Protecting against national-scale power blackouts

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Protecting against national-scale power blackouts
Daniel Bienstock, Columbia University
Collaboration with Sara Mattia, Universitá di
Roma, Italy Thomas Gouzènes, Réseau de Transport
dElectricité, France
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Recent major incidents

• August 2003 North America. 50 million people
  affected during two days New York City loses
  power
• September 2003 Switzerland-France-Italy. 57
  million people affected during one day Italy
  loses power
• Other major incidents in recent years in Europe
  and Brazil
• The potential economic and human consequences of
  a prolongued national-scale blackout are
  significant

Were the blackouts due to insufficient generation
capacity?
No they were due to inadequately protected
transmission networks
U.S.-Canada task force The leading cause of the
blackout was Inadequate System Understanding
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A power grid has 3 components
The transmission network is the key ingredient in
modern grids Modern transmission networks are
lean and, as a result, brittle
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An inconvenient fact
The power flows in a grid are controlled by the
laws of physics
When analyzing a hypothetical change in a
network, the behavior of the power flows must be
computed -- it cannot be dictated

• Two popular methodologies
• AC power flow models
• DC (linearized) flow models

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Summary

• AC models for computing power flows
• Account for both active and reactive power
  flows
• Fairly accurate
• Non-convex system of nonlinear equations
• Computationally intensive, Newton-like methods
• Solution methods tend to require a good initial
  guess
• Heavy data requirements

• DC models
• Linearized approximations of AC models
• Much faster
• Usually preferred by the industry for
  large-scale analysis

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How does a blackout develop?

• Individual power lines fail due to
• External effects fires, lightning strikes,
  tree contacts, malicious agents (?)
• Thermal effects an overloaded line will melt
  -- usually requires several minutes
• (protection equipment will shut it down first)
• The physics and engineering underlying line
  failures are well understood

Individual line failures
system collapse
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A model for system collapse
Initial set of externally caused faults Several
lines are disabled
The network is altered new power flows ensue
flows in some of the lines exceed the line ratings
Further line shutoffs
New network new power flows
Cascade !
(sometimes)
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Simulation
Round No. of shut-off lines No. of connected components (islands) Demand served ()
1 2 1 100.0
2 8 3 100.0
3 17 8 87.66
4 20 16 82.72
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What we are doing

• Proactive planning how to economically
  engineer a network
• so as to ride-out potential failure scenarios
• Each scenario is an interesting
  combination of
• externally caused faults.
• Example from industry N k modeling

• Reaction planning what to do if a significant
  event materializes
• From a theoretical standpoint, very intractable
• Multiple time scales

The adversarial model
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Proactive model
We can upgrade a network in a number of ways.
Examples
Upgrade individual lines
Add new lines
Join/split nodes
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Integer programming approach

• 0/1 vector x each entry represents whether a
  certain action is taken, or not
• x has an entry for each line of the network
• example a line parallel to a certain line is
  added, or not
• total cost cT x, for a certain cost vector c

Problem find x feasible, of minimum cost
What is feasible?
In each scenario (of a certain list), the
network augmented as per vector x survives the
cascade
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Solution approach game against an adversary
Maintain a working model M, which describes
conditions that a protection plan x must
satisfy This model may be incomplete
Solve the problem FIND x OF MINIMUM COST THAT
SATISFIES THE CONDITIONS STIPULATED BY M, with
solution x
Add this algebraic statement to M
Is x adequate in all scenarios?
In some scenario, x does not suffice. State
this fact algebraically
YES - DONE
NO
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Solution approach Benders decomposition
Maintain a working formulation Ax ? b of
inequalities valid for feasible x
Solve the problem Minimize cTx subject
to Ax ? b, x 0/1 With solution x
Add ?Tx ? ? to Ax ? b
x feasible?
Find a valid inequality ?Tx ? ? with ?Tx lt ?
YES - DONE
NO
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• Simple example
• we protect power lines
• a 0/1 variable x per each line
• the grid survives a cascade if 70 of demand is
  met
• if the grid survives two rounds then it survives

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• First round after initial event
• lines 1 7 shut off
• 5 islands, 80 of demand is met

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• Second round
• lines 8 13 shut off, 15 islands
• 61 lt 70 of demand met, collapse

x1 x2 x3 x6 x11 x12 x13 ? 1
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Experiments, and lessons

• Algorithm converges in few iterations,
• even with thousands of scenarios
• But each iteration is expensive because of the
  need to simulate scenarios to test if a certain
  network is survivable in the worst case, all
  scenarios must be simulated

And where do the scenarios come from?
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A model for system collapse, revisited
Initial set of externally caused faults Several
lines are disabled
The network is altered new power flows ensue
flows in some of the lines exceed the line ratings
Further line shutoffs
New network new power flows
Research topic can this process be efficiently
approximated?
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How are scenarios generated?

• Today N-1 analysis

• It can prove too slow on large networks
• Many of the scenarios are uninteresting
• The generalization N k analysis is
  prohibitively expensive

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A different technique

• Stochastic simulation
• assign a fault probability to each network
  component, and simulate the entire system

• We are dealing with extremely low probability
  events
• The interesting scenarios have very low
  probability, which will likely be incorrectly
  estimated
• And in any case we will generate many
  unimportant scenarios

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Ongoing work adversarial problem
Problem find a smallest initial set of faults,
following which a cascade occurs
Enumerating all k-subsets for k ? 5 is
computationally infeasible for large grids
Approach we are using combination of
approximate dynamic programming and integer
programming
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One approach

• Adversary enumerates sets of k (small) lines
  at a time
• Adversary chooses the best set according to an
  appropriate merit function
• Examples number of overloaded lines, nonlinear
  function of overloads (e.g. exponential), cost of
  flow under nonlinear cost function

A difficulty problem is not monotone
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Braess Paradox
Example if we cut lines a, b, and c the
system cascades but if we cut a, b, c,
and d it does not
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Solution approach game against an adversary
Maintain a working model M, which describes
conditions that a protection plan x must
satisfy This model may be incomplete
Solve the problem FIND x OF MINIMUM COST THAT
SATISFIES THE CONDITIONS STIPULATED BY M, with
solution x
Add this algebraic statement to M
Can the adversary collapse the system protected
by plan x?
In some scenario, x does not suffice. State
this fact algebraically
YES - DONE
NO