[Marika (OP)]

Rudolf wants to know:

When I drive with a vehicle up a steep hill, (constant grading) and at a constant speed, why does the vehicle start to accelerate by itself when getting near to the top? Sometimes more than 200m before the top?

What do you think?

[Ophiolite]

I cannot picture any road on which the grade does not gradually level out as you approach a summit. So your description of a constant grade appears to me to be faulty. If the grade is gradually reducing and you are maintaining the same pressure on the accelerator then an increase in speed is to be expected.

[evan_au]

When I drive with a vehicle up a steep hill...why does the vehicle start to accelerate by itself

If Rudolph were driving a car with cruise control enabled, the described behavior can be caused by the
 Proportional/Integral/Differential (PID) controller inside the cruise control.

This means that the amount of fuel going into the engine is controlled by several factors:
- The Preset speed, when cruise control was engaged
- Proportional component: determined by the difference in desired & actual speeds
- Integral component: determined by how long the desired & actual speeds have been different
- Differential component: determined by how quickly the desired & actual speeds are diverging

It is the Integral component that can produce the described behavior:
- The Proportional and Differential components merely try to get you back to the Preset speed, but always lag behind where you would be if you had kept a constant speed
- The Integral component tries to eliminate this lag, and will result in acceleration as you go up the hill (if your engine has enough power)
- Often coinciding with the hill levelling out at the top
- So you break the speed limit as you come over the crest of the hill
- Some more modern vehicles claim to be able to sense changes in road gradient, so they start increasing fuel flow before the hill starts, and reduce fuel flow before reaching the top of the hill. This must be quite economical with mass-produced smartphone accelerometers and barometric pressure sensors.

See: https://en.wikipedia.org/wiki/PID_controller

[alancalverd]

Real roads are not, generally, sawtooths but sinusoids, so if you apply constant power the car will accelerate as it approaches the top.