I'm looking to build a battery pack from lithium-ion 18650 cells, 13s16p (parallel first) to achieve around a 50V (nominal) battery pack. I realize there are probably charge solutions out there with the proper voltage and BMS which can be used to charge the entire pack with balancing and protection; however, my idea is to use a single adjustable buck CC power supply on each parallel cell pack to keep it charged to 4.2V while in series, and then power all of these supplies with a single larger amperage AC-DC 12V supply which can be powered with 120V AC.
I'm wondering if having all of the individual battery chargers powered by the same supply will be a conflict while the packs are wired in series (the inputs would all share a common ground). Does each circuit need to be segregated with a transformer? Is there another way to separate them? Other pitfalls I'm not thinking of?