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From SIAM News, Volume 36, Number 10, December 2003
The Power Grid as
Complex System
Sara Robinson
Last August 14, a series of seemingly unrelated events conspired to produce
a massive power blackout, affecting an enormous swath of the northeastern
United States and Canada. Shops and businesses closed, public transportation
ground to a halt, and the economy lost billions of dollars. With their
constituents clamoring for an explanation, politicians sprang into action.
They mobilized task forces charged to determine exactly what happened
and how recurrences could be prevented.
But even as politicians focused on the single event, scientists were
considering the problem from a broader perspective. As an example of what
is often called a complex system, the power grid is made up of many components
whose complex interactions are not effectively computable. Accordingly,
some scientists have found it more useful to study the power grid's macroscopic
behavior than to dissect individual events.
Ian Dobson, an electrical engineer at the University of Wisconsin, Madison,
and his physicist-collaborators have been examining power-outage data
from a complex systems point of view. Their findings suggest that the
power grid may be a self-organized critical (SOC) system, a system that
perpetually steers itself toward a dynamic equilibrium, where small perturbations
have long-range effects. If the power grid is indeed an SOC system, then
large power outages are more likely than traditional risk analysis predicts,
Dobson points out. Moreover, addressing the triggers of the August 14
outage, without heeding the larger picture, might do little to prevent
similar events in the future and could even make matters worse.
Percolation and Forest Fires
An explanation of self-organized criticality starts with a simple model,
called percolation, devised by statistical physicists to capture the notion
of a phase transition.
Imagine a square grid on which a unit square is colored black with probability
p. For what values of p will there be a path of black
squares from one side of the board to the other?
Let R(p) be the probability that there is a path across
the board-a spanning cluster. Clearly, if p is close to zero,
R(p) is close to zero, and if p is close to
one, R(p) is close to one as well. Surprisingly, as
the dimensions of the checkerboard grow, R(p) makes
a dramatic transition from low to high values---a phase transition---at
a critical value p = 0.5927. Phase transitions abound in nature.
Examples include the transition of water to ice and the transition of
an outbreak of disease into an epidemic.
One property of percolation is that the distribution of cluster sizes
becomes a power law---that is, it scales proportionally to an inverse
polynomial---right at the critical point. The probability of very large
clusters is much larger in a power law distribution than in a Gaussian
distribution, whose exponential tail makes the probability of far-larger-than-average
clusters extremely low. Thus, for percolation, a power law at the critical
point means that clusters of all sizes appear, with a greater number of
very large clusters than would be expected from a normal distribution.
To observe a phase transition in a laboratory, a scientist typically
holds an external parameter, such as temperature, at the critical value.
Some systems, however, have an internal dynamic that holds them perpetually
in a phase transition. To see how this can be, consider the following
variation on the percolation model.
Imagine an empty grid on which black squares, representing trees in a
forest, start to spring up at a steady rate. Initially, the forest is
sparse, but as time goes by, clusters begin to form, with small clusters
eventually merging into larger ones. Suppose that lightning strikes individual
trees at random, at a lower rate than that of tree growth. As each tree
burns, sparks fly out and the fire spreads to neighboring trees, continuing
to burn until the tree's local cluster is engulfed. In any spanning cluster,
one of the trees will eventually catch fire, and the entire cluster will
burn. As the system passes through the percolation threshold and the dimensions
of the grid go to infinity, major forest fires ignite across the grid,
bringing the system back to the brink.
Tree growth and forest fires are opposing mechanisms, conspiring to keep
the system in a dynamic equilibrium at or near the critical threshold.
Indeed, the distribution of the sizes of fires for such a model follows
a power law, suggesting, by analogy to the percolation model, that the
system is at or near the critical point.
Because the distribution of fires follows a power law, the probability
of very large fires is relatively high. Attempts to snuff out small fires,
moreover, can enable large clusters to develop faster, leading to a greater
probability of large conflagrations.
This conclusion is supported by practical data. In a 1998 paper, Cornell
University geologist Donald Turcotte, with visiting scholar Bruce Malamud
and Gleb Morein, then a graduate student, analyzed data sets for forest
fires around the world, including in Yellowstone National Park. They showed
that the distribution of the fires followed a power law and suggested
that real forests might be SOC systems. Before 1972, Yellowstone had a
policy of suppressing small forest fires. The researchers suggested that
this policy was an enabling factor in a fire that devastated the park
in 1988.
Another classic example of self-organized criticality---given in the
1987 paper by Per Bak, Chao Tang, and Kurt Wiesenfeld that introduced
the concept---is called the sand pile model. Here, grains of sand are
dropped at random sites on a grid. Eventually, small cones of sand build
up; as the slope of a cone reaches a critical value (dependent only on
the physical properties of the sand itself), one additional grain of sand
will cause an avalanche that redistributes sand throughout the grid. This
process keeps the slope of each small pile of sand just below the critical
threshold.
Cascading effects are a feature of both the sand pile and forest fire
models. One grain of sand or one spark can set off a chain of events that
affects the entire system.
From the Forest to the Grid
It's not hard to see the parallels between forest fires and sand piles
and the electric power grid, and indeed, this is the focus of a series
of papers by Dobson and his co-authors---Benjamin Carreras, David E. Newman,
and others. (The papers can be found on Dobson's Web site, http://eceserv0.ece.wisc.edu/~dobson/home.html.)
