yes algae will grow under artificial light, there are lots of ways people keep it in control but I've never heard that anyone was able to totally eliminate it.
H2O2 is a common additive used to control mold / fungus / algae and in the right concentrations harmless to your plants.

Well, unless you make the system hostile to plants, I believe you have to accept you're going to have algae.

Sorry for my late response. Thanks ajames54 and dbarron.

I think I have to accept the algae there. I might need some H2O2 to keep my system clean. :)

As a side note and a late one at that, you can
put a barley straw mat in the reservoir and allow it to float. It will inhibit
algae growth.

I would just be careful
with H2O2 because you can mess up bacteria in your system. I know there have
been some posts in the past knocking the use of it.