Yes. All good quality studio flash heads have a number of capacitors (cheaper ones may only have 1 or 2)
Power is reduced in one of 2 ways, either switching out capacitors or reducing the voltage to the capacitors. Switching out capacitors is by far the best method but people expect to be able to set the flash power to any setting they want, switching out capacitors doesn't allow this level of fine control so, even if the flash head switches out capacitors, heads typically have a voltage reduction control (potentiometer) in place too.
So, when the power is reduced the voltage is reduced too, and it takes longer for the electrical current to flow through the flash tube. Cheaper more cheaply made flash heads, with a small number of capacitors, rely on a potentiometer heavily, so the flash duration can increase dramatically at low power settings.