Is there a formula to determine what temp would be most economical efficient to cut over my Heat pump to backup propane based on energy costs?

I see "rules of thump" all over the place from 45 to 30 deg to leave it running.
But I wanted to know the math (if its not that complicated.) so I can change the cut over point as energy prices change.

input data. 

Heat pump:
3 ton 2 stage
SEER 16
HSPF 9.7

Gas heat: (converted to propane)
2 stage 96K BTU
96.1% efficient


Electric price is .13 Kwh.
propane price is $1.50 per gallon


System works great.  Currently I allow AUX (propane) to run up to 34 deg (it turns off heat pump when gas comes on)
and I have the compressor to cut out at 25 deg.

The thermostat (ecobee smart with Remote sensor manual) also controls the advances between stages.

It sounds like you are looking to calculate the heat pump economic balance point which of course depends the specific heat pump capacity/COP curve and the specific building heat load curve. Anyhow, this might get you thinking/started:

Economic Balance Point Formula

Thanks, I will try the math part when I get home

COP47=3.88
COP17=2.74
K= 3412 BTU/KW
F=96.1 (AFUE - 96.1%)
E= .13 / KWHR
C=1.50 gallon
B=91600 gallon

compared to your document 4/82, COP has gone way up 
But so has C and E

my calculations come to around 0 deg, which makes me realize there are other factors that need to be considered.

using the loads from my Manual J calculation, and the output tables from the heat pump manufacture (and some other tools I have) I see at 0 deg F my Heat pump output is well below the building load of the home.
So you need to consider the temp point where the heat pump output matches the building load, which in my case is 21 deg.
because I can only use 5 deg numbers (20, 25, 30, etc) in the temp field which I set as my compressor lockout, I think I will stick with 25 deg (where the heater run time is 92% when on heat pump).

Thanks for the math though, its always good to exercise the head.

Don't forget that the heat pump has to periodically go into defrost mode (essentially runs in reverse like an air conditioner with the outside fan off) when the outside coils are below freezing to remove frost build up. Your heat pump COP is <1 while in defrost mode because it removes heat from the home to heat the outside coils. On most heat pumps with electric backup heat, they generally turn on the heat strips during defrost mode to keep from blowing cold air inside. I'm not sure about heat pumps with gas furnace backup. If it kicks in the gas furnace during defrost mode, this can result in short operating cycles on the gas furnace which would reduce efficiency and negatively affect equipment longevity.

I would probably use a heat pump-to-gas furnace transition temperature of 30F or 35F just to keep the wear and tear down on the heat pump, even if the operational cost was slightly more.

Yes, I would agree with Arkie. This formula does NOT account for maintenance cost (only fuel cost). With heat pumps, reducing wear and tear is likely more important than anything else as the associated maintenance cost can quickly destroy the ROI operational cost benefit analysis.

When going into defrost mode, the outside unit triggers the gas furnace to start stage 1. 

Here are some figures that come out of one of the Geo Thermal applications I have  (it allows you to compare air/air heat pumps to geo solutions).
Compressor lock out at 25 deg.
Heat pump does 86% of heating = $382.00 per year
Aux does 14% of heating = $89.00 per year
total $470.00

Compressor lock out at 30 deg
Heat pump does 71% of heating = $310.00 per year
Aux does 29% of heating = #179.00 per year
total $489.00

question about the wear on the heat pump, when you have an electric backup heat pump, don't they run the compressor any time there is a call for heat, regardless of the temp, they just turn on additional electric heat at lower temps?

climatemaster.com has a modeling program intended to compare different products so ASHPs are one of the selections. So you could download that (free) put in your system and try different balance points.

question about the wear on the heat pump, when you have an electric backup heat pump, don't they run the compressor any time there is a call for heat, regardless of the temp, they just turn on additional electric heat at lower temps?

I have a heatpump, a carrier that has a carrier thermostat. It uses a backup gas coil when temps get cold (like they are here in Chicago). I have been very busy working to finish the rest of the house but I can tell you the supplemental coil is used when an internal trigger is set on the thermostat and the thermostat shows this on the screen. It only happens when the temps get very cold (teens or less - I"m guessing - I only noticed it when it was minus 6 one morning. In the teens it was off, so at some point between 15 and 0 the supplemental kicks in). I have radiant slab in the basement at 65 (air temp) which provides the "base" heat for the home and the HP only takes it up from there to 70 (3400 SF). THe HP runs, but not exhaustingly.