# Battery Aging Model

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#### Stoaty

##### Well-known member
I am starting this thread to discuss various battery aging models. Surfingslovak developed a battery aging model referenced in the Wiki:

Since some of the numbers didn't fit with Nissan's claims (even after they mentioned the "glideslope" in Arizona to 76% battery capacity remaining, and the fact that only 7500 miles were assumed for Arizona), I decided to see if I could tweak the model to fit TickTock's graph:

Here are my results (you may have to make text smaller in browser to fit on screen):

Summary:

Model fits the data from Nissan quite well using following assumptions:

Calendar loss first year without aging factor - 6.5%
Calendar loss slows with square root of time

Cycling loss per 12500 miles without aging factor - 1.5%
Cycling loss is linear

Boston aging factor adjusted to 0.75
"Normal" aging factor adjusted to 0.90
Phoenix aging factor adjusted to 1.35

Aging factors relative to "Normal" are:

Boston - 0.83
Normal - 1.00
Phoenix - 1.50

Note: while it might seem that my selections were random, or that many combinations could give the same results, I found that changing any of the values from this set of numbers caused increasing errors in predicting the numbers found on the graph. I make no claim that these numbers are good predictions of reality, only that they fit Nissan's figures closely. Comments, discussion, suggestions for improvement, etc. are welcome. If anyone wants to play with the actual spreadsheet, let me know.

I checked the model that Surfingslovak created, and to my surprise he came up with the exact same calendar loss and cycling loss figures. The reasons that his results differ from mine:

1) Different aging factor was used - mine were derived empirically rather than calculated. Mine could be correct depending on the activation energy, as tbleakne pointed out.

2) His model uses 7500 miles as the baseline for all cities; I used 7500 miles for Phoenix, 12500 miles for "Normal" and Boston. I believe this is an error on his part, and leads to results showing more degradation than Nissan predicts. However, I could be wrong.

I'm not sure it is worthwhile trying to fit a model to the curves which Nissan gave to TickTock. Frankly, I don't expect to see any LEAFs in Phoenix with 76% capacity left after five years, regardless of miles driven. Why? Because I think Nissan's faith that the degradation will taper off after a year or so is only valid in a usage regime where cycling dominates degradation. In the cases we have in Phoenix where calendar life is the dominant (or at least A dominant) force, I expect the degradation will be closer to linear. Additionally, as these batteries age, the daily DOD increases. This is another effect which I don't think is being properly accounted for.

As usual, I hope I am wrong about this. OTOH, Nissan's track record for making predicitions in Phoenix is quite poor to date.

Of course, I think all the data needs to be corrected for 12,000 or 12,500 miles. Just because Nissan pulled some number out of their posterior, it doesn't reflect the average driver.

RegGuheert said:
I'm not sure it is worthwhile trying to fit a model to the curves which Nissan gave to TickTock. Frankly, I don't expect to see any LEAFs in Phoenix with 76% capacity left after five years, regardless of miles driven. Why? Because I think Nissan's faith that the degradation will taper off after a year or so is only valid in a usage regime where cycling dominates degradation.
From what I have read, calendar life loss does follow the square root of time, so presumably the calendar life loss will slow. If it doesn't the Phoenix batteries are definitely cooked. I added an additional factor to my model, miles per kwh. Nissan stated that their assumptions were based on the LA4 drive cycle, which is why they focused in on highway driving as a factor in the Phoenix Leaf accelerated losses. Based on 100 mile range and usable capacity of 21.381 kwh, I calculated that the miles per kwh for that drive would be 4.68. If one is driving less efficiently, more cycling of the battery will be involved. Using a figure of 3.5 miles per kwh for TickTock (I have no idea what his longterm average is, just a made up number), my model predicts for his car (84.7% capacity measured at Casa Grande after 1.3 years):

Calendar loss - 10.01%
Cycling loss - 3.16%
Cycling loss corrected for less efficient driving - 4.22%
Total capacity loss - 14.23

Predicted capacity - 85.8%

Difference from actual capacity - 1.07%

This doesn't take into account solar loading, which I imagine Nissan may not have included in their model. I don't know if TickTock's Leaf spends a large portion of its time in direct sun (if he reads this post, he can certainly chime in with more info).

