I had the same issue when I first purchased my rental. I was doing a quick walk through when I notice an orange heavy duty power cord stretched through the fence and into the external outlet. I followed back to my neighbors back door and walked around to the front to confront them about it. I knocked on the door and no one answered, I walked back around back and magically the cord had vanished.
The next day I made a surprise visit to the property and caught the neighbor outside. When I asked him about it, he claimed that he was embarrassed and he only did it because his power was cut off and he didn't want the food in his freezer to go bad. I have him the benefit of the doubt, but told him when I received the bill, if I saw something I didn't like he would be reimbursing me for the overages. The bill didn't change but a few pennies, and since then he has been my unofficial security guard, calling me if so much as a squirrel climbs on the porch. He also helped out my letting my contractors use water seeing as the water in my property was off at that time. A favor I probably wouldn't have received if I took a more aggressive approach. This was my experience, hope it helps.
