The Wrong Side of my Car

The blog that wants to go obsolete

Cycling Hill Misery Index

A few years ago I moved from Milford to Birkdale, and I have always felt that in Milford I could easily get around on a bicycle, whereas in Birkdale an acoustic bicycle is pretty much useless. It is really hilly.

So here I introduce the Bicycling Hill Misery Index, an estimate of how annoying hills are. A value of 0.0 means flat as a pancake. 1.0 means an area is so hilly that it doubles the effort required to go the same distance. By looking at the slopes of streets, we can put this on a map.

Extra effort required

The first part is estimating how much extra effort it takes to ride up and down a hill, versus the flat.

A while ago we talked about hill climbing for casual cyclists and we introduced an approximation of how much power it takes to ride a bicycle:

• rolling resistance: m · g · C_rr · v: we assume 0.005 for the friction coefficient.
• air resistance: (12 · ρ · C_d · A) · v²_rel · v: The product on the left is assumed to be 0.35. v_rel, the relative wind is the speed relative to the air, it increases with stronger headwind.
• potential energy on slopes: m · g · slope · v.

These formulas give us a curve that tell for every speed how much effort is required.

Then we have to make some arbitrary choice of how fast we are going to ride. In our example we selected approximately 60 W to go 17 km/h *1, or about 3.5 Wh/km

We have to make some further assumptions about how much extra effort we’re going to do when going upwind or uphill. At some arbitrary point we’re going to walk the bicycle uphill. If we have a tail wind we could go faster, but we would probably just expend less effort.

What I did is make some curve of how much power I’m willing to expend to stay above a certain speed. Most people will push more power uphill than on the flat, that is why they often don’t make it to the top. But it is also often necessary to just go fast enough to stay upright. By the time you walk, the speed has almost no influence on how much total effort you have to do to climb the hill.

Baseline

For the baseline I could simply use that 3.5 Wh/km, the amount of power needed when there is no slope and no wind. But that would be cheating, usually there is some wind. So I took the average of varying wind speeds ranging from 25 km/h tail wind to 25 km/h head wind. This gives a somewhat higher baseline figure of 4.2 Wh/km

Walking

If the hill is too steep you will have to walk. That is the second green curve. Unfortunately walking is less efficient than riding (that is why we have bicycles in the first place). And you’re now walking in a somewhat awkward position to push that bicycle.

I don’t know how much extra power to add, but I’ve seen a mention that walking on the flat uses about 60 W, or about 10 Wh/km. So for walking, I will add this value to the value obtained from our calculation. (this is quite arbitrary, if you know the right answer here drop a comment)

Hills

For a given slope, we repeat this process twice, uphill and downhill. The ratio of the average to the baseline tells how much extra effort it takes to cover the same distance. Finally subtract 1 to get our misery index because we want a misery index of 0 for the flat.

For low slopes, the misery index is very close to 0. On these low slopes you get the extra energy you use going uphill back when going downhill. You can use less effort or coast, but you won’t go really fast.

On steeper slopes, the uphill part takes much more energy, but here when going downhill you will speed up much more, so most of this energy is burned off by your air resistance and even braking. Going uphill might require 4 times as much energy, but you can at best get 1 times back. If you’re lucky you can coast for a short while after the downhill slope, but your air resistance is so big compared to your momentum that this doesn’t give you much extra.

At around 9% the penalty for walking starts to kick in, first for strong headwind, then for all wind speeds. Such gradients are serious barriers for casual cyclists, and will make it physically impossible for families with young kids to go out for a bicycle ride.

Mapping slopes on streets

The choice of which streets to include is somewhat arbitrary. I left out motorways and major arterials, because obviously. I also left out what Auckland Transport calls minor and access roads. Most of the time you’ll be cycling on the more major roads, because those are usually the roads that follow a sensible route from A to B. I did not do any attempt to include off-road bike paths, because frankly, they are so rare in Auckland that I don’t think they will influence this a lot. To capture a reasonable amount of detail the roads are split into segments of around 100 m.

We can cross-reference the map with the Auckland 1 m DEM to calculate the slope for each piece of road. There will be some artefacts around bridges, because the DEM dips down to the surface under the bridge. We can get rid of the worst of these by throwing away any road segments with a slope of over 25%. Bridges are anyway such a tiny fraction of the roads that I don’t think it will mess up the data.

Some roads get double counted because the 2 carriageways show up separately in road data. I don’t know what to do about it so I will just live with it.

We use our misery index table to map every piece of road to a misery index value. The area covered by the map is divided with a 0.5×0.5 km grid, and every road segment contributes to the average misery index in a grid cell.

Because you usually want to go a few kilometres on a bicycle, we then blur this grid a bit. This blurring is weighted, the more length of road in a cell, the more it will impact the neighbouring cells.

When this is all said and done, we end up with this little image.

Aw, after all that work. Is that all we have to show for it?

But with some tinkering in QGIS we can turn it into the map above.

Caveats

Wind

Wind introduces a similar ‘misery index’ as hills, in the sense that you have to spend a lot of extra effort going upwind, that you don’t quite get back going downwind. I’m not totally sure which wind speeds are typical in Auckland.

Generally, higher winds increase the base line, the amount of effort necessary to cycle on the flat. So if you calculate for higher winds, for the same hilliness the hill misery index will be lower.

Energy versus effort

Energy is a somewhat crude approximation for effort. There a few more reasons why going uphill is so tiring.

• You need to put a large continuous force on your pedals, no matter how slowly you ride. While on flat stretches the force you need will go way down, even when going upwind. Using a low gear helps, but once you reach slopes over 5% you will need lower gears than what you find on most bicycles.

• You will get hot. Normally on a bicycle you cool down by moving through the air. When climbing steep hills you go too slow to get this cooling effect. This will be much worse on warm days.

• 120 W isn’t exactly twice as hard as 60 W. A lot of people will get tired within minutes at 120, while they can do 60 for a long time. *2

Observations

Torbay and Birkdale are particularly hilly. I would conclude that the ‘bikable range’ more or less halves in these areas. Bicycle advocates like to point out that there is no such thing as ‘too hilly’, but I think these hills are genuinely a barrier to riding a bicycle to many people. However this amount of hills is a bit unusual. *3

One of the most salient features on the map is the big flat area in South Auckland. However this is also an area with very low cycling uptake. This tells you there are worse barriers to cycling than hills. For example poor street design (particularly at intersections), cultural taboo, lack of bike parking, and prevalence of bike theft.

For those who like looking inside sausages

Good news: I was going to make the source code for this map available if there was a non-zero amount of interest.

So the Cycling Hill Misery Index is now on GitHub: bicycle-hill-misery-index.