Has anyone made predictions for what future decades of energy generation might look like? Specifically any ideas for how big if any a return to nuclear will affect everything else?
Back in the day, the utility would look at population growth which correlates well with electrical demand growth and did 1,5,10, & 20 year projections which told them if they needed to build new generators or transmission lines. They looked at expected fuel costs and federal/state policy changes too. One common simulation they run is called "contingency analysis" or "CA" for short. You start with a model of your grid at peak winter and summer loading. You then take each transmission line out of service in the model (1 at a time) and then run a loadflow calculation for the modified model(basically solving a massive set of nonlinear equations) to determine the amount of power flowing over all equipment. Any overloads are logged in a master list and studied to see if they can be fixed. There are also studies to look at voltage and many other things.
Today, studying the grid is a lot more complex and time consuming due to things like renewables and energy markets. For example, a long time ago before optimization software was available for large models, you would probably most likely have a list of which resources to run when and it didn't deviate much throughout the year. You would simply call on a gas plant to help you get over peak and that is about it. Now, things like probability are becoming very deeply ingrained in planning in a way they never were before.
In essence, folks working for utilities, government entities, state commissions...etc do this all the time.
It has trigonometric identities in all the equations (sin & cos) if you do the full AC method. We all generally use the Newton-Raphson iterative method for the large sparse models we have. Gauss-Seidel is great for small models, but takes ages to converge as your model size increases. There is a "Decoupled" version of the Newton-Raphson that zeroes out some terms that usually don't matter in the Jacobian to speed things up considerably. There are some other solver types being researched, but Newton-Raphson is fast, good at converging, and is well understood. The last and very popular method is the DC loadflow where you make a lot of assumptions and linearize the entire thing. This isn't used for important studies by itself, but is embedded in lots of things such as market commitment and dispatch as it is extremely fast and almost never diverges. There are also other useful sensitivities to derive from it.
Sorry to give you a wiki reference, but I had literally just read the power flow study page [1] earlier today:
"The problem is non-linear because the power flow into load impedances is a function of the square of the applied voltages"