Blackouts

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Produced by P.Jaques at the Centre for Maths in
Industry, Massey University, as part of a 2008
New Zealand Sciences, Mathematics and Technology
Teacher Fellowship funded by the New Zealand
Government and administered by the Royal Society
of New Zealand.
Blackouts
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• USA and Canada, August 14, 2003 50 million
  people lose power for up to 2 weeks
• Denmark and Sweden, September 23, 2003 2.4
  million people lose power for 7 hours
• Auckland, June 12, 2006 230,000 people lose
  power for 8 hours
• Costa Rica, April 19, 2007 whole country (4
  million) loses power for over 1 hour
• Florida, February 26, 2008 4 million people lose
  power

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Causes of Blackouts

• Problems that the system designers did not expect
  to have to cover
• Operator errors
• Incorrect installation of equipment
• A combination of problems that the power system
  could not cope with
• Example Denmark and Sweden 2003
• a power station stopped generating,
• 5 minutes later there was a double fault,
• some lines and plant were also unavailable due to
  routine maintenance

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Recent Research

• Electricity networks are complex systems that are
  run near their limits so the failure of a small
  part may cascade to bring the whole system down
• Network owners seek to reduce the chance of
  failure in most probable areas first so
  unlikely events, which may be extremely serious,
  may not get much attention

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This means that the probability of a large
blackout is not normally spread out as you
might expect
As engineers try to avoid small blackouts by
improving the systems, the probability drops on
the left but the total must still be 100, so
the relative probability on the right rises (so
large blackouts are more likely than you might
expect)
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So the best model may well be a power law, as
this graph demonstrates
Stays higher than the others
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Transpowers Question

• What is the expected cot of a BIG blackout?

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To answer this we need to find the probability of
a BIG blackout

• Find data about blackouts
• Calculate the probabilities from the data
• Draw a graph to see if there is a pattern
• Try to fit a curve to find a formula
• Extrapolate the curve to answer the question

The problem with this is that the data we have,
especially for NZ, has very few very large
blackouts in it. So the best idea turns out to be
to look for the underlying distribution using the
data, and use that to make predictions.
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Find data about blackouts
Keep in mind that the blackouts are usually not
independent one small blackout here may cause
another small blackout there, and so on for
this reason, the data must first be cleaned so
that each connected series of blackouts is
considered as just one blackout event.
Example of data from NERC, United States
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Log-Log Graph of Probability against the Size of
the Blackout
This is a log-log graph, but the axes are labeled
with the actual values instead of the logs of
those values
Are you happy that a line fits this data well?
Why?
skip the theory log-log graphs
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Do any of our distributions look like these, on
log-log axes?
Straight line! Why?
So the straight line on the previous slide
suggests that a power law is a good model
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Probability Model

• Curve P(x) 391.62x-1.6286
• Use this model to find the probability of all of
  New Zealand losing power for an hour (x300000
  MWminutes)
• Probability 0.000 000 47
• How confident in this answer do you think
  Transpower can be?

Photo of Comet McNaught, Glenfield Jan 2007, by
Ashley Hopkins (www.cometmcnaught.net/mcnaughtphot
ogallery.htm)
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So What?

• In this case, the work done to solve this problem
  led to a shift in thinking at Transpower.
• Now people are planning to cope with a large
  blackout, not just trying to make the probability
  zero.

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References

• http//en.wikipedia.org/wiki/List_of_power_outages
  
• Mr Conrad Edwards, Transpower
• EEA Paper 2005 Is a Large Scale Blackout of the
  NZ Power System Inevitable?
• EEA Paper 2006 Allowing for Extreme Events in
  Transmission Planning
• Carreras BA, Newman DE, Dobson I in IEEE
  Transactions on Circuits and Systems 1 Vol. 51,
  Nr. 9, Sept 2004
• ftp//ftp.nerc.com/pub/sys/all_updl/oc/dawg/distur
  b06.pdf

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Questions?