First, your question is not solely a physics question, but it's also about policy and economics.
Apologies for the length of this answer, but this happens to be my daytime job and field of expertise, so I get a bit carried away.
There might be regional differences and some policies and pricing principles that I mention might differ a bit from country to country.
I'll address the different headlines first, then give some additional general comments.
A new transmission line is being built from country A to country B,
which will increase country B's ability to import electricity from
country A by X MW.
This one is a simple statement of fact. The same transmission line will obviously also increase country A's ability to import electricity from country B by X MW.
One thing to keep in mind here is that one of these countries will often produce more than they need, while the other produces less. So the headline in country A will read:
A new transmission line is being built from country A to country B,
which will increase country B's ability to export electricity from
country A by X MW.
Also, many cross-country interconnections are DC, not AC. The converters at each side of the DC transmission line can determine the direction of the power flow. The direction of the power flow will (in general, but not always) be determined by the prices on each side of the cable.
Electricity prices in (say) Northern Europe are on the rise because
power companies there prefer to export their electricity to (say)
Central Europe where there is a high demand and therefore better
prices per MWh, leaving their home markets with less supply.
This is both correct and wrong at the same time. This is not about physics but economics. In Europe, electricity is traded on Nordpool. All electricity suppliers say what they are willing to sell their electricity for, while all distributors1 say how much they want to buy the next day. This is supply and demand, and everyone will pay the price the "last" producer demands.
The Northern European producers don't sell to Central Europe, they sell the amount of power they get paid to produce. If it turns out that the producers in Northern Europe are willing to produce more electricity for the given price than domestic consumption, then there will be a production surplus, and power is exported.
Their home markets will only have less supply if the total energy is "fixed". This is the case for Norway, where the energy is stored as water in large reservoirs. If they produce more than the domestic consumption then they will "use up" the water they could otherwise have kept for later. This is not the case for wind and sun power plants for instance, as today's production won't affect tomorrow.
Because of [powerplant maintenance / low hydro reservoirs / declining
foreign relations / whatever else the reason], country A will reduce
its electricity exports to country B by Y MW.
This has been done by some countries in Europe already, and many more are considering it due to the European energy crisis.
There are several ways to do this:
- Open a breaker thus disconnecting a transmission line.
- One less line = less power in total
- Tweak the settings of converters on DC lines.
- You can control the flow however you like.
- Alter the power angle, thus controlling the power flow.
- This is less common since it's not as simple in a dynamic system (well, it is, but a lot of equipment is old)
Electricity prices vary greatly from region to region in country C
because most of the generation happens in a different region than most
of the consumption, and transmission capacity between regions in that
country is inadequate.
Europe is divided into "price areas". Some countries are divided into several price areas, due to "bottlenecks" in the grid. Norway for instance is divided into 5 price areas, and Sweden is divided into 4.
The reason why prices differ is that producers in one region are willing to produce a lot of power at a low price, leaving that area with a surplus. In a neighboring region, the producers demand a much higher price. The producers in the first region will produce as much as they can and the surplus will be transported to the other region. In an "ideal" grid, the prices in both regions would now be the same. However, the lines between the regions have limited capacity, meaning that even though the producers in the first region are willing to produce a lot of energy, they have to limit their production.
Now, remember supply and demand. The producers in the first region were willing to produce a lot of power at a low cost, but the demand wasn't high enough. They must limit their production to the regional demand plus the maximum export capacity. High supply and low demand give low prices.
The neighboring region imports as much as it can from the first. The rest must be produced in their own region at the price the producers demand. The price they demand is high, but the consumers are willing to pay the price. Low supply and high demand give high prices.
Side note: You might have heard of "bottleneck income". Bottleneck income is income the transmission system operators get when there is a bottleneck in the system. Remember that the producers in the first region were willing to produce at a low cost, and the price in the neighboring region was high. What happens is the producers in the first region sell their power for the price in their region, and the consumers in the neighboring region buy their power at the price in their region. The consumers pay more than the producers get paid. The surplus, called bottleneck income" goes to the TSOs.
You say:
They are not in a position to say that "today we'll sell X MWh" or
"tomorrow we'll cut our delivery to B by 50%" – not in a direct "hands
on the water tap" way at least.
Well, that is exactly what they say. We're willing to sell X MWh for y €/MWh tomorrow.
You are right of course. It's impossible to know tomorrow's demand. That's why there's a "flexibility market" where producers are willing to produce more or less power for a given price. The system operators will then "tell" certain producers to produce more or less power to maintain a stable frequency (power produced = power used).
How do grid and transmission line operators regulate the amount of
power flowing in their networks, both to utilize the network optimally
The network is not utilized optimally when it comes to minimizing losses. It is a market optimization. The producers aim at maximizing their profits, while the system operators impose certain restrictions to maintain balance in the grid.
You can read more about grid balance here.
