Power line rating conversion
you need to define what \$P_{i,k}^{\max}\$ is exactly, because as you state it. It can become confusing.

if you grab the current \$f_{i,k}^{\max}\$ and multiply it by \$V_{i}\$ as phasors you can get the power that is going out of the inode given that you defined the current in that direction.

after you compute the apparent Power S this way you can just take the real component using the cosine of the phasor angle and that gives 

$$
f_{i,k}^{\max}V_{i}=S
$$

Now I am not considering this a three phase system, but if it is just add a sqrt of 3 

now the resulting phasor of S will have an angle and the module multiplied by the cosine shall give you.

\$P_{i,k}^{\max}= |S|Cos(\phi)\$

Now this defines the maximum real power that goes out of the node, you could also talk about a Pmax related to the losses on the line... or you could define Pmax as the maximum real power at the receiving node k, you can use that voltage instead.

You need to think about how you want to define this, all 3 values I mentioned are related, but not the same.