 2075.0 40.4640 15 0.0090 3075.0 27.6855 0 0.0015 4575.0 6.86

EENS 2 401.70



Loss of Energy IndicesEENS with Units 1, 2 and 3

The expected energy produced by Unit 3= EENS2 - EENS3= 401.7 - 64.08 = 337.6 MWh




Expectation(MWh)

0 75 0.60528 0.0 0.0010 65 0.27936 44.4 12.4220 55 0.02522 177.8 4.4825 50 0.04656 286.1 13.3230 45 0.03036 475.0 14.4240 35 0.00864 1119.4 9.6745 30 0.00194 1575.0 3.0650 25 0.00078 2075.0 1.6255 20 0.00144 2575.0 3.7160 15 0.00036 3075.0 1.1175 0 0.00006 4575.0 0.27

EENS 3 64.08

ee353 notes no. 2 - lole&loee

Documents