In 1960s nasa was awarded its first patent for a fake sun. Since that time many more related patents have been acquired. With the help of bausch and Lomb they now have a working fake sun and is in use. There are still problems to work out as you've probably noticed.

Basically, a large reflector is placed in orbit. Then, a combination of lasers and?? Are used to charge it. You may sometimes notice they are hexagonal. This proved to be their best design.

Since a single sun only illuminates a relatively small area if you happen to be near the edge you will see two sun's, the real and fake.

As to why I have no idea. The only possible reason I can think of is, if the earth tilts too far, maybe some of our crop land will need extra light??