13 volts is not acceptable. The proper range of charging voltage is between 13.75 and 14.75 volts. The fact the voltage does increase above 12.6 volts indicates the rest of the system is likely working, but the alternator has a bad diode. You did not list the engine size so I do not know which alternator you have, but a typical one for an 2004 model would be around 100 amps. With one bad diode of the six, all you will be able to get on a full-load output current test is exactly one third of the rated current. In my example, 33 amps is not enough to run the entire electrical system under all conditions. The battery will have to make up the difference until it slowly runs down.
A professional load test will indicate whether there is a bad diode. The second clue during that test is "ripple" voltage will be very high.
Do not worry about the ABS light. The computer knows it cannot run the solenoids when system voltage is low, so it shuts the system down, and turns on the warning light to tell you.
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Sunday, August 6th, 2017 AT 11:24 PM