Engine Starts

Since Prius places a high priority on emission reduction, the short answer is no.

Heat is needed for the catalytic-converter to properly cleanse emissions. So it will run the engine to maintain that temperature, even if it means sacrificing some gas to do it.

Fortunately, other things take place as long as the engine is running. But there is still a penalty for short trips. (Of course, there is a heat requirement for traditional vehicles too... only the effect isn't as pronounced since it's just an RPM difference.)
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Park the car. Turn off headlights. Turn off climate control if vehicle is unoccupied. Lock the door with the key. If the trip is going to be more than 10-15 minutes then turn the car off. You will have to unlock the door with the key AND press the unlock button on the armrest to enable SKS.

 

:welcome: Welcome to PriusChat, Crusader Rabbit!
One thing you can do to help is to plan your errands and go to the farthest first. This gets everything warm so your new starts will be shorter, but Jim's idea of leaving your car in READY is the only way to avoid the start-up penalty.

 

As said above, the Prius goes through a cycle of "warm the catalytic converter" every time its' started, even if the cat. is warm. If you don't shut it off, it won't go through that cycle.

Even going through the cycle, it's only running the engine for 20-40 sec. at idle. That doesn't really use that much fuel. Typically Prius burns about 1-3 l/hr at idle. So that translates to about 3-4 cc of fuel per start. You -might- be able to see this on your fuel economy, but I suspect there are larger variables that will swamp it out.