A Charge of Battery

This week, I have a question from my mother Sally about their Chevy Bolt: I was thinking about the most efficient way to drive the Bolt on a long trip: Do you drive more slowly, so that your miles/kWh are more and you don’t need to charge as much? When you do charge do you stop frequently and only charge at the highest rate (the rate declines as the battery % is higher, e.g. 42kWh, declining to 26kWh), or do you stop as little as possible and stay until you’re at 80%?

As Sally outlines, there are two states the car will be in during a trip: charging and driving. The amount of time it takes to add a certain amount of charge to the battery depends on how much charge is in it – As it fills up, it gets harder to add more. During charging then, we can write

that is, the rate of charging is proportional to the space remaining. While driving, we have to push against a wind resistance that depends on our speed:

Drag typically depends on the square of velocity, meaning that increasing speed can quickly become counter-productive.

Now, both these equations I've written are just proportionalities, which need specific scales and shapes. Normally I'd have to make up something based on wild assumptions of a clueless physicist, but lucky for me, Bolt drivers are my kind of crazy, and have measured the behavior of their cars in detail!

For the first equation, I found this chart from InsideEVs:

and for the second, I found one from ChevyBolt.org:

Since these are just graphs, I had to do a bit of fitting, but I'm pretty happy with the results. To answer Sally's question, I chose a bunch of different driving speeds and max/min charge levels. We drive the car at the given speed until the battery hits the minimum charge level, then we stop and charge until it gets to the maximum charge. We continue for 1000 miles, then check the average speed during the trip, and the number of stops.

First, I plotted a heatmap of the average speeds for the different max/min charge values, planning to mark the point where we get the best speed and the fewest stops. Unfortunately, it turned out both of those always occurred with a min charge of zero, so it's easier to see them as lines:

I found this a bit difficult to take in, so I tried plotting only the zero min charge case for each speed:

This shows some really interesting results: Because of the difficulty in getting up to the battery's capacity, high max charge can result in spending most of your time charging, even at high speeds. However, if your max charge is too low, you have to stop over and over, even if it's only for short time.

Thanks for a great question, Sally!