Not a nitpick at all...I wish I could talk about this more! Let me know if you want a good book list on this. There are several out there with full Matlab code

Not a nitpick at all...I wish I could talk about this more! Let me know if you want a good book list on this. There are several out there with full Matlab code to accompany the equations. 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.