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No. Permits are issued by your local Air Resources Board and the units are usually stationary in nature. Registrations allow for portable engines and the associated equipment to travel within multiple districts without having to be permited in each area.
Warning.... if you don't have either a permit to operate or a valid PERP registration you can be cited by an Air District Inspector!
Engines that qualify for PERP registration are:
50-99hp with EPA/CARB Tier 2 Engines (until December 31, 20007)
100-749hp EPA/CARB Tier 3 Engines.
750+hp with EPA/CARB Tier 2 Engines.
Tier Zero engines can not be registered.
All other engines not meeting current Tiers but operating in CA beetween March 1, 2004 and October 1, 2006 CAN BE registered but higher fees may apply.