As the debate over global warming rages on, there’s one thing we can all agree on. It’s stupid! It’s like wondering if the can of varnish you drank was half-full or half-empty. Pollution is bad. Period.

This leads to several conclusions I’m sure you’ve heard ad nauseam, so I’ll jump straight to the point: renewable energy, and sustainability in society as a whole, is paramount to our long-term survival as a country, and as a species.

Renewable energy has come a long way in the last half-century. Photovoltaic cells have become cheaper and more efficient; wind turbine design has improved power output, reduced noise, and expanded the range of acceptable wind speeds; and new sources of energy like geothermal show promise.

In general, renewable energy can be filed into two categories: that which generates at a fairly constant rate (hydroelectric, geothermal), and that which generates intermittently, subject to outside factors (wind, solar).

The electric grid in the United States is powered overwhelmingly by coal. The Regional Transmission Operators (RTOs) who run the grid since the federal government deregulated the industry like on-demand power, because the load on the grid is not constant by any means. It goes up during the day, down at night, has a higher average in the summer due to air conditioning, etc. Being able to generate power as needed makes managing the grid simple.

Sadly, this same reasoning leads the RTOs to dislike most renewable sources—they’re harder to integrate into the grid. You can predict the average power output of a wind farm or solar site, but at any given moment they may be well above or well below this average. If they’re above, it’s no problem—just turn off some traditional generation. If they go below, though, you need to turn on generation, which takes time. A lot of time.

For this reason, Generation Companies (GenCos) typically have spinning reserve: generation that can be ramped up quickly to meet sudden grid deficits. This typically comes in the form of gas turbines or coal plants running at not-quite-maximum capacity. Paradoxically, adding renewable power to the grid increases the need for spinning reserve, and thus the need for central generation.

If there was a way to level the output of these clean power sources, it would be much simpler to add them to the grid. Once the grid constraints are overcome, the economics of renewables would make them a very attractive option to the GenCos. For example, wind power is very fast to install, easy to maintain, is cheaper than nuclear and gas power, and is priced competitively with coal.

Leveling of output requires energy storage. While there are some interesting and creative ways of doing this (Compressed air energy storage and pumped hydroelectric, just to name a couple), there is also one very obvious way: batteries. But simply building giant banks of batteries would require tremendous amounts of space and money. Is there some unused source of electric energy storage we’re missing?

There is, and it’s in the light vehicle fleet. There are approximately 191,000,000 cars in the USA. If each of these was outfitted with a 15 kilowatt plug and a high capacity battery, the grid could draw 2,865 gigawatts of power from these batteries—over six times the average load on the national grid. The storage is there, so how to we use it?

Unfortunately, the cost of retrofitting cars with the kind of batteries and control systems necessary to make this scheme work would be astronomical. Fortunately, we are on the edge of a new frontier in automotive propulsion: plug in hybrid-electric vehicles, fuel cell cars, and of course electric cars are all ideally suited to Vehicle-to-Grid, or V2G, since they already have state-of-the-art Li-Ion batteries with much better capacity/weight than the traditional Lead-acid batteries in most cars today.

It may be easier to implement large scale renewable power generation and low emissions vehicles at the same time, rather than separately. The grid storage afforded by V2G makes renewable power far more attractive, while the fees paid by RTOs to owners of these new vehicles for use of their batteries would help offset their otherwise higher price.

Now that the framework for V2G has been laid, we have to consider a vast array of implementation obstacles, all of which must be modeled with graph theory and game theory.

The graph theory is a consequence of geography: electric current cannot be transported an arbitrary distance due to line losses. RTOs must somehow weight storage according to its distance from load, and at some distance, the storage will simply become inaccessible.

The game theory arises from myriad concerns of both RTOs and car owners. RTOs need to make sure that the grid supply equals the grid load at all times. They can do this by drawing upon spinning reserve or the new V2G storage, each with their own costs.

There are rules about how they may draw on V2G, though. Cars are parked 23 hours a day on average, but you can’t draw power from a car while it’s driving. You also can’t draw batteries all the way down at any time, since this can cause lasting damage to the battery and leave the motorist stranded.

How low can the batteries be drawn? If you’re sure the motorist isn’t going anywhere while he’s at work, you could draw them almost to death and let them recharge while he’s on the highway heading home, in the case of HEVs, which run on gasoline when the engine is at its most efficient, aka, highway driving). If this is the strategy, though, and the car is driven around during lunch, the vehicle will be forced to rely on the gas engine, which is less efficient at city driving than the electric one.

For electric vehicles, the question becomes more complicated. How much charge needs to be left in the battery at the end of the day depends on how far user needs to drive to get home. Will he take the car out soon after reaching home? Will he be able to charge it overnight every night? And how will he weight all of these concerns against the money paid to him by the RTOs?

Presumably, all of this information can be gathered from the car owners with known statistical deviation, such as how often they go out for lunch and on what days. These statistics, along with the rules discussed earlier, define a game played by the various motorists and transmission companies, and it’s played on a nation-wide graph. Given this vast body of data and the complexity of the grid network as is, how can we best use this newfound electrical resource?

I don’t know. I’m just an undergrad. But this is perhaps the most useful and broad application of graph theory in the world today.

Primary sources:

Kempton, W., and Dhanju, A., 2006, Electric Vehicles With V2G: Storage For Large Scale Wind Power, available at http://www.udel.edu/V2G/docs/KemptonDhanju06-V2G-Wind.pdf

and

IEEE Power and Energy Magazine, Vol. 3, Number 6, Working with Wind: Integrating Wind into the Power System