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Five Stages of Prius Hybrid Operation
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S1:

In Stage 1 the ICE is cold. If not prevented by use of the EV switch, the ICE will start a few seconds after the car is powered On. Until the system reaches the S1/S2 transition temperature, the ICE will run continuously. This accounts for the anomalous observation of electric regeneration or no arrows (coasting) at the same time as a low instantaneous mpg reading. No power is being drawn from the ICE, but it is running. Also in Stage 1 you will notice that battery power is used for even mild acceleration. In Stage 1 the HSD prefers not to draw power from the ICE if it is not necessary. You cannot enter EV mode while in Stage 1.

S2:

When the system temperature reaches 40 degrees C. (104 F.) the computer goes to Stage 2 operation. The original Japanese poster gave more detail than I clearly remember, but it appears that in Stage 2 the ICE will shut off if it is not needed to provide power. At a stoplight, in S2, the ICE will stop immediately and fairly smoothly. However, the car will not go into golf-cart mode in Stage 2. It can draw power from the battery or it can recharge the battery, but will not use the battery without the ICE. I have not ascertained if you can enter EV mode from S2.

S3:

When the system temperature reaches 73 C. (163.4 F.) the computer goes into Stage 3a. In S3a the ICE will not shut off unless you come to a full stop for 5 or 10 seconds. Paradoxically, it is more reluctant to shut off than it was in S2, and in S3a it shuts off very roughly. You will feel the car shudder. But once it does shut off at a full stop in S3a, it will then be in Stage 4 operation.

Once the car is in S3a, if you reach 34 mph without having come to a full stop for 5 or 10 seconds, the car will go into S3b operation. S3b is identical to Stage 4, except that if you slow down below 34 mph and apply the brakes, you will be back in S3a. In S3 you can engage EV mode providing that you meet all the conditions for EV mode.

I have observed that at the full stop in S3a sometimes the ICE stops after 5 seconds, and other times runs for 10 seconds before stopping. I do not know what determines the difference. I have also observed more than one rough shut-down in a trip, suggesting to me that perhaps the ICE may sometimes stop roughly in S2, and may not stop immediately when you let off the accelerator in S2. I repeat that I do not clearly understand Stage 2 operation.

Coming to a full stop while in EV mode does not seem to trigger S4. You must be in normal mode when you stop in order to enter S4 operation.

S4:

Stage 4 is normal, full hybrid operation. The Prius is most efficient in S4. The car will go into golf-cart mode at the computer's discretion. You can encourage it to do so by lifting your foot off the accelerator pedal for a moment, and then pressing on the pedal very lightly. This is known as "feathering" the pedal. Feathering does not force the car into golf-cart mode, nor is it necessary for the car to go into golf-cart mode. It merely encourages the car to do so a little bit sooner than it would otherwise do on its own. The benefit of feathering the pedal is questionable, since the computer is deciding when it thinks Golf-Cart Mode is most desirable. Increasing the time spent in Golf-Cart Mode may only increase the efficiency losses involved in charging the battery and drawing power from it. On the other hand, encouraging Golf-Cart Mode when the SOC is very high may be useful, as it creates head room in the battery to accept charge wheen needed.

Above 42 mph the ICE must spin, and will generally provide power. It is possible for the ICE to spin without providing power to the car above 42 mph, and this has been observed. But since the ICE is more efficient at these higher speeds, Golf-Cart Mode above 42 mph (Super Golf-Cart Mode) is unusual. So the maximum speed in EV Mode is 34 mph, cutting out at a tiny fraction over 34 mph, and the maximum common speed in golf-cart mode is 42 mph. You can go faster in non-EV golf-cart mode than you can in EV mode.

Credit to Daniel @ PriusChat for the original writeup.