1. "Power Management" task relates to RTCI. If you want to restart, power off computer, this task will checks available protocols in Default Connection Profile, such as AMT, WMI, etc to perform this action.
You need to have enable protocols such as AMT, WMI, etc. on your managed clients to be able to power on them, even if they are powered off.
If you want to use such technologies, you need to have enabled these protocols in Default Connection profile and then perform "Get Out Of Band Inventory" task to scan clients for available protocols.
2. "Power Control" task works through Task Server and Client Task Agent on client PC and it doesn't use "Default Connection Profile" protocols to perform these actions.
3. "Tickle Client" task also doesn't use "Default Connection Profile" and works between NS, TS and Client Task Agent. Here is more detailed description about how works "Tickle Client" task,