FPR is basically a valve controlled by manifold pressure.
It opens at a set pressure when there is no boost and allows the fuel from the fuel line to enter the fuel rail. By circulating the fuel from the rail inlet to the rail return and opening at a set pressure, it keeps the said pressure constant into the rail (if you try to remove it, it will splatter petrol everywhere

).
The stock injectors' fuel flow is calculated for the said rail pressure. AUQ injectors can flow about 317cc at 3bar (and up to 395cc at 4 bar).
If the car is aggressively modded, ECU will force the injectors to run at their upper limit (in normal operation they run about 85% of their maximum flow) and the mixture may be lean.
All transverse 1.8Ts have 3bar FPRs. Some longitudinal 1.8Ts (Audi A4 comes to mind) have 4bar FPR, despite running just 150-170bhp stock power.
4bar FPRs are usually not a necessity on Stage 1/2 AUQs and AMKs/BAMs, if the car is not driven very hard and the climate is not extremely hot. Revo mapping calls for a 4bar for safety.
Due to closed loop ECU operation, the fuel pressure does not influence horsepower too much. Higher fuel pressure means more fuel flow, more fuel flow means richer running, ECU reads Lambda sensor and reduces the fueling, so the car runs just like before. But when throttle opens, suddenly more fuel is needed, and ECU will make the injectors' flow to peak. Having more fuel available allows a safety margin.
An adjustable FPR allows best tuning of the mixture, but it's more expensive.
Other compatible FPRs are:
Bosch
3.5bar - OEM equipment for 1999-2004 BMWs, from 320i to X5 4.4, but not BMW V12s
Code: 0280 160 597 (no longer in production)
0280 160 504
Other manufacturers equivalent, from dearest to cheapest: Pierburg 7.21548.51.0, Vaico V20-0499, Hoffer 7525014, Meat&Doria 75014.
Bosch
3.8bar - OEM equipment for 2.5 V6 Vectra B and 3.2 V6 Omega B
Code: 0280 160 642
Bosch
4.0bar - OEM equipment for longitudinal 1.8Ts (Passat / A4)
Code: 0280 160 575
There are also 4.5bar compatible FPRs, which are out of question, since the max pressure composed of boost + FPR pressure has to max at 6bar.
The advantage in fuel flow is computed like this: ((new FPR pressure / original FPR pressure) - 1) divided by 2. Because half of the fuel is taken by the return line. That is, compared to a 3bar, a 3.5bar provides 8% better flow, a 3.8bar provides 13% and a 4bar provides a bit more than 16%.
I'm doing an experiment right now with a 3.5bar FPR (running a Stage 2 AUQ which makes slightly above 230bhp / 264 lb-ft at the flywheel). Installed today, a very hot day (35°C). Sought to give slightly better fueling under boost and also to find an answer why the idle and low speed operation were always better and smoother on wet days, even when the car had been stock.
Right now, the car has made just a few dozens of miles in city traffic on the 3.5bar FPR, but ever since it has been fitted, the idle is better and smoother. To be seen if there is an advantage in the future.
~Nautilus