Controller and method for a power converter. For example, a controller for a power converter includes: a first terminal configured to receive a first terminal voltage; a second terminal configured to receive a second terminal voltage; a comparator configured to receive a first threshold voltage and the second terminal voltage and to generate a comparison signal based at least in part on the first threshold voltage and the second terminal voltage; and a switch configured to receive the first terminal voltage and the comparison signal, the switch being further configured to be closed to allow a current to flow out of the second terminal through the switch if the comparison signal is at a first logic level; wherein the comparator is further configured to: receive a first reference voltage as the first threshold voltage if the first terminal voltage is smaller than a second threshold voltage.