How exactly does a solar flare cause a power line surge? There are occasional large scale electrical blackouts caused by solar flares. 
Quebec Solar Flare Blackout 
My question is: what affects the severity of the problem?
It obviously depends on the energy of the flare, which we can do nothing about.
But does it also depend on the length of the power line, the material it is made from, and/or the current or voltage on the line, at the time of the flare? 
Does it affect high tension transmission lines more than say, low current carrying telephone lines?
And, most importantly of all, what can utility companies do, if anything, to mitigate the effects of the flare?
 A: If the flare drops energy into the atmosphere, it sets up a changing electric field.  The strength of the electric field is measured in $V/m$.  This means that the longer the distance, the greater the potential difference.
If you have a short wire, the potential between one end and another is not very large.  But a long wire can connect points with much larger potential differences.  This can drive a high current through the wire that damages connected equipment.
Low-current lines can be similarly affected, but may also contain low-current fuses that protect equipment.  A power line will be designed to carry a lot of current before protections kick in, and a high (but unplanned) current can do more damage.
A: In my understanding the major problem with solar flares are the (near) DC currents that get induced in power lines (phone lines have DC blocks, so that's not a problem). Since both ends of a power transmission line are terminated by a transformer, the DC current will induce a DC magnetization in the transformer cores. Since the transformers are run close to the saturation magnetization of the core, the DC shift in magnetization will start saturating the transformer during part of the ac cycle, leading to a strong increase in current, which produces large I^2R losses. In effect it's not the deposited flare energy that destroys these transformers and/or triggers the overload circuits but the fact that they start acting like magnetic amplifiers, drawing a large multiple of the flare energy from the power plants.
The solution to this (rare) problem lies in better protection circuits for power networks, more robust transformer design, the addition of DC blocking circuits and, ultimately, in decentralized power production. 
