One traction battery, plus a 12 V battery. Software controls keep it charged between around 30 to 70% for long life. Deep discharges and overcharging shortens battery life. Most have experienced this with laptops and cell phones. A PHEV is a regular hybrid with a larger battery, nothing more. Two advantages over a "regular" hybrid, the main one is that you can plug in to charge the battery using your relatively cheap home power instead of the ICE recharging the battery. The other one, relatively rare, is you can recapture more energy going down a really long steep hill than the smaller traction battery before the car will turn on the ICE for engine braking instead of overcharging the battery with regen.
Two huge downsides of PHEVs. The main one is that they cost a lot more. All us taxpayers subsidize the tax rebates on PHEVs (plus enforce extra cost on manufacturers to make such cars to bring down their total fleet mpg). Otherwise there would be very few sales at all.
Eventually those subsidies will stop, and either demand (likely from fuel prices well north of $4) will keep these cars in production, or the cost of manufacture will drop (which is happening with battery cost in a significant way). But the extra stuff needed to plug in will always make such cars cost more than "regular" hybrids.
They also weigh a lot more, which is why they get less mpg than regular hybrids when not being plugged in - like on a long trip.
2018 Kia LX HEV Metal Stream with Advanced Tech