I wrote a paper earlier about the problems associated with VFDs. The problems aren't the controllers, it's the installation and misapplications associated with them.

Instead of spending a lot of time explaining the advantages of them on fire pumps, you can rummage through everything here.
http://www.mastercontrols.com/Variable_Speed_General.html#Variable_Speed_General

They offer a unique advantage of constant pressure output on city mains with wide pressure swings for 20 - 50, 60 PSI, utilizing the 20 PSI minimum without overpressure at 50 - 60 PSI. This eliminates high-rise valves and fittings on lower floors or PRVs. They offer the same advantage with extremely tall, narrow storage tanks, utilizing the lower levels for suction design. Sprinkler designers will probably find it a big savings on material and labor on the heavier and more expensive fittings without one in certain locations depending on the pumps churn pressure at the city's normal pressure..

In Houston Texas, we use small 2500 GPM break tanks with complete refill at 150% so they don't offer the hydraulic advantages like in the aforementioned situations.

Are they approved for fire pumps?

Tony, click on the link above, it's to a Master Fire Pump Controller VFD as an example since they've done a lot of work already on the advantages. Master was the first one out.

Thank you Mike, we just order two pumps with Master controllers, i will check is it possible to have them with variable speed, there are many advantages. I was not mention that master offer Fm listed variable speed controllers.
In the past i use some variable speed controller in foam system, but the local authority don't have any specific limitation and requirements on the pump controller. Otherwise many of our project required FM or VdS listed pumps, somehow i miss this product.
Thanks you again.

Tony,

On VFDs, NFPA 20 requires that the motor be specifically listed for fire pump service, specifically RATED for INVERTER DUTY and the Service Factor is not allowed to be used.

PS: Typically the pump people should have all of that taken care of when you order the pump but it never hurts to check. People putting the wrong motors on fire pumps when energy efficient motors first came out resulted in the motor listing requirements in the first place.