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The problem of getting power to startup equipment is one thing, but there's another cool problem that this article covers but I don't think explained very well.
A power grid is a massive distributed physical system. The energy input must exactly match the energy output at all times.
But what happens when the energy ins and outs are not balanced? The answer is that a balance is found somehow. Physics demands it.
If there is excess power on the grid and no electrical load, that power comes back to the generator (s). The turbines or whatever driving the generators produce more torque than the retarding torque from the generator coils, so they speed up. The AC grid frequency is mostly maintained by the rotating generator speed (3000 rpm for 50 Hz), so that goes up too.
Conversely, if there is excess load and not enough power, electrical drag from the generator coils exceeds the torque from the turbine (or whatever), and the generator slows down. The operator has to burn more fuel, or pull out control rods, or open more water gates, to get the speed back up.
So what is the black start challenge here? You have to go from 0 W to whatever GW the grid normally runs at. Normally when a generator plant is switched onto the grid, that gen represents a small fraction of the total grid power, so the disturbance to the grid is small. But coming back from a black start that's not true. Going from 1 plant online to 2: you could be doubling the power level. This means you have to switch on loads (possibly many km away) at the exact same time you switch in the power. If the disturbance is too much, various equipment will trip off the grid as the AC frequency careens out of control.
it's also made worse by the fact that many appliances are turned on in blacked out areas, so when these are brought online, there's spike in load power that then tapers off pretty quickly. this can be managed by powering on areas on in smaller chunks, as small as single blocks
this is PSA to turn off appliances off during blackout (biggest ones are things that deal with heat: mostly air conditioning/heat pump and all kinds of heaters)
also in Spain specifically some capacity was met by spinning generators from nuclear or hydro or gas powerplants, but also there's a lot of photovoltaic generation, which doesn't vary frequency with imbalances in load. after loss of power nuclear reactors that were running at that time are out for a day because of xenon poisoning, and it looks like first nuclear powerplant went online again only yesterday
maybe they'll update inverters in at least some of generating stations so that blackstart with PV assistance would be possible in future, because from what i understand most of these are grid-following inverters. this might require policy changes and tighter control of PV powerplant by grid operator. hydropower is most useful in starting from complete blackout condition because all power that is needed is just what it takes to turn valves + some communications and remote switching to make sure it goes to other powerplants
for now no one knows what really happened, but i do hope that investigation will allow for figuring out what went wrong and preventing similar failure in the future
it looks like Portugal had it even worse - their power generation dropped to zero (until 15:00), disconnected from Spain, started hydro and pumped hydro to bring up gas and wind power (22:00 to midnight), then solar, then when they figured their shit out connected to Spain again and lent them a couple gigawatts (bigger grids are more stable so it's good for both)