Hello, thanks for updates! Is it intended to spill fragments when storage full and how to use them when research stopped?

Yes, it's normal for them to spill if you have an excess of fragments you can use them as fuel or remove few power leeches to reduce production
As a fuel, they'll return exactly the same power it was used to make them

Will it affect charging speed?

If your question is will it affect the Tesseract's charging speed the answer is: The charging speed is only affected by power supplied by power leeches the more of them you have the faster it will recharge
if your question is will it increase vehicle speed the answer is yes it provides a pretty good boost to your speed

I mean faster charging make faster fragment production

Yes, all tesseract's excess of energy is converted into fragments each fragment produced drains 100MJ from the tesseract if you give it 500MW beyond its consumption it will produce 5 fragments per second, if you give it 50GW it will produce 500 fragments

Ok, now I understand this. Thanks for answer!