Hi Steve,
Good questions. Like others have recommended, check out the podcast series. We’ve tried really hard to elaborate more on how the system works, and to explain why the system gives the results that it does.
Without getting too deep into the nitty gritty, if you push for a hard sprint under fatigue, you might very well see an increase in PP. The assumption that the system is making is that if you could do that sprint effort under fatigue, then your starting Peak Power has to be higher than currently recorded in the system. Personally, I’ve never really hit my Peak Power, as hitting it will require almost the perfect scenario of inertia, resistance, and cadence/gearing. Unlike other systems, we don’t display your Power Duration curve as what you have done, but rather what you could do on a good day. Hitting your numbers below ~1 min on the power duration curve will require near perfect execution and a good cadence selection (being at a non-optimal cadence increases fatigue more quickly than being at the ideal cadence - think about trying to do a 1 min effort at 40rpm vs at 100 rpm).
Re: TP, the best way to get a valid reading for TP is to have an effort near MPA for extended periods of time. I typically find a micro-interval workout good for this (such as the Ronnestad workout (30/15’s), tabatas (20/10’s) or billats (30/30’s)). They allow you to sufficiently draw down MPA and keep it suppressed for extended periods of time, which makes for a good way to tease apart your signature - in particular HIE and TP.