I'm in a similar situation to you. Using Ubuntu 16.04 LTS as a web server.
The server and a Raspberry Pi are both connected to the router via ethernet cable.
The RPi is running continuously. I have installed and configured fail2ban for extra security. The router is configured to allow SSH access from the internet to the RPi. When I want to start my power hungry IBM server. I SSH connect (mosh actually) to the RPi and run the following Python program:
""" Send a WoL packet to the specified MAC address
NB: Won't work on OS/X. The AF_PACKET is specific to Linux.
Needs to be run with sudo.
PREAMBLE = bytearray((0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF))
# Need to configure the following for the correct MAC of the target
MAC = bytearray((0x00, 0x14, 0x85, 0xa1, 0x43, 0xde))
pkt = PREAMBLE + 16*MAC
sock = socket.socket(socket.AF_PACKET, socket.SOCK_RAW)
# Configure the correct interface name
If you don't have a static IP address, then you probably need to configure dynDNS or something similar and your domain addresses.