You have it fairly correct, and the induction motor is fairly forgiving when it comes to synchronization because of slip. The last one of these I worked on did not have a Plc, although it could have used one. Basically it was relay logic with a few specialized relays to handle synchronizing and power monitoring. Additionally, it was controlled by a PI control off of Forebay elevation which controlled how far the needle valve opened (this was a pelton turbine).
So basically, once you get going, you monitor power to make sure you are still pushing energy out to the grid, you monitor rpm (which will be roughly proportional to power), and you monitor Forebay elevation to be sure you aren't taking water out too fast. Now, if you have a large Forebay, you may not use the level to drive your needle valve, but we another variable such as a time of day schedule or another external dispatch source.
With an induction motor generator, it sort of self governs as long as you are pushing the turbine past synchronous frequency. The harder you push, the higher your power output becomes. Don't forget that the motor still needs overload and short circuit protection as well, and most people monitor bearing temp and vibration if it is an unattended plant. Some other permissives to check for: penstock pressure, tailrace water elevation (too high and you should stop the unit), ground fault on your high voltage line, and also any kind of relay protection required by the utility like good voltage/frequency present, differential current, etc.
