Simulations of three JET H-mode discharges in a power scan are carried out using the JETTO integrated modeling code with predictive core and pedestal models, which include the effect of ELMs. It is found that current-driven peeling modes trigger the ELM crashes in these discharges and, as a result, yield an explanation of the increase in pedestal height with heating power. After each ELM crash, the bootstrap current density and the related pressure gradient rapidly increase with increasing heating power, while the total current density rises only slowly because the total current density is impeded by a back EMF. Hence, as the heating power is increased, the pedestal pressure can rise to higher values during an ELM cycle before the current density reaches the level required for destabilization of the current-driven peeling modes. A stability analysis using the MISHKA code is carried out in conjunction with these simulations. The analysis includes infinite-n ideal ballooning, finite-n ballooning and low-n kink (peeling) modes.