Pilot protection for active distribution networks based on the phase-space trajectory of quasi-power

•The quasi-power theorem is introduced into active distribution networks and the specific application method is proposed.•The phase space theory is used to explore the characteristics of quasi-power.•High-speed protection criterion is designed to shorten the duration of faults.•Protection criterion for immune to synchronization errors is designed to reduce the requirement of data synchronization and save the cost of extra synchronization equipment.•The protection criterion adopts a floating threshold with excellent protection performance.