Edison base electrical fuses were widely used for household electrical circuit protection prior to the introduction of circuit breakers and are still found in some older homes. When electrical fuses blow, it is important to replace the fuse with a fuse of the same amperage rating to continue providing protection to circuit. Often times, the correct replacement fuse is not immediately available and a U.S. penny could be used as substitute. Substituting a penny for a fuse did solve the immediate problem of no power in the circuit, but created a new and possibly more serious problem of no circuit protection.

Through a friend of a friend, I was introduced to Texas A&M University research associate Alexander Roitershtein Ph.D. (Alex) who among other pursuits, studies Markov Chains. A Markov Chain is a model used in the study of stochastics to describe a sequence of events where the outcome of the events are dependent only on the outcome of the previous event.

The Penny

To introduce Markov Chains, many times a series of coin tosses with two possible outcome states (heads or tails) is used. There is also a third possible outcome state of the coin coming to rest standing on edge. Because of the angular momentum of the coin when it is tossed, the chance of it coming to rest standing on edge is infinitesimally small. This outcome state is usually not considered, but it is still possible.

The possibility of the coin coming to rest, standing on edge is much greater for coins with flat edges versus coins with irregular edges. Since angular momentum weighs heavily into this outcome, small coins with flat edges like U.S. pennies and nickles are more likely to produce this outcome than larger coins with flat edges like the Israeli (NIS) half-shekel.

The Fuse Box

The electrical grid is the interconnected network that delivers electric power from the power stations where it is generated to consumers. The electrical grid is designed to be resilient and to prevent widespread outages by providing multiple paths to the consumer from power stations. Single point failures do occur and most times the outage is localized and affects only a small number of consumers.

Large scale outages happen infrequently. Compared to the number of hours that the power is on, the number of hours it is off across a large geographic area is quite small. Sort of like a tossed penny coming to rest, standing on edge just once, after thousands of coin tosses. Even following a seemingly cataclysmic transformer explosion (pictured at right) at the Con Ed facility in Astoria, it did not result in a major power outage.The photograph was taken fifteen miles away in Rockaway Beach.

Let’s examine the cascade of events that led to two major power outages that affected New York City.

1965 Northeast Blackout

The 1965 Northeast Blackout illustrates a Markov Chain of protective relays tripping and breaking the interconnected electrical grid and causing power stations to shut down.

1. An improperly set, protective relay at the Beck Power Station in Ontario tripped and cut off power to Southern Ontario.
2. With power cut off to Southern Ontario, other protective relays tripped resulting in the Beck Power Station shutting down.
3. Within minutes, the electrical grid in the Northeast with protective relays tripping and the subsequent loss of generating capacity cascaded through the grid. This effectively broke the grid to the point that it consisted of disconnected islands.
4. The subsequent load imbalances caused nearly every power station in the now, disconnected grid to shut down.
5. Eleven minutes after the first protective relay tripped, New York City was plunged into darkness.

1977 New York City Blackout

Although the chain of event leading up to the 1977 New York City Blackout was initiated by a series of lightening strikes and not faulty hardware, it still can be described in terms of a Markov Chain.

1. The first lightning strike takes the Indian Point Power Station North of New York City offline.
2. The second lightening strike takes two major transmission lines North of New York City offline.
3. The power company (Consolidated Edison) tries to bring additional generator online, but the generators fail to start.
4. A third lightning strike takes two more major transmission lines North of New York City offline.
5. Approximately thirty minutes after the first lightning strike, the power company begins reducing line voltage to reduce the load on the system.
6. The only remaining major transmission line from North of New York City began to sag from being overloaded and shorted out to ground.
7. Under the heavy load from an increasingly isolated New York City, the Long Island Lighting Company disconnected their transmission line that fed power to New York City.
8. Protective relays on the transmission line from New Jersey tripped and disconnected power coming from New Jersey.
9. The power company was unable to meet the power demand on its own and was in the process of disconnecting consumers when the largest generator in the system (known as Big Allis) shut down.
10. Approximately one hour after the first lightning strike, the entire New York City electrical grid shut down.

In the paper entitled: Markovian influence graph formed from utility line outage data to mitigate cascading Alex and his colleagues use stochastic modeling to examine how cascades as those outlined above can be mitigated.

The entire paper can be accessed here.