I have a friend that would like to allow their dog to roam a large, unfenced, yard during the day but an "invisible" electric fence would be too prohibitive for the terrain.
Are there any alternatives, like GPS trackers for dogs that have an "electric fence" feature, that will train the dog to not stray from a defined area? And how safe would they be?
A basic search online against "GPS electric fence" using any given search engine reveals several brands of electric fences using GPS technology vs. using wireless technology. Here is a video explaining the technology.
In my experience with GPS, there is too much delay, variance, false positives, inaccuracy, and unreliability. That being said, I have NO experience with GPS-based electric fences and therefore can only speak from the following GPS-related experience:
I am friends with several parole officer-types who monitor GPS-based ankle-bracelet-wearing convicts. The convicts check in from certain places at certain hours, to confirm they are at home or at a job, but not anywhere they shouldn't be. I have witnessed several times when my parole officer friends are alerted (by text/phone) that a convict is not where he is supposed to be at a given time, and they have to contact/locate the convict. Many of these times, it is the GPS mis-alerting where the bracelet is located, or the bracelet isn't sending a signal, and my friends have to call and/or drive to confirm the convict location.
Perhaps it is the quality of the GPS device being used, but witnessing how potentially inaccurate they can be, I would not rely on GPS for ensuring containment of my dog. I wouldn't want to allow an accuracy error of 30 feet for my dog near a street, let alone the 10 foot error radius mentioned in keshlam's answer.
Electric dog fences are really only safe for dogs that don't really need fencing. An excited dog who sees another dog, a squirrel, a cat, is liable to run right through the electric fence, and will then be stuck outside it unable to get home again. They can be very dangerous for this reason. They also don't protect your dog from anything that might come onto your property - so if your dog is out there on his own, another dog might come in and attack him.
If your dog is capable of learning not to run away from home, you can teach him without shocking him with electricity to do that, as long as you aren't going to leave him out there alone for hours.
Give him lots of exercise, lots of social contact, lots of training and in between all that, plenty of company and quiet downtime, and there's a decent chance he will not want to leave your home. Sit outside with him and practice. Give him a command to leave your property, and reward him for waiting on the boundary. Here's a good video showing how to do this : https://www.youtube.com/watch?v=OuTh47i3hOY
If your dog is young and silly, or an enthusiastic chaser, and there are Way Too Fun things out there for him to run after, do not rely on an electric fence. It will not be safe. Use a leash and lots of supervision.
There is no invisible solution that will protect a dog being left outside unsupervised for long periods, and it would really be best to find a different management approach if that is your problem.
I looked into GPS-based collars about a year ago. These were not specifically for training the dog to stay, but are designed to alert the owner if a dog strays outside of its designated area, and to help the owner locate the straying dog. All of these devices had the same flaw: GPS monitoring is battery-intensive, and so the collars all user a periodic check instead of a continuous one. This was commonly around every 10 minutes. IMHO, these are useless because a dog could be a mile away in those 10 minutes.
There are some wireless invisible fences available, and these might be an alternative. I do not have any experience with this product: Havahart fence
GPS normally has an error radius of 10 feet or so. There are ways to narrow that down much more tightly but that needs additional hardware. Just not the right technology for this application.
