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I'm assuming when you say fault you're referring to something like an earth ground fault. In that case fault currents will be sustained indefinitely (until a breaker trips or something else fails), while inrush current will peak and then decrease over time. So when looking at current waveforms you have to look at a window containing multiple cycles. The duration of the window will depend on the typical duration of the inrush current.
Inrush current is time limited and in high power DC system have various circuit to limit, by resistor(s) with relay or similar. Inrush current must be associated with turn on of equipment, so there are one first different situation to separate from fault current. May be another solution to differentiate them by monitoring di/dt assuming that fault current have high di/dt that inrush current. Such di/dt circuite is used in some electric protection devices also.
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