Caveat: I am not a power engineer, so the following comments are deliberately approximations - there are subtle effects which I carefully ignore!
Couple of points worth mentioning. The skin effect is dependant on a number of factors, including the line frequency.
At DC, there is no skin effect and there is uniform current carrying capability across the whole conductor.
At 50Hz, skin depth is about 9mm, i.e. the current travels in the outer 9mm of the conductor, so if your conductor is less than 18mm in diameter, this doesn't matter. However, for grid power distribution, you need very high currents, even at 400kV plus you need the physical strength of a larger cable to handle the spans between pylons, so cables are larger than 18mm o.d. and thus skin effect becomes an issue for long high power transmission lines. Special cables are used where there is an outer sheath of low resistivity material, e.g. copper, and an inner core which is strong, e.g. made of steel.
HVDC is very efficient, not just because you can link non-phase-locked grids, but because there is very little power loss in the cables due to corona discharge losses. You do have AC/DC -> DC/AC conversion losses at each end of the cable, but modern solid-state systems MMC converters are very efficient so there is often an overall benefit to using HVDC over HVAC in many circumstances.
It's because of I2R losses, which are the same for AC & DC, that we use high voltages to reduce the current necessary to deliver a given amount of power - P=IV, so for a given P, if you dramatically increase V, the required I (current) drops proportionally. As resistive power loss is proportional to the SQUARE of I and R (the resistance of the line) is fixed, reducing I is a GOOD THING.
Corona losses are calculated using Peek's Formulae - I'm approximating here as the formulae is very specific and complex. With a typical AC grid system, you lose between around 3.5kW/km (good weather) and 8.4kW/km (stormy weather) per phase . The UK grid has around 25,000km of cables in total (approximate sum of all high voltage AC distribution). Assuming all this is single phase (it's not - most of it is 3 phase), on a dry day roughly 87.5MW is being wasted and on a stormy day around 210MW is lost PER PHASE. So the total is more like 3 times that and then times by 24 to get to daily power loss in kWh or MWh as appropriate. Plus that excludes traditional I2R (I squared R) power losses, i.e. the power lost due to the resistance of the cables.
That's a LOT of lost power, but it's still the easiest way to distribute lots of power as AC allows the use of simple transformers to step up and down.
There are other issues too with HVAC and HVDC that I conveniently ignore here - HVAC corona can effect the quality of the underlying AC, introducing amongst other things, noise (both RF and on the line) and harmonics. HVDC conversion produces a stepped approximation to a true sine wave and thus can be rich in harmonics which have to be removed before onward transmission.
It was a fire at the Sellindge end of the cross-channel HVDC cable's DC/AC converter that caused a big outage and will reduce cross-channel capacity by around 50% for many months.
