Do I need multiple Power Plants?

There's one additional reason to use two power plants, and that is making more money with each train, to increase their effective ROI ratio. With the default trainset, refits are typically expensive or not possible, so you would have trains hauling empty loads back in a simple setup. With two plants, you can haul coal 'both ways'.

Suppose you identify two pairs of power plant and coal mine, close together, with both pairs far apart. You could build two small train lines, but since you're paid 'by the square', it can be far more lucrative to have your train line go:

Coal mine #1 ----> Plant #1 -> Coal mine #2 ----> Plant #2 -> 

where longer arrows represent greater distances. You can get (almost) twice the income over a simple one-way empty, one-way full connection. Almost, since you will probably need a small distance between the Plant and the next Mine, and since full trains are heavier and accellerate slower.

By the way, this is a great way to jumpstart your company in openTTD, and typically regarded as the optimal opening move on most maps/settings. Here's a link to a more in-depth tutorial.