BMW Questions? Ask a BMW Repair Expert for Answers.
So below are you faults listed I don't think it is vanos related. I have two ideas on what could be the issue but first I need some info.
Do you live in a cold weather area?
Does it always run rough when cold?
Is there any smoke from the tail pipe when this happens?
How long has it been doing this?
P0300 Random/Multiple Cylinder Misfire Detected
P0313 Misfire Detected with Low Fuel
Misfire Cylinder 1 With Fuel Cut-off
P0171 System too Lean (Bank 1)
P0174 System too Lean (Bank 2)
My first thought is a clogged cyclonic separator would cause this issue but it is more common in colder weather like weather in the 20s. I would still check to make sure a telltale sign is to remove the oil cap on the valve cover and look for yellow gunk/slime on the cap and just inside the cap. Below is a picture for reference.
My next thought is a vacuum leak because of your lean mixture faults (I listed the faults so it is easier for me to read them all at once). I would check the intake boot near the throttle body and idle valve for cracking, I will attach a picture below so you know what to check for.
Another Idea is a fault with the fuel system, either a weak fuel pump or a it is bleeding down overnight or a fault with the pressure regulator. You will need a fuel gauge to check this.
My last thought could be a MAF but it is really uncommon.
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