Transcript Identifying Major Reliability Event Days for Distribution

Major Event Day Classification
Rich Christie
University of Washington
Distribution Design Working Group
Webex Meeting
October 26, 2001
October 26, 2001
MED Classification
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Overview
•
•
•
•
•
MED definitions
Proposed frequency criteria
Bootstrap method of evaluation
Probability distribution fitting method
Comparison
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MED Classification
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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?
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Existing MED Definition
(P1366)
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.
•
•
•
•
•
Reflects broad range of existing practice
Ambiguous: “catastrophic,” “reasonable”
10% criterion inequitable
No one design limit
No standard event types
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MED Classification
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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.
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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)
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Probability of Occurrence
• Frequency of occurrence f is probability of
occurrence p
f
p
365
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Reliability Threshold
• Find MED threshold R* from probability p
and probability distribution
pdf
f(ri)
p(ri > R*)
R*
Daily Reliability ri
• MEDs are days with reliability ri > R*
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MED Classification
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Reliability: SAIDI/day or
CMI/day?
CMI
SAIDI 
NT
(SAIDI in mins)
• If total customers (NT) is constant, either one
• If NT varies from year to year, SAIDI
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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?
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Bootstrap Example
• Take daily reliability data (3 years worth)
SAIDI in
mins/day
Day of Week
Friday
Friday
Friday
Friday
Friday
Friday
Friday
Friday
Friday
Friday
Friday
Friday
Friday
Friday
October 26, 2001
Date
April 03
April 10
April 17
April 24
August 07
August 14
August 21
August 28
December 04
December 11
December 18
December 25
February 06
February 13
Year
1998
1998
1998
1998
1998
1998
1998
1998
1998
1998
1998
1998
1998
1998
MED Classification
CMI (x10^6)
0.002961
0.000324
0.016986
0.012815
0.011563
0.035424
0.002589
0.02517
0.000759
0.003969
0.004133
0.012265
0.064379
0.03099
SAIDI/Day
0.00984
0.00108
0.05643
0.04257
0.03842
0.11769
0.00860
0.08362
0.00252
0.01319
0.01373
0.04075
0.21388
0.10296
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Bootstrap Example
• Sort by reliability (descending)
Day of Week
Saturday
Tuesday
Wednesday
Thursday
Saturday
Sunday
Monday
Thursday
Saturday
Tuesday
Saturday
Tuesday
Sunday
Friday
October 26, 2001
Date
March 18
August 29
May 27
March 11
June 17
October 08
May 18
August 31
December 11
July 21
July 03
May 30
November 28
July 07
Year
2000
2000
1998
1999
2000
2000
1998
2000
1999
1998
1999
2000
1999
2000
MED Classification
CMI (x10^6)
5.508412
1.745452
1.559164
1.525844
1.31193
0.997673
0.957878
0.951024
0.902157
0.659672
0.57174
0.546133
0.520744
0.505345
SAIDI/Day
18.30037
5.79884
5.17995
5.06925
4.35857
3.31453
3.18232
3.15955
2.99720
2.19160
1.89947
1.81440
1.73005
1.67889
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Bootstrap Example
• Pick off worst N·f as Major Event Days
N = 3 yrs
f = 3/yr
MED = 9
Day of Week
Saturday
Tuesday
Wednesday
Thursday
Saturday
Sunday
Monday
Thursday
Saturday
Tuesday
Saturday
Tuesday
Sunday
Friday
October 26, 2001
Date
March 18
August 29
May 27
March 11
June 17
October 08
May 18
August 31
December 11
July 21
July 03
May 30
November 28
July 07
Year
2000
2000
1998
1999
2000
2000
1998
2000
1999
1998
1999
2000
1999
2000
MED Classification
CMI (x10^6)
5.508412
1.745452
1.559164
1.525844
1.31193
0.997673
0.957878
0.951024
0.902157
0.659672
0.57174
0.546133
0.520744
0.505345
SAIDI/Day
18.30037
5.79884
5.17995
5.06925
4.35857
3.31453
3.18232
3.15955
2.99720
2.19160
1.89947
1.81440
1.73005
1.67889
MEDs:
98: 2
99: 2
00: 5
R* =
2.19 to
3.00
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Bootstrap Results
Freq f
MED/
yr
3
4
5
6
October 26, 2001
1998
MED
s
2
3
4
4
1999
MED
s
2
3
4
5
2000
MED
s
5
6
7
9
MED Classification
R*
low
R*
high
2.19
1.73
1.56
1.25
3.00
1.81
1.60
1.42
14
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?
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MED Classification
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Bootstrap Data Size
• Order statistics result, probability of exactly
f new values in n new samples greater than
k’th value of m samples
f =3
0.2
p (f MEDs)
p f |m ,k ,n
 m  n 
  
