We have done a lot of investigation on injectors and engine balancing. In this case, let's start with the most obvious. It could be low compression in a cylinder. Compression test the engine.
I'll not forget the fella that came into the shop one day, after we just got a performance engine built and it was purring along in the shop with the hood up. He looked at the oil dipstick boot, which was standing stock still. He pointed at the dipstick, turned to me and said,"I didn't know they could do that..."
The first thing to wonder is if the imbalance is new or if it's been doing this for some time. Just from a standpoint of rotating assemblies, VW really doesn't care about how well the TDI piston/ rod balance is. "It's only going to be running at about 2500rpm it's whole life." So, balancing is not really a big issue at the factory, unless you decide to make it one. Regularly, I have found the imbalance between reciprocating sets (1 & 2, 3 & 4) are often 8 grams out of balance. We balance to .1 gram. The crank is generally quite close. So, balancing is a matter of matching weights between pistons and rods, separately, then together. It does make a lot of difference for the 'feel' of the engine.
The flywheels are rarely balanced well. Maybe 10% are balanced 0.0grams. Not going to get into the depth of it, but we balance every single flywheel and resurface it 'correctly'. The humerous ones... We have balanced 3 flywheels that the drilling was done 180 degree out! We had to double-depth drill on the opposite side for correction. It would help if they were to check their work.
The injectors can have a great deal to do with what goes on at idle. Some will try to blame the injection pump, but rarely will you find that one injector will receive a different volume of fuel, as the pump wears very evenly.
But the injectors, there is a fine mess to deal with and that can very likely, be your issue. We examine injectors and dynamically balance them by using the VCDS, Engine module, block 13 and 15 to set the flow rate. You can check to see the variance between cylinders in block 13, Idle balance compensation and we also look at block 15, which the 3rd block shows liter per hour (LPH) fuel usage or FE (fuel economy).
If there is a large variation in fueling between the idle balance numbers, that will equate to an idle vibration. The whole purpose of the idle balance compensation is for the ECU to vary the fueling of each injector, so at idle, it runs smoothly. However, my contention is that the closer to 0.00mg/ str (milligrams per stroke) each cylinder shows for block 13, the more efficiently the engine will run. You want each cylinder receiving the same amount of fuel.
For example: If you have #3 cylinder showing 1.50 and #'s 1, 2, and 4 showing -.50-.75, then what is happening is #3 is demanding MORE fuel and 1, 2 and 4 are reducing fuel to compensate for an injector that is a problem. The engine's speed is measured through the crank sensor and if it runs too fast, the next injector's fueling is reduced to compensate. Too slow, fuel is added to speed up the crank.
The maximum block 13 deviation for an injector is 1.80 to -1.80. If you have a reading like that, you need to fix the injector(s) at fault. Or, on a rare occasion, the crank sensor wheel is bent. That is a whole story in itself... but maximum deviation can be disaster for you engine.
The other block I mention, 15, show the fuel consumption... FE this is a 'virtual sensor', as it does not actually figure the total fuel used, but uses different readings from the engine's modules to estimate fuel usage. So, it can vary, but will show a good baseline. Normally, a well-tuned set of injectors on a good engine should have an idle FE of .3-.5 lph.
Then, raising the idle speed to just under 1600 rpm (the idle balance compensation turns off at 1600 rpm), you can get a reading for the second stage of your injectors. Since this is where you operate your engine, this is the more important number for engine fueling and balance. If your injectors balance numbers remain skewed, you should take steps to remedy.
For example, if one injector show a highly negative number, as read with the engine running at 1596rpm, that mean fuel is being reduced. But once the engine is operated above 1600rpm, the idle balance is turned off. You get whatever the nozzle produces. Without compensation, the nozzle that was reading a highly negative number, now is overfueling. That can cause loss of fuel economy at least, and at worst can overheat the cylinder and melt pistons.
To set IQ, start with what you KNOW is too low of an IQ. 2.0. Do a road test on flat pavement with a minimum 45 mph speed limit, 3rd gear, 35 mph, accelerate to Wide Open Throttle (WOT). Remain if 3rd gear for the duration of this test. When your speedometer reads 40mph, using the stop watch feature most phones have, count 3 seconds. Check speedometer. Let's say it hit 55 mph. Now, raise your IQ 1 point with a hammer mod. Move the IQ lid toward the right side of the vehicle. Run road test again. Are you still reading an assumed 55mph? Raise IQ again. Continue road test until you lose performance in the 3 second count. Say it dropped to 53mph. Revert to the previous IQ reading that maintained 3 second count speed. That is how I optimize IQ settings.
"The least amount of fuel to maintain acceleration speed"
I covered a lot of ground. I hope you can pick out what is helpful. Questions? Email, PM, call...