Wind farms provide electricity with no direct emissions. However, their output cannot be forecasted perfectly, even a short time ahead. Consequently, power systems with large amounts of wind power may need to keep extra fossil-fired generators turned on and ready to provide power if wind farm output drops unexpectedly. In this work, I introduce a new model for estimating the uncertainty in short-term wind power forecasts, and how this uncertainty varies as wind power is aggregated over larger regions. I then use this model to estimate the reserve requirements in order to compensate for wind forecast errors to a 99.999% level of reliability, and an upper limit on the amount of carbon dioxide that would be emitted if natural gas power plants are used for this purpose. I find that for regions larger than 500 km across, operating reserves will undo 6% or less of the greenhouse gas emission savings that would otherwise be expected from wind power.