k
 k  f 

nk  f  nm 


n  k  f 
f =5
0.15
f =10
0.1
0.05
0
5
10
15
M , years
• 5-10 years of data looks reasonable
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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.
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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*
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MED Classification
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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
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MED Classification
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Examine Histogram
40
Bin Count
Bin Count
1000
Data: 3 years,
anonymous
“Utility 2”
20
0
0
0
10
20
0
10
20
r, SAIDI/day
(b)
r, SAIDI/day
(a)
• Not Gaussian (!)
• Not too useful otherwise
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MED Classification
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Probability Plot
• Order samples: e.g. ri = {2, 5, 7, 12}
• Probability of next sample having a value
less than 5 is
k  0.5 2  0.5
pk rk  

 0.375
n
4
• 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
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Probability Plot for Gaussian
Distribution
20
Sample Valuer k
15
10
5
0
-4
-2
-5
0
2
4
Estimated Value x k
• Not Gaussian (but we knew that)
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MED Classification
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Sampled Value ln(r k )
Probability Plot for Log-Normal
Distribution
-4
-3
-2
-1
4
2
0
-2 0
-4
-6
-8
-10
-12
1
2
3
4
Estimated Value x k
• Looks good for this data
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Probability Plot for Weibull
Distribution
Sampled Value ln(r k )
5
0
-10
-8
-6
-4
-2
-5
0
2
4
-10
-15
-20
Estimated Value x k
• Not as good as Log-Normal
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Stop at Log-Normal
• Good fit
• Computationally tractable
– Pragmatically important that method be
accessible to typical utility engineer
• Weak theoretical reasons to go with lognormal
– Loosely, normal process with lower limit has
log-normal distribution
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Some Other Suspects
•
•
•
•
Gamma distribution
Erlang distribution
Beta distribution
etc.
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Fit Process
• Find log-normal parameters
1 n
   ln ri 
n i 1
1 n
2







ln
r

i
n  1 i 1
Example:
 = -3.4
 = 1.95
Leave out ri = 0,
but count how many
• ( and  are not mean and standard
deviation!)
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Fit Process
• Find R* from p
Solve
pdf
f(ri)
p(ri > R*)

p
 x
R*
R*
Daily Reliability ri
October 26, 2001
ln x  2
1
2
e
2 2
dx
For R* given p
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Fit Process
F R*   1  p
• Or!
F(r) is CDF of log-normal distn
 ln R*    

F R   



*
R  exp  1  p   
*
October 26, 2001
1
 is CDF of standard normal
(Gaussian) distribution
-1 is NORMINV function in
ExcelTM
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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
p
pˆ 
1  p 0
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Fit Results
Freq f
MED/
yr
3
4
5
6
October 26, 2001
pˆ
0.00831
0.01104
0.01380
0.01656
R*
1998
MED
1999
MED
2000
MED
Total
MED
3.148
2.552
2.157
1.873
2
2
3
3
1
2
2
3
5
5
5
5
8
9
10
11
MED Classification
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Result Comparison
Freq f
MED/
yr
3
4
5
6
Bootstrap
R* lo
2.19
1.73
1.56
1.25
Bootstrap
R* hi
3.00
1.81
1.60
1.42
Fit
R*
1998
MED
1999
MED
2000
MED
Total
MED
3.148
2.552
2.157
1.873
2 (2)
2 (3)
3 (4)
3 (4)
1 (2)
2 (3)
2 (4)
3 (5)
5 (5)
5 (6)
5 (7)
5 (9)
8 (9)
9 (12)
10 (15)
11 (18)
Bootstrap MEDs in parentheses
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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
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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/
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MED Classification
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