In an off-grid system, do the batteries connect directly to the inverter? or do they need to pass by the charge controller before connecting to the inverter? (as suggested in the figure shown in the Chapter 1). Thanks!
It can be either way. If the connection is made through the charge controller, which is usually the case, it is really just a pathway for the battery to be connected to the inverter on a typical system. Even when connecting to the battery through the charge controller, the conductors are sized based on the battery voltage and the inverter power.
Whatever you program to be the lowest "shut-off" voltage of the inverter input will determine the maximum current on the inverter input circuit. After taking this course, there is a good chance you will see a test question relating the size of the battery inverter input circuit being sized based on the lowest voltage setting. This is in 690.8(A)(4) Stand-Alone Inverter Input Circuit Current.
When it comes down to it, you will have unique equipment from different brands and installation manuals that you will study and ask the manufacturer questions about.
I have a small Primus wind turbine which can be used with a PV system. The manufacturer recommended to connect the wind turbine directly to the battery with or without a PV system also charging the battery. The wind turbine is specially made for charging a battery directly (charge control built in).