I can't get putty to authenticate via an ssh key.
Server refused our key
This also looks promising, but vsphere CLI has been deprecated. I have installed the new solution esxicli, but I don't have a clue how to use it or create the scheduled task.
The best way to accomplish this is via vSphere CLI, you need a Windows or Linux machine for this or you can also use the vSphere Management Appliance (free product)
1. create a cronjob or a scheduled task on the Windows/Linux client with this command
"vicfg-hostops.pl" -a poweroff -f --server=***ESXIHOSTNAME/IP*** --username ***root*** --password ***YOUR PASSWORD***--operation shutdown"