I think the trick is to tune to system and changeover temp to always use the cheaper source of heat. That is one advantage of a dual-fuel system. As mentioned before, at my gas and electric rates the HP would still be cheaper to run down to 0°F. The only limitation is the capacity of the HP at those temps.
I'm not sure about the inverter units, but from what I found with the two-stage units the COP is the same between the two stages. Cost per BTUh is the same, but capacity is reduced. COP varies with outdoor temperature. Also, power consumption actually drops as outdoor temp drops, so it uses less amps at lower capacities.
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