Here's another way to look at it. No amount of riding at 13mph is ever going to suddenly have your average-speed jump to 15mph. You can ride 5000mi or 50,000mi @ 13mph and that's going to be as fast as you can go. You'll have trained your body to ride really well at 13mph forever. You can probably even ride 500miles non-stop @13mph, great.
But if you want to average 15mph, then you gotta put in the intervals @20mph every once in a while and do some sprints @30mph as well. Yah, it'll be a lot of suffering (not pain in my book), but it'll be worth it. Eventually with this type of training, your average will creep up surely.