The pilot of a home-built; kit aircraft; while on an instructional flight; reported an engine failure; and subsequent loss of battery power; and all communications. This resulted in landing without receiving a clearance from the tower.

Narrative

First flight with instructor in Aircraft X and still learning how it worked. Throttled to idle to practice slow flight and stalls. Engine died but within gliding distance so transitioned to practicing with the engine off and headed back towards airport. As we neared the edge of the class D; we talked to tower and they told us where to proceed to our desired runway but there would be traffic landing on the cross runway. As we answered that we would comply our radio died. We then noticed that the battery was low and that we hadn't reduced the electrical load when the engine stopped. We tried to restart the engine but the battery was too low. Even though we didn't get permission to land; we had no other good option and proceeded to the area that tower told us. We maid a good landing and stopped well short of the cross runway and exited the runway.Lesson learned:Use engine shutoff checklist even when engine diesReduce electrical load when engine isn't runningMonitor battery voltageBe prepared with backup communication (tower phone number saved and carry backup radio).