Welcome. My name is XXXXX XXXXX would be glad to assist.
Under normal operating conditions, an electric fence will draw power when an object touches between the power side and ground, yes. If the item does not break loose, the power will still flow, yes. Normally a weed or other object will actually burn in 2 or burn off completely and the power will stop.
There could be an instance, where something large is touching and constantly drawing the power and that would keep drawing until removed. It is difficult to say how much actual current is drawn to relate it to total home electric bill.
Best thing to verify the entire run and make sure all is clear and all lines are intact for an entire month with visual checks in order to see what happens on the bill.
Now, you could also, read your meter everyday and start a written log. Watch the numbers daily and if you see a spike in the number, just think of what may have taken place to create before thinking the fence did it. Stove uses a lot of power, air conditioners as well. so, just keep things in mind that may have made a large change otherwise, I would check the fence.
I really would not expect a fence to increase a bill much at all, since it uses a low current to shock animals. Tht small amount produced should be minimal on any usage.
Most chargers are set at the maximum watts they produce, so reading the nameplate will give you that information. There are some (rarely used) that produce what is needed to clear the line. Those however can be dangerous to livestock and why they are not used normally.
If you check plate and find it uses, say 20 watts of power. Take that 20 watts, and you could run it for 50 hours to equal 1000 watts or 1kw. Check your electric bill and see what they charge per KWH and that is what it costs for that 50 hours.
That is an example and may not be exactly what yours is, but probably close.
Let me know if any other is needed, just reply. Thanks