headlights dimmer than they should be need help

No I put the HID in thinking that the bulbs were just getting weak and there was no change after I put them in. We are going through all the wires from the headlight to the plug and my father in law is a electrical engineer he is going to start testing with a voltmeter. My wiring runs from the bulb to a connector and then it goes to the other side with a y that it runs along the frame all the way up to the cab. We are starting to check the wiring from the headlight to the plug and if nothing there then we are going from the y and start tracing it all the way back. Where are the relay all located at? I have only seen the 2 under the dash one is for the left side the other is for the right side

 

With the truck not running just turning on the headlights I have 12 to 13 volts at the light socket. We unplugged the light and tested it on the plug itself. We didn't start it and test it but we are thinking that that is pretty decent for the lights being on for awhile and not starting the truck.