I'm not good with electrics and that kind of thing (so it may be a very daft question) but, especially for storing meds, would it be a different electricity supply to the house? If so there could be less chance of realising if the electric had gone off for some reason, more noticeable quicker if it would affect the house too?
Apparently fridges are good for a while though if shut, I panicked about my meds when there was a widespread power cut affecting my local area, I was at work and had a load of injections in my fridge. Had it not been on the local news website though I'd probably have had no idea that my injections could have been at risk. They have put me on some kind of priority list for any power outages and for updates, now they know I am storing medication which has to be kept under a certain temperature. They were very good about it, send me text updates too. Might be useful for you too if your local one does the same. It was Electricity NW here, not my electric provider.