With data provided by NERC, the North American Electric Reliability Council,
the researchers analyzed a 15-year time series of transmission system
blackouts. Using three measures of blackout size, they demonstrated that
the distribution of blackout sizes follows a power law, indicating, they
say, that the power grid may be a self-organized critical system hovering
at or near the critical point. As further evidence, they showed that the
power data is, by several measures, similar to data from a sand pile model.
For the sand pile model, the counteraction of two processes---the localized
addition of sand and the pull of gravity---is what keeps the system hovering
at a dynamic equilibrium near the critical point. The researchers suggest
that a corresponding pair of forces work to keep the electric power system
in a near-critical balance. One is the yearly growth of about 2% in the
amount of power coursing through the grid; the opposing force is the human
response to blackouts. Each blackout, Dobson says, exposes bottlenecks
in the grid, which the power companies address when they add capacity
to the grid. That process, in turn, allows the grid to handle greater
power loads, which exposes the system to new blackout threats.
Viewed another way, the power grid operates within margins: Each power
line and each generator have a region of safe operation. When the power
load exceeds the margin for a line, the line trips out and the power redistributes
itself throughout the network according to the impedances of the surrounding
lines. When the grid is run with low loading of power (which is economically
impractical), a single event, such as a line tripping out, is unlikely
to cause others; events are independent, and the distribution of event
sizes has an exponential tail. Once loading is high, however, small events
have a high probability of cascading and spreading into large outages.
Extremely high loading is thus impractical as well.
To bolster their theory that load growth and engineering response to
blackouts lead to self-organization of the power system, Dobson and his
colleagues have devised power system simulation models that, incorporating
cascading blackouts, slow load growth, and engineering response to blackouts,
can self-organize to criticality.
Power Laws
One problem is the lack of a clear set of criteria for identifying a system
as self-organized critical. Power laws are an indication of a possible
SOC system, but many non-SOC systems show power laws, too.
On the Internet, for example, numbers of links to Web sites follow a
power law. One plausible explanation is that visitors to a site usually
arrive via a link, and some fraction of these visitors add their own links
to the site. This differential growth rate, according to the size of the
sites, leads to a power law. The same is true for paper citations and
other multiplicative random processes.
A recent theory known as HOT, for highly optimized tolerance, gives another
mechanism for generating power laws. The theory is essentially this: Going
back to the forest fire model, suppose that the trees are not allowed
to grow randomly, but rather are interspersed with fire breaks. If lightning
strikes uniformly throughout the forest, the risk of large fires will
be minimized by cutting the forest into equal-sized chunks, surrounded
by breaks. If lightning strikes more frequently in some areas than others,
the goal of preventing large fires can be best accomplished by creating
chunks with sizes inversely proportional to the rate at which lightning
strikes that area.
Jean Carlson of the University of California at Santa Barbara and John
Doyle of the California Institute of Technology have shown that such a
model exhibits power laws and suggest that the same might be true of other
engineered systems optimized by external agents. In a recent paper, M.D.
Stubna and
J. Fowler of Cornell showed that a modified version of the HOT model can
be made to fit the NERC power data, but also pointed to challenges in
mapping the HOT model to power systems.
SOC appears to be a better model than HOT for power systems, according
to Dobson. For one thing, he says, HOT assumes that the system is engineered
for global optimization, whereas the power grid is shaped by a complex
combination of engineering, economic, and political forces.
Even showing that data follows a power law can be difficult if the data
set isn't large enough. Dobson concedes that his time series isn't long
enough for a conclusive demonstration that the blackout sizes follow a
power law. "We did what we could on real data, given its limitations,
and what we have strongly suggests a complex system near criticality,
but it's not proof," he says.
How, then, can a SOC system be identified? The only convincing demonstration,
according to Mark Newman, a professor of physics at the University of
Michigan, would be to show that not only the external manifestations of
the system, like avalanche sizes, but also the internal dynamics resemble
those of an SOC system. Dobson believes that his group's simulation models
provide evidence that the internal dynamics of the power grid are SOC.
SOC and Public Policy
Suppose for the moment that the North American power grid is a self-organized
system, perpetually hovering near equilibrium. What are the implications
for public policy?
First, large power outages, as the stimulus for expansion, may be intrinsic
to the system. "It's an appalling thing when the lights go out, but
at the same time that's part of the self-organization," Dobson says.
"People rush around and invest in trying to do something about it,
and this weighs in as part of the dynamics." It is also harder to
identify one entity as the cause of a blackout when the large-scale pattern
of blackouts follows such a regular pattern.
It's possible to address the number of blackouts without changing the
overall dynamics of the system, Dobson points out. One possible system,
he suggests, would have blackouts of all sizes at half their current probability,
with the overall frequency pattern still following a power law, although
the gain would have to be balanced against the cost of implementing such
measures.
The possibility that the power grid may be SOC highlights a need for
caution when locally addressing problems. "You don't want a policy
that lowers the frequency of small blackouts and inadvertently increases
the probability of large ones," Dobson says. Seeking to determine
the effects of more, better-trained grid operators, his group has done
simulations with crude models. The result was a decrease in the number
of small blackouts and an increase in the number of large ones.
Still, Dobson points out, if nothing is done to upgrade the system, it
will move past criticality and many more large blackouts will occur. It
seems that effective policies may need to address the global dynamics
of the grid, and not just local bottlenecks.
For now, Dobson's goal is better monitoring of the system from real data
to determine how close it is to criticality. Understanding and monitoring
the complex systems dynamics of blackouts, he believes, will be key factors
in limiting the damage from major blackouts.
Sara Robinson is a freelance writer based in Pasadena, California.
©2003, Society for Industrial and Applied Mathematics
Updated: LBH
1/13/04
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