Major Event Day Classification

October 26, 2001 MED Classification 1

Major Event Day Classification

Rich ChristieUniversity of Washington

Distribution Design Working Group Webex Meeting

October 26, 2001


Overview

• MED definitions• Proposed frequency criteria• Bootstrap method of evaluation• Probability distribution fitting method• Comparison


Major Event Days

• Some days, reliability ri is a whole lot worse than other days– ri is SAIDI/day, actually unreliabilty

• Usual cause is severe weather: hurricanes, windstorms, tornadoes, earthquakes, ice storms, rolling blackouts, terrorist attacks

• These are Major Event Days (MED)• Problem: Exactly which days are MED?


Existing MED Definition (P1366)

• Reflects broad range of existing practice• Ambiguous: “catastrophic,” “reasonable”• 10% criterion inequitable• No one design limit• No standard event types

Designates a catastrophic event which exceeds reasonable design or operational limits of the electric power system and during which at least 10% of the customers within an operating area experience a sustained interruption during a 24 hour period.


10% Criterion

A B

Same geographic phenomenon (e.g. storm track) affects more than 10% of B, less than 10% of A. Should be a major event for both, or neither - inequitable to larger utility.


Proposed Frequency Criteria

• Utilities could agree, with regulators, on average frequency of MEDs, e.g. “on average, 3 MEDs/year”– Quantitative– Equitable to different sized utilities– Easy to understand– Consistent with design criteria (withstand 1 in

N year events)


Probability of Occurrence

• Frequency of occurrence f is probability of occurrence p

365fp


Reliability Threshold

• Find MED threshold R* from probability p and probability distribution

pdff(ri)

Daily Reliability ri

p(ri > R*)

R*

• MEDs are days with reliability ri > R*


Reliability: SAIDI/day or CMI/day?

• If total customers (NT) is constant, either one• If NT varies from year to year, SAIDI

TNCMISAIDI (SAIDI in mins)


Bootstrap Method

• Sample distribution is best estimate of actual distribution

• In N years of data, N·f worst days are MEDs– R* between best MED and worst non-MED ri

• How much data?– More better– How much is enough?


Bootstrap Example• Take daily reliability data (3 years worth)

Day of Week Date Year CMI (x10^6) SAIDI/DayFriday April 03 1998 0.002961 0.00984Friday April 10 1998 0.000324 0.00108Friday April 17 1998 0.016986 0.05643Friday April 24 1998 0.012815 0.04257Friday August 07 1998 0.011563 0.03842Friday August 14 1998 0.035424 0.11769Friday August 21 1998 0.002589 0.00860Friday August 28 1998 0.02517 0.08362Friday December 04 1998 0.000759 0.00252Friday December 11 1998 0.003969 0.01319Friday December 18 1998 0.004133 0.01373Friday December 25 1998 0.012265 0.04075Friday February 06 1998 0.064379 0.21388Friday February 13 1998 0.03099 0.10296

SAIDI inmins/day


Bootstrap Example• Sort by reliability (descending)

Day of Week Date Year CMI (x10^6) SAIDI/DaySaturday March 18 2000 5.508412 18.30037Tuesday August 29 2000 1.745452 5.79884Wednesday May 27 1998 1.559164 5.17995Thursday March 11 1999 1.525844 5.06925Saturday June 17 2000 1.31193 4.35857Sunday October 08 2000 0.997673 3.31453Monday May 18 1998 0.957878 3.18232Thursday August 31 2000 0.951024 3.15955Saturday December 11 1999 0.902157 2.99720Tuesday July 21 1998 0.659672 2.19160Saturday July 03 1999 0.57174 1.89947Tuesday May 30 2000 0.546133 1.81440Sunday November 28 1999 0.520744 1.73005Friday July 07 2000 0.505345 1.67889


Bootstrap Example• Pick off worst N·f as Major Event Days

N = 3 yrs

f = 3/yr

MED = 9

MEDs:

98: 299: 200: 5

R* =2.19 to3.00

Day of Week Date Year CMI (x10^6) SAIDI/DaySaturday March 18 2000 5.508412 18.30037Tuesday August 29 2000 1.745452 5.79884Wednesday May 27 1998 1.559164 5.17995Thursday March 11 1999 1.525844 5.06925Saturday June 17 2000 1.31193 4.35857Sunday October 08 2000 0.997673 3.31453Monday May 18 1998 0.957878 3.18232Thursday August 31 2000 0.951024 3.15955Saturday December 11 1999 0.902157 2.99720Tuesday July 21 1998 0.659672 2.19160Saturday July 03 1999 0.57174 1.89947Tuesday May 30 2000 0.546133 1.81440Sunday November 28 1999 0.520744 1.73005Friday July 07 2000 0.505345 1.67889


Bootstrap Results

Freq fMED/

yr

1998MED

s

1999MED

s

2000MED

s

R*low

R*high

3 2 2 5 2.19 3.004 3 3 6 1.73 1.815 4 4 7 1.56 1.606 4 5 9 1.25 1.42


Bootstrap Data Size Issue

• How many years of data?– New data revises MEDs– Ideally, one new year should cause f new MEDs

(i.e. 3, in example with f = 3 MED/yr)– What is probability of exactly 3 new values in 365

new samples greater than the 9th largest value in 3*365 existing samples?