TonyWilliams said:
Of course, I think all the data needs to be corrected for 12,000 or 12,500 miles. Just because Nissan pulled some number out of their posterior, it doesn't reflect the average driver.
I totally agree. I used the 7500 mile number to develop the model, not because I think that is a valid assumption. Using 12500 miles per year, my model predicts a 29.75% loss of capacity at 5 years in Phoenix--and that doesn't include the effects of driving faster, which would lead to more cycling.

After seeing what's expected best case scenerio, worst case scenerio . . . . I must admit I am one of the VERY lucky ones - still getting 100 miles per . . . and 280 gids (knock on wood)

RegGuheert said:
Using a figure of 3.5 miles per kwh for TickTock (I have no idea what his longterm average is, just a made up number), my model predicts for his car (84.7% capacity measured at Casa Grande after 1.3 years)
4.44 is my overall efficiency (from the dash) since I started logging last October. Nissan claims my accelerated degradation is just due to higher then 7500/year milage (i.e. more battery cycles).

TickTock said:
RegGuheert said:
Using a figure of 3.5 miles per kwh for TickTock (I have no idea what his longterm average is, just a made up number), my model predicts for his car (84.7% capacity measured at Casa Grande after 1.3 years)
4.44 is my overall efficiency (from the dash) since I started logging last October. Nissan claims my accelerated degradation is just due to higher then 7500/year milage (i.e. more battery cycles).
My model changes only slightly with that figure: difference is 1.97 between predicted and actual. My model slighly under-predicts at one year, is very close at two years, and very slightly over-predicts at 5 years. That's the closest I could get it. Of course, it might help to have the actual numbers from TickTock's graph (hint, hint). I estimated them by looking at the graph. Also, wondering how much you parked in the sun, TickTock. Based on the Prius study of solar loading, parking in the sun full-time can cause a 10% (low solar load) to 24% (high solar load like Phoenix) increase in battery aging compared to avoiding parking in the sun completely. The solar loading depends on geographic location. Of course, the Leaf may be designed in such a way that the solar loading factor would be less.

Stoaty said:
TickTock said:
RegGuheert said:
Using a figure of 3.5 miles per kwh for TickTock (I have no idea what his longterm average is, just a made up number), my model predicts for his car (84.7% capacity measured at Casa Grande after 1.3 years)
4.44 is my overall efficiency (from the dash) since I started logging last October. Nissan claims my accelerated degradation is just due to higher then 7500/year milage (i.e. more battery cycles).
My model changes only slightly with that figure: difference is 1.97 between predicted and actual. My model slighly under-predicts at one year, is very close at two years, and very slightly over-predicts at 5 years. That's the closest I could get it. Of course, it might help to have the actual numbers from TickTock's graph (hint, hint). I estimated them by looking at the graph. Also, wondering how much you parked in the sun, TickTock. Based on the Prius study of solar loading, parking in the sun full-time can cause a 10% (low solar load) to 24% (high solar load like Phoenix) increase in battery aging compared to avoiding parking in the sun completely. The solar loading depends on geographic location. Of course, the Leaf may be designed in such a way that the solar loading factor would be less.
I almost never park in the sun (usually get one of the covered parking spots). For a couple of months I was plugging in at work and taking advantage of the pre-cool feature before I left the office (but very rarely actually charged there - only if I had a far errand over lunch). Since the AC condenser seems to dump the heat into the cavity above the battery, I wouldn't be surprised if that contributed.
Here are the numbers I managed to jot down. Keep in mind *I* didn't see the actual numbers either and read these values off the graph. I would guess I am within 0.5%.
• Age Nom Boston Phoenix
0.5 ??? 94% 92%
1.0 92% 93% 89%
2.0 88% 90% 85%
5.0 80% 84% 75%

TickTock said:
I almost never park in the sun (usually get one of the covered parking spots). For a couple of months I was plugging in at work and taking advantage of the pre-cool feature before I left the office (but very rarely actually charged there - only if I had a far errand over lunch). Since the AC condenser seems to dump the heat into the cavity above the battery, I wouldn't be surprised if that contributed.
Here are the numbers I managed to jot down. Keep in mind *I* didn't see the actual numbers either and read these values off the graph. I would guess I am within 0.5%.
• Age Nom Boston Phoenix
0.5 ??? 94% 92%
1.0 92% 93% 89%
2.0 88% 90% 85%
5.0 80% 84% 75%
Thanks. Those numbers are pretty close to what I guessed. I figured out a way to tweak my model by applying an empirical correction factor to almost exactly match the graph you jotted down from Nissan. The observed corrections needed to come up with the right numbers plotted against the number of years are linear, with a correlation coefficient of 0.9995. It needs a bit more work, but I think I can improve the predictive value. After that, I need to plug in more data to see if it matches up.

