I find flat bars unbearable for anything except short rides due to the lack of multiple hand positions. However, 10 to 15 miles is not very far so there may be something else at play...
Many people position their seats too far forward in an attempt to get closer to the bars, resulting in a bike fit problem. The problem is that sliding your seat forward puts more weight on your hands. If you have too much weight on your hands (i.e. you fall forward if you take pressure off the bars) then your seat needs to be moved backwards. That may also mean changing the stem to make it have less forward reach. It's difficult to diagnose whether this is a bike fit issue without seeing pictures of you riding your bike. You could also go to a good LBS and talk to someone experienced in bike fitting.