The Power Grid: Fertile Ground for Math Research

October 31, 2003

Sara Robinson

With the ongoing restructuring of the North American power grid comes the ability to sell and transport power hundreds of miles from its production source. The goal is merely to lower power prices by increasing competition, but as the events of August 14 showed, increasing the complexity of a system can have unintended consequences.

On the morning of August 14, power produced by generators near New York City was flowing across Pennsylvania and Michigan to feed consumer demand in Toronto.
Shortly after noon, a series of seemingly disparate events unfolded: A 375-megawatt generating plant in central Ohio went offline for unknown reasons; an hour later, an enormous 785-megawatt plant north of Detroit went down, followed by a large plant in northern Ohio. Then, at about 2:00 P.M., a brush fire forced a high-voltage transmission line carrying many megawatts of power from southwest Ohio to northern Ohio to disconnect itself.

The electricity coursing through the grid seamlessly adjusted to the losses, rerouting itself through the network in accordance with the laws of physics. Thus, these generator losses, while large, were invisible to the millions of consumers in the Midwest.

Meanwhile, human system operators, too, were attempting to stabilize the new configuration. Operators cannot direct the flow of power along a particular pathway, but they can make adjustments that influence power flows indirectly, using computer programs to measure the state of the system and stepping up generators or shutting down lines. But those efforts were hampered by a lack of information. The new system in the Midwest region, implemented only recently, gave its operators limited data about the state of the network.

Still, the system seemed to have righted itself until just after 3:00 P.M., when three more high-voltage lines in Ohio overloaded and shut down. Again, the remaining grid adjusted, with power flows rerouting themselves through southern Ohio, Pennsylvania, and Michigan. Those lines weren't designed to carry such high loads, however, and soon began to overload and disconnect. The system began to lose its ability to self-stabilize, and voltage dropped precipitously, knocking out the remaining connections between eastern and northern Ohio. After successfully sustaining so much damage, the grid finally succumbed in a small way. Akron and surrounding areas went dark.

Nevertheless, the system had managed to confine the problems to a small region, severing connections between that region and the rest of the grid in an attempt to stabilize the remaining network. But shortly after 4:00 P.M., three more big lines overloaded and tripped, blocking pathways from southern and western Ohio into northern Ohio and eastern Michigan. A few more lines and generators later, and the Midwest region collapsed: Ohio, Michigan, and Ontario went out.

As Toronto went down, the vast feed of electrical power flowing from far away New York hit a dead end. Trapped, with nowhere to go, it could not reroute. Instead, it boomeranged back toward New York in a giant shockwave. The electric billboards in Times Square flickered and went out.

With a large chunk of Canada and the U.S. still shrouded in darkness, scientists, engineers, and politicians went into high gear, trying to trace the exact sequence of events that led to such a large-scale disaster. As this issue of SIAM News goes to press, however, experts still haven't put their finger on the triggering event that launched the avalanche. "All that is known is how the dominoes fell," says Massoud Amin, director of the center for development of technological leadership and a professor of electrical and computer engineering at the University of Minnesota.

The Modern Grid: Liability and Opportunity
A highly interconnected and integrated power network has obvious advantages: Because no one piece is essential, the network has a built-in back-up system; because power can be shipped from wherever it is produced most cheaply, the network is also cost-efficient. But as the events of August 14 demonstrate, the interconnection also comes at a price: The effects of a few problems in one region can propagate into others, potentially cascading into a massive failure.

The long-distance, bulk power transfers that have come with restructuring have had wide ranging effects on the system. Resulting complex phenomena include widespread power outages and huge spikes in prices, Amin says; in the latter half of the 1990s, the number of large-scale power outages (those affecting 50,000 or more customers) increased by 41%. The way to combat the complexity is with more sophisticated feedback, he suggests. With sufficient information to determine what is happening in real time, grid operators would be able to contain a cascading outage, or perhaps prevent one altogether.

