This seemed to be the best place for this...

[MCBC]

So I have recently taken up cycling, and started using a route mapping tool. I noticed this tool listed elevation and road grades. It said one short 50 foot segment of my route had almost a 40% grade. I thought about how the average grade, since I started and ended at the same place, would be zero, but that I had obviously made more effort than if I had ridden the same distance at a 0% grade.
I am wondering if anyone knows a formula that will figure out the normalized average grade. I have heard about and seen calculations for normalized power, but road grade seems to be untouched by this.
I am looking for this formula to give me an indication of the normalized power on my route without me having to buy a power meter. Something that "normalizes" the data using the distance (d) and grade (g) of segment 1, d and g of segment 2... d and g of segment n.
Sorry if this isn't the best forum for this, but seeing as road grade is mostly trig, it seemed like the best place for it.

 

[MCBC]

An example to explain what I am talking about

So, I created an example to show what I am trying to show/figure out. I posted it to another forum, which is why some of the wording may seem out of context, but I changed almost everything that was tied to previous conversation and wasn't necessary to explain the problem. So, here it is:

"Let me take this hypothetical circuit.
The grade for segments 1 and 2 is 2%. The grade for segments 3-7 is 4%. The grade for segment 8 is 2%. The grade for segment 9 is 0%. The grade for segment 10 is -2%. The grade for segment 11 is -4%. The grade for segments 12-14 is -10%. The grade for segment 15 is -4%. The grade for segment 16 is 0%. The grade for segments 17-19 is 4%. and the grade for segment 20 is 2%.
I am going to use this calculator to get speed and duration calculations. I am also going to ignore momentum for the time being. The power is at a constant 125W.
Segments 1 and 2 take me 8.4 seconds total to complete at a speed of 8.1 mph. Segments 3 through 7 take me a total of 32.8 seconds to complete at a speed of 5.2 mph. Segment 8 takes me 4.2 seconds to complete at a speed of 8.1 mph. Segment 9 takes me 2.6 seconds to complete at a speed of 13.3 mph. Segment 10 takes me 1.8 seconds to complete at a speed of 19.4 mph. Segment 11 takes me a total of 1.4 seconds to complete at a speed of 24.8 mph. Now, I want to stay below the speed limit of 25 mph. So for segments 12-14, I will reduce my power to 0, because negative energy does not go back into my body (this is where the average gradient fails). Instead, I will apply about 700 watts of energy through the brakes (I don't have to apply such energy, as brakes are more efficient than this). Segment 15 takes me 1.4 seconds to complete at a speed of 24.8 mph and a power of 125 watts. Segment 16 takes me 2.6 seconds to complete at a speed of 13.3 mph. Segments 17 through 19 take me a total of 19.7 seconds to complete at a speed of 5.2 mph. Segment 20 takes me 4.2 seconds to complete at a speed of 8.1 mph.
Adding up the time in seconds that I applied 125W of energy gives me a total number of 79.1 seconds, and thus, 9887.5 joules expended.
The same distance at the average grade of 0% (since it is a perfect circuit) would take 51.3 seconds (6412.5 joules expended), instead of the 79.1 seconds it took with the hills. This is obviously incorrect, because it took longer to go up the hill and as such, required more energy.
Let me do the same course backwards.
Segment 20 takes me 1.8 seconds to complete at a speed of 19.4 mph. Segments 19 through 17 take me a total of 4.1 seconds to complete at a speed of 24.8 mph. Segment 16 takes me 2.6 seconds to complete at a speed of 13.3 mph. Segment 15 takes me 6.6 seconds to complete at a speed of 5.2 mph. Segments 14 through 12 take me a total of 44.5 seconds to complete at a speed of 2.3 mph.Segment 11 takes me 6.6 seconds to complete at a speed of 5.2 mph. Segment 10 takes me 4.2 seconds to complete at a speed of 8.1 mph. Segment 9 takes me 2.6 seconds to complete at a speed of 13.3 mph. Segment 8 takes me 1.8 seconds to complete at a speed of 19.4 mph. Segments 7 through 3 take me a total of 6.9 seconds at a speed of 24.8 mph. Segments 2 and 1 take me a total of 3.5 seconds to complete at a speed of 19.4 mph. Again, this is disregarding momentum.
This direction took me 85.2 seconds of work to complete (versus 79.1) because it had the steeper hill. As such I used 10650 joules.
So, my question is, is there a formula that will allow me to figure out the 85.2 vs. 79.1 vs. 51.3 simply by using the road grade and the distance it continues at that grade?"

I realize now that this may not be the best forum to put it on. But I figured clarifying what I mean might help in some way.