Ahh yeah, it's a little confusing just looking at it. It's actually pretty simple though. Lemme see if I can explain it clearly..
1. You have your plain old main power grid with batteries right?
2. You create a second, tiny grid with just one pole. Use shift to place it with no wires near one of your main grid poles. Pop down a single battery so it overlaps both coverage areas like this:
The cool thing is that battery will be doing double-duty; it will drain down with your main grid, as well as handling the additional drain from the inserters/lights we're going to be adding. It dies first!
3. Now let's pop down an inserter near the edge of the tiny grid (the small poles blue coverage area). Let's also give it a smart chest to keep full. If that battery every runs empty, this inserter will stop and the chest will empty out.
4. Now let's add in another inserter to empty the chest. It needs to be in your "main" grid so it can keep running when the tiny grid runs dry. Let's also pop down some conveyers to circle the items back around to the filler inserter:
5. Done. Add in 6-12 items to the chest (enough that it doesn't get empty while running normally) and watch it go:
6. So we have an inserter that shuts off before your main grid runs dry, and a smart chest that will send the signal we can use when it empties out. Now just wire your boilers' Smart Inserters to turn run if "Item < 1" like this:
When the filler inserter runs out of power, the smart chest gets emptied, "Item < 1" and your boilers turn on just before the power cuts out! Works every time, even with varying loads and main battery banks that aren't big enough yet!
If you want your steam to cut in sooner, add more drain to the tiny grid with some lights. If you have a huge battery bank in your main grid and want backup power to turn on a little later, add another battery or two between both grids. A good way to see how it's working and tune the system is to right-click your main-grid pole and then hover over the stranded battery with your mouse. You can compare the total "Accumulator Capacity" on the left to the "Energy" reading over on the right for the single accu: