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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