A start-stop or idle-stop method for a heavy-duty hybrid vehicle that turns off the fuel supply while maintaining the crankshaft rotation of the internal combustion engine when the vehicle stops or, optionally, when the vehicle travels downhill, travels in a noise sensitive location, travels in an exhaust emissions sensitive location, or operates in an emergency situation. The stop-start or idle-stop method automatically turns on the engine fuel supply to restart combustion when the vehicle starts accelerating, is no longer traveling downhill, is no longer traveling in a noise sensitive or exhaust sensitive location, is no longer in an emergency situation, or has dropped below the minimum energy storage restart level.