– What number of years of existing data maximizes this?


Bootstrap Data Size

• Order statistics result, probability of exactly f new values in n new samples greater than k’th value of m samples

fknmnfn

km

fknkp nkmf ,,|

0.05

0.1

0.15

0.2

0 5 10 15

M , years

p(f

MED

s)

f =3

f =5

f =10

• 5-10 years of data looks reasonable


Bootstrap Characteristics

• Fast• Easy• Intuitive• Saturates

– e.g. if f = 3 and one year has the 30 highest values, need 11 years of data before any other year has an MED, or exceptional year must roll out of data set.


Probability Distribution Fitting

• Should be immune to saturation• Process:

– Choose a probability distribution type– Fit data to distribution– Calculate R* from fitted distribution and p– Find MEDs from R*


Choosing a Distribution Type

• Examine histogram– What does it look like?– What doesn’t it look like?

• Make probability plots– Try different distributions– Parameters come out as side effect– Most linear plot is best distribution type


Examine Histogram

• Not Gaussian (!)• Not too useful otherwise

0

1000

0 10 20

r, SAIDI/day(a)

Bin

Cou

nt

0

20

40

0 10 20

r, SAIDI/day(b)

Bin

Cou

nt

Data: 3 years, anonymous “Utility 2”


Probability Plot• Order samples: e.g. ri = {2, 5, 7, 12}• Probability of next sample having a value less

than 5 is

• Given a distribution, can find a random variable value xk(pk) (pk is area under curve to left of xk)

• If plot of rk vs xk is linear, distribution is good fit

375.04

5.025.0

nkrp kk


Probability Plot for Gaussian Distribution

• Not Gaussian (but we knew that)

-5

0

5

10

15

20

-4 -2 0 2 4

Estimated Value x k

Sam

ple

Val

ue r k


Probability Plot for Log-Normal Distribution

• Looks good for this data

-12-10-8-6-4-2024

-4 -3 -2 -1 0 1 2 3 4

Estimated Value x k

Sam

pled

Val

ue ln

(rk)


Probability Plot for Weibull Distribution

• Not as good as Log-Normal

-20

-15

-10

-5

0

5

-10 -8 -6 -4 -2 0 2 4

Estimated Value x k

Sam

pled

Val

ue ln

(rk)


Stop at Log-Normal

• Good fit• Computationally tractable

– Pragmatically important that method be accessible to typical utility engineer

• Weak theoretical reasons to go with log-normal– Loosely, normal process with lower limit has

log-normal distribution


Some Other Suspects

• Gamma distribution• Erlang distribution• Beta distribution• etc.


Fit Process

• Find log-normal parameters

• ( and are not mean and standard deviation!)

n

iirn 1

ln1

n

iirn 1

2ln1

1

Example: = -3.4 = 1.95

Leave out ri = 0,but count how many


Fit Process

• Find R* from p

pdff(ri)

Daily Reliability ri

p(ri > R*)

R*

dxex

pR

x

*

2

2

2ln

21

Solve

For R* given p


Fit Process

• Or! pRF 1*

*

* ln RRF

F(r) is CDF of log-normal distn

is CDF of standard normal (Gaussian) distribution

pR 1exp 1* -1 is NORMINV function in ExcelTM


Fit Process

• What about ri = 0?– It’s a lumped probability p(0) = nz/n– Probability left under curve is 1-p(0)– Correct p to

01ˆ

ppp


Fit Results

Freq f MED/yr

p̂ R* 1998 MED

1999 MED

2000 MED

Total MED

3 0.00831 3.148 2 1 5 8 4 0.01104 2.552 2 2 5 9 5 0.01380 2.157 3 2 5 10 6 0.01656 1.873 3 3 5 11


Result Comparison

Freq fMED/

yr

Boot-strapR* lo

Boot-strapR* hi

FitR*

1998MED

1999MED

2000MED

TotalMED

3 2.19 3.00 3.148 2 (2) 1 (2) 5 (5) 8 (9)4 1.73 1.81 2.552 2 (3) 2 (3) 5 (6) 9 (12)5 1.56 1.60 2.157 3 (4) 2 (4) 5 (7) 10 (15)6 1.25 1.42 1.873 3 (4) 3 (5) 5 (9) 11 (18)

Bootstrap MEDs in parentheses


Method Comparison

• Bootstrap simpler• Bootstrap limits number of MEDs• Bootstrap can saturate - fit doesn’t• A good fit for most of the data may not be a

good fit for the tails


Conclusion

• Frequency criteria (MEDs/year) is at root of work

• Two methods to classify MEDs based on frequency - strengths and weaknesses

• Reliability distributions may not all be log normal

• White paper and spreadsheet at: http://www.ee.washington.edu/people/faculty/christie/

Documents

Major Event Day Classification