Amin likens an effective system for managing the grid to a pumped-up version of Deep Blue, IBM's chess-playing program that beat Gary Kasparov. As in chess, power grid operators must be able to see several moves ahead, sorting through millions of possible scenarios to choose an appropriate response. But simulating the power grid is, in some ways, a much harder problem than chess, Amin says.

With thousands of generators and hundreds of thousands of miles of transmission lines, the problem requires the solution of nonlinear, stochastic differential equations sitting in an extraordinarily high-dimensional space. What's more, Deep Blue had plenty of time to compute each move, while scientists hope to do power grid simulations in real time or faster.

Restructuring poses economic challenges as well. Power markets must be designed to suit the properties of the commodity; mistakes can lead to price spikes like those that devastated California two years ago. Complicating things further, each region of the grid is making up its own rules for restructuring and proceeding at its own pace. Following the 2001 pricing debacle in California, other parts of the country are moving more slowly, says Vijay Vittal, a professor of electrical and computer engineering at Iowa State University. Some areas of the South, for instance, still haven't begun a restructuring process.
Of course, each of these challenges provides opportunities for research scientists-electrical engineers and control theorists as well as physicists, economists, mathematicians, and computer scientists.

"It's interdisciplinary: You can pick your subject and it applies to power systems," says Ian Dobson, a professor of electrical and computer engineering at the University of Wisconsin, Madison. "You need a lot of smart people to work in a coordinated way to have a reliable power system."

For our part, SIAM News will run a series of articles that give a taste of the range of challenging power-related mathematical problems. One article will take a closer look at power-market economics in the context of the market manipulation that played havoc with electricity prices in California in 2001. Another will look at how power collapses, such as the one in August, can be studied as large-scale, complex system phenomena. Still another will look at the physical and computational challenges of simulating the power grid in real time. Readers with further suggestions for articles are encouraged to contact the author (

This first article concludes with a basic overview, including the physics of electric power transport, the organization of the restructured power grid, and the operation of power markets.

The Constraints: Direction, Protection, and Control
Imagine that you're the proprietor of an ice cream shop that has a mandate to provide neighborhood children with all the ice cream they can eat. You're paid a flat rate per ice cream cone by the children's parents at the end of the month. Now suppose that you have to fulfill your mandate without an in-store freezer. All the ice cream you think the children will consume at a given time on a given day has to be ordered in advance, with adjustments via last-minute deliveries. Such are also the constraints of the power grid.

Electric power, unlike Internet packets, cannot be stored or shipped along specific routes; instead, it flows through all the circuits available according to their impedances within the network. This is why grid operators cannot manage power flows directly, but rather influence them by making systemic adjustments.

Electric power, unlike Internet packets, cannot be stored or shipped along specific routes; instead, it flows through all the circuits available according to their impedances within the network. This is why grid operators cannot manage power flows directly, but rather influence them by making systemic adjustments.

The interconnections are further subdivided into 142 "control areas," each with its own control system, which monitors data and maintains operating parameters at the correct levels. Each time a consumer plugs in a hairdryer, the flow of current causes an infinitesimal drop in frequency, which must then be restored to 60 Hz by the control system. The local generators, too, must be kept in constant synchrony with all the other generators in that interconnection.

To keep generators, transformers, or power lines from carrying too much power and overheating, the grid also contains protection systems. When these devices reach their safe limits of operation, they automatically shut down and the power flow is diverted. Similarly, if voltage drops catastrophically across a region, protection systems will isolate that region and protect the rest of the grid.

A control center typically receives data about power loads and flows every two minutes. The control software then calculates the state of the system, including the voltages and phase angles for every node of the system. The software also evaluates hundreds of what-if scenarios so that regional operators can better manage the system, adjusting for downed power lines or bringing in standby generators. As the August 14 blackout demonstrates, however, evaluating one level of what-if scenarios is not enough.

