Lights dim while vehicle is running

My SUV has had this issue where all the lights dim while the engine is running regardless of load (I have noticed it has an irregular idle when the battery voltage drops). It is not constant but it is a regular occurrence. The charging system light will come on every now and then for about a minute or so and then it will go away. I have had my charging system looked at and after multiple tests on it, Pepboys gave up on it. It does not do this with the vehicle off and all the lights on. I originally suspected the alternator but after all those tests on it, I was told the alternator was not the problem. I do not have any issues with start up. I could let it sit for a few days and it will start up no problem. I have made cross country trips in it and even though it has this issue, it has not died on me or left me stranded anywhere. Oh, and no check engine light. Any ideas?Thank you for your time.