Do I need multiple Power Plants?

Solution 1:

While there are mods available that add mechanics regulating the demand side (that means, adding limits to the amount of coal for example, that a power plant can process in a certain timespan), in vanilla OpenTTD any processing plant can process unlimited amounts of goods.

However, delivering the goods to the most distant plant will yield the maximum amount of profits, so you might need to supply more than a single one. There is an in-game chart that shows the profit per good by (days in transit vs. distance travelled), and you can see that especially coal profits a lot from large distances.

Normally, you would start building around a single plant, and connect multiple mines to that, in order to reuse existing infrastructure. Then later re-route trains to go to different power plants that have a larger distance.

As Private Pansy correctly mentions, in the late game you need to optimize your track layout in order to get it more efficient. That means removing any "wait until fully loaded" orders and usually packing your tracks with the maximum amount of trains affordable. This is less important for coal, but highly important for other goods (say, fruits in the desert setting).