For a synchronous generator (a 3600 rpm, 60 Hz generator), if it has engine varying in rpm due to an unstable rpm govenor, its causes its AC output frequency to vary. This can be a problem for inverter to track. A sudden heavy load surge on generator requires the carborator throttle to be raised and there may be a momentary time delay in responseding to heavy load and therefore a dip in engine rpm. The inverter may release from generator under this condition. Inverter will reconnect to generator when its freq/phase, and AC input voltage stabilizes again.
It is important that hybrid inverter maximum AC input current limit user setting is set before connecting generator. If inverter's AC input current limit is set too high, based on wattage spec of generator, when inverter syncs and closes connect relay to generator, the inverter can jump on genertor with a load up to the max limit setup on the inverter. If this exceeds generator capability the generator will bog down, slump in rpm, causing inverter to immediately release from generator. This process will just keep repeating at a rate of every minute or two.
The inverter tracks slowly for a number of reasons. First, you really don't want AC phase to make abrupt changes for loads like AC motors as there will be current surges in AC motor. Second, the inverter PWM chopping by inverter to create a sinewave needs to be recomputed for voltage, frequency/phase shift and it must do this in small continuous increments without an abrupt jump in the increments. This requires a lot of recalculations, sensor checking, and adjusting MOSFET PWM chopping sequences. If generator wobbles more than about 0.3 Hz per second rate the inverter will not be able to track it. Inverter may refuse to connect to generator because it cannot achieve a continuous sync with generator or will release from generator if wobble happens after inverter has connected to generator. Input frequency range spec on inverter is not the same as acceptable AC input frequency wobble rate. Whatever frequency the AC input is, within acceptable range of inverter spec, it must also be relatively stable at that frequency.
Inverter detects voltage level and zero crossing to determine how to match AC input. It does not directly check waveform distortion of AC input source which may not be great for a cheap synchronous generator. Also, load on inverter AC out may have a load that has a power factor less than one. This means the output current is not necessarily in phase with output AC voltage so this is another complication on inverter sensor current measurements.
When zero crossing and voltage is tracked to the best ability of the inverter the final criteria for maintaining connection to AC input is peak current imposed on inverter. If the AC input wobbles in phase, drops or rises too much in voltage level, or has too much waveform distortion there will be current surge peaks in the inverter. When these current surge peaks get too great the inverter will open connect relay which releases tie between AC input and inverter. Inverter must then supply all AC output load.
Final tough problem for inverter is detecting when AC input goes away. When on grid for AC input the voltage is usually pulled down when grid goes down causing a momentary overload on inverter causing it to immediately release from grid AC input.
For a generator running out of gas there usually begins a sputtering surge of engine rpm that causes inverter to disconnect. There will be some momentary backfeed current into generator during this process so it is better if you avoid letting generator run out of gas, instead disconnecting generator electrical connection to inverter first, then stop generator and refuel generator.
Toughest thing for hybrid inverter to detect is an AC input breaker opening. Zero AC input current is allowed under certain conditions so inverter may not immediately open AC input connect relay. Usually the internal node current sensors in hybrid inverter feeding info to microcontroller will allow the microcontroller to figure out that there should be some current coming in or out of AC input based on AC out load and inverter power production, so it makes the decision to re-open input connect relay. This may not be immediately after opening AC input breaker however.
The inverter running in parallel with AC input gives a lot of options. These options are features within firmware of inverter and may not be supported by all hybrid inverters. For example, battery power through inverter can be summed with generator AC input power allowing more AC output loads than can be individually supported by generator or inverter alone. You can store PV power in batteries then push it to grid at a later time. For time-of-use grid tariffing, you can load shave power taken from grid, supplimenting AC house loads from battery power to avoid heavy grid power consumption during time of high grid power tariff.
No different than connecting to the grid to charge the batteries or do grid support or grid tie.