Calculating coefficient of performance (COP) for Heat Pump Here I am considering the following cycle;

I know that the coefficient of performance for a heat pump can be calculated by;
$$COP=\frac{Q_h}{W}$$
Where $Q_h$ is the heat exhausted into the hot reservoir. I am running into a problem with this... From here I know that the work $W$ for this cycle is given by;
$$W=nkT_h\ln{(V_f/V_i)}-nkT_h\left(1-\frac{V_i}{V_f}\right)$$
And then given that;
$$\frac{V_f}{V_i}=\frac{T_h}{T_c}$$
$$W=nkT_h\ln{(\frac{T_h}{T_c})}-nkT_h\left(1-\frac{T_c}{T_h}\right)$$
Where we take $T_c$ to be the temperature at point 1. What I am confused about is when we calculate, correct me if I'm wrong, but isn't the process 2-3 the only segment that releases heat into the hot reservoir? and thus, $Q_h=Q_{2\to3}$? When I carry out the calculations, I then get;
$$Q_h=nkT_h\ln{(\frac{T_h}{T_c})}$$
If I then calculate the coefficient of performace of this heat pump, I obtain the relation;
$$COP=\frac{\ln{(\frac{T_h}{T_c})}}{\ln{(\frac{T_h}{T_c})}-\left(1-\frac{T_c}{T_h}\right)}$$
Which in fact is greater than the carnot COP for a heat pump,
$$COP_{C}=\frac{T_h}{T_h-T_c}$$
Have I gone somewhere in my calculations? or am I missing something else?
 A: Note that the step $1 \to 2$ takes heat from the hot reservoir. (It cannot be from the cold reservoir, as this would require heat flowing from cold to hot.) Since the goal of the cycle is to send heat to the hot reservoir, accounting for this lowers the efficiency.
A: 
Have I gone somewhere in my calculations?

Yes, I believe some of your equations are incorrect. First of all, it appears you have used the wrong equation for work. You have used.
$$W=nkT_h\ln{(V_f/V_i)}-nkT_h\left(1-\frac{V_i}{V_f}\right)$$
This is the work done for the reversible heat engine cycle described in your original post. The isothermal expansion work is positive (done by the gas) and the isobaric compression work is negative (work done on the gas). However, for the heat pump the cycle reverses. Work done during the isothermal compression process is negative (done on the gas) and during the isobaric expansion process is positive (done by the gas). I believe equation should then be:
$$W=nkT_h\ln{(V_i/V_f)}+nkT_h\left(1-\frac{V_i}{V_f}\right)$$
It also appears you used the wrong equation for the heat rejected in the isothermal compression. Your equation is:
$$Q_h=nkT_h\ln{(\frac{T_h}{T_c})}$$
Which would yield a positive value for $Q_h$, whereas it should be negative as heat is out of the system. I believe it should be
$$Q_h=nkT_h\ln{(\frac{T_c}{T_h})}$$
Since it appears you have used the wrong equations for both work and heat, that would lead you to an in incorrect equation for the COP. I suggest you review the calculations.

or am I missing something else?

I believe so. I think there is a fundamental problem with considering your cycle as a refrigeration/heat pump cycle. To my knowledge (which is admittedly limited re refrigeration) all such cycles normally involve two fixed temperature environments, the environment to be cooled and the environment to be heated. That is the basis of the Carnot COP that you are comparing your cycle with.
In your cycle, if it is to be considered reversible as your previous heat cycle post, it appears there are multiple temperature environments, which makes it unclear what the objective (environment for the desired heat transfer) is.
Heat is transferred out in both the isothermal and isochoric processes. Which is the desired heat transfer to be used in the COP?
During the heat absorption (isobaric expansion), heat is obtained from a higher (than $T_c$) temperature environment. Is this the same environment where heat is rejected during the isothermal compression? If so, then it would seem it must be subtracted from $Q_h$ reducing the COP as pointed out by @knzhou.
Hope this helps.