The Restructured Grid: Competition and Headaches
For most of the last century, U.S. power needs were supplied by federally regulated, local monopolies. In exchange for the right to do business without competition, these companies were charged with the task of supplying all existing and future customers in their service areas at a price determined by state regulators.

Control of the transmission lines was shared by all the local monopolies, which cooperated more than they competed. Rates charged to retail customers were chosen so as to cover the utilities' allowed expenses. What this meant, however, was that in regions that built a lot of power plants, prices grew much higher than in regions without much construction.

In an attempt to keep prices down, a 1970s directive created, for each region, separate generating companies from which the local utilities had to buy power at regulated prices. Thus, the tasks of generation and distribution were beginning to separate. The Federal Power Act of 1992 extended this process by mandating open access to power transmission lines, allowing companies to buy and sell power outside their regions. (Some of this background information is from an excellent article by Thomas J. Overbye in the May-June 2000 issue of American Scientist.)

In the late 1990s, utilities relinquished control over transmission networks to Independent System Operators-nonprofit, regional organizations that manage control areas and run power markets. The ISOs are responsible for maintaining the reliability of their control areas and must pay penalties if they operate beyond their limits. This mandate requires different types of contracts with generation companies. Under one type of arrangement, generators are paid by ISOs to remain on "spinning reserve"; if pressed into service, they are paid at a higher rate for the power they generate.

Following restructuring, the former monopolies focus primarily on distribution, making contracts with end-users and buying (and sometimes selling) power in the market.

The changes following the 1992 act are often referred to as "deregulation" of the electricity market; according to Frank A. Wolak, a professor of economics at Stanford University, this is a misnomer. Only the generation aspect of power has been deregulated, he says; transmission and end-user prices are still regulated.

In countries further along in the restructuring process, consumer prices vary depending on the power-market price, Wolak says. Such a market structure doesn't necessarily require expensive new power meters: One way to include demand-side feedback is to let prices vary by the time of day. Consumer feedback could have prevented the problems in California, Wolak points out, because consumers would have begun conserving power as prices got higher. Instead, consumer prices were capped and distribution companies were forced to meet steady demand at ever-growing prices.

In the U.S., "consumers are passive participants," Wolak remarks. "There isn't much benefit to restructuring unless demand side is involved." Another problem with American-style restructuring is that it's not clear who will pay for upgrades to transmission systems. As a result, construction of new power lines has ground to a halt and loads on existing ones are increasing.

The Power Market: An Exercise in Volatility
A third problem lies in the structure of the power markets. In many regions of the grid, "the original market design was done in an ad hoc way, without accounting for the engineering aspects and physical constraints of the electrical grid," says Vittal, of Iowa State.

Because it takes time to switch on and ramp up a power generator, distribution companies must estimate their power needs days or hours in advance. To do so, they use historical data for the day of the week, the time of year, and the weather forecast. Then, in the "day-ahead market," they buy power 24 hours in advance for every hour of the following day. They can also turn to an "hour-ahead market" to satisfy last-minute power demands at spot market prices, or (as was done more recently in California) they can purchase long-term contracts with individual generation companies.

Typically, the ISO serves as the power-market auctioneer, matching buyers and sellers. The auction mechanism works like this: First, the auctioneer figures out the total demand for power; at the same time, those selling power enter sealed bids indicating how much power they will sell and at what price. The bids are then entered, in order from lowest to highest, until the demand is met. The price for the last batch of power required to meet the demand then becomes the price that everyone pays.

It's easy to see that this system will lead to very high prices in the event that demand comes close to supply. Two years ago in California, the power producers figured this out and withheld power supply from the market. (For more on this subject, watch for one of the upcoming articles in SIAM News.)

So that's the modern power grid: an immensely complex system with many constraints. To date, despite its tremendous potential, the system has fallen victim to misguided public policies: poor design and insufficient funding. But with large-scale outages draining the economy of billions of dollars, willingness to invest is likely to grow, making the power grid a highly fertile ground for scientific research.

Sara Robinson is a freelance writer based in Pasadena, California.

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