There is a coal power plant near me which in recent years I saw it running only during winter.

I think what he was saying is that with a peaker plant you have an extra source of power. So if you have a low wind/low irradiance period - you can rely on peaker to provide more power, but by this stage batteries are long gone.

Is that a local question for Vermont, a nation question for the US, or a global question?

If the question is global, then yes, there are, although as renewable energy has increased globally so has the need for peaker plants to handle the variance in the grid, thus plants that historically has only been used for seasonal variance are now used all year round to handle weather variance. An example of that is the oil power plant in south of Sweden.