TonyWilliams said:
Of course, I think all the data needs to be corrected for 12,000 or 12,500 miles. Just because Nissan pulled some number out of their posterior, it doesn't reflect the average driver.

7500 miles is the average number for Leaf drivers in Phoenix, probably a bunch of wealthy retired geezers.

OK, after tweaking my model it now mathematically reproduces the curves shown in TickTock's graph with very small errors. I am guessing that the additional tweak could represent calendar life loss between the time of manufacture and and the time of delivery to the customer, but I haven't had a chance to change the model to see if that is the correct explanation. That would certainly explain the fact that, for example, at one year the model was off by approximately the same fixed amount for each city. That implies something that was happening to the cars while they were in the same environment. After tweaking, the model now predicts TickTock's loss to within 1% (predicted - 85.7% capacity retained, actual - 84.7%). There is one more piece of data that I need from TickTock (and from other Casa Grande cars if I can get it). How much did you charge to 100%, and how long did your Leaf sit at 100% typically until you drove it?

Here is a graph showing the results of my model:

#### Attachments

• Battery Aging Model Graph.png
92.2 KB · Views: 117
hill said:
After seeing what's expected best case scenerio, worst case scenerio . . . . I must admit I am one of the VERY lucky ones - still getting 100 miles per . . . and 280 gids (knock on wood)

Gary beginning to think we are simply seeing a much higher than expected variance between pack characteristics straight from the factory. i propose, your balance was a tighter match than average. in the new pack stage; micro differences may extrapolate to extreme differences in longevity

a bit OT; but do you have covered parking at work?

Herm said:
TonyWilliams said:
Of course, I think all the data needs to be corrected for 12,000 or 12,500 miles. Just because Nissan pulled some number out of their posterior, it doesn't reflect the average driver.

7500 miles is the average number for Leaf drivers in Phoenix, probably a bunch of wealthy retired geezers.

And they magically knew that is who would buy the car, and how far they would drive? Is each owner required to hit "OK" every time?

No, they pulled it out of their posterior to cover their posterior. Surely, it was based on 12,500 miles, and then when they saw things weren't going to work well, they "reindexed" 76% and 7500 miles.

Stoaty said:
From what I have read, calendar life loss does follow the square root of time, so presumably the calendar life loss will slow.
From what I can find in posted literature, that type of degradation is what researchers use when they are fitting calendar life to data measured over the first 10 DAYS of the cell's life. In each case where researchers have actually MEASURED calendar life for many months, the degradation is at least linear and in many cases the degradation accelerates with time.

I have previously posted links to a paper that did measurements back in 2003. Here is a more recent study: 2007 Calendar Li-ion Pouch Cell Calendar Life Study - Zhang and White. Granted, this is still on a different chemistry, but it shows the same effect. Interestingly, the degradation is linear unless the temperature is above 15C: at 25C and above it rolls off a cliff at some point. The researchers suggest there may be different mechanisms active at the higher temperatures.

I will say that the data we have for the LEAF battery so far does not look very promising, either. Some LEAFs with low miles have been well-pampered and still appear to be degrading quickly.

So I will ask, does anyone have *measured* data for pouch Li-ion cells showing calendar degradation that is linear for many months or years?

the difference in chemistry cannot be ignored. even for the chemistry Nissan uses, it can be tweaked to favor cycling, power output or temperature resistance.

another thing about a lot of batteries is that "talk time" is favored so batteries are purposefully charged slightly over recommended levels which in essence, creates an artificial linear degradation curve. granted its cellphones but the idea is still out there

I realized there is one additional factor in TickTock's case that led my model to slightly under predict his capacity loss: his car has been through most of 2 hot summers during the 1.3 years of ownership. My model is only accurate for a whole number of years, and will have slight inaccuracies for partial years. Aging will be different during different seasons, which is not easy to account for. One would have to calculate an aging factor for each month of the year (not something I plan to do in general, although I might end up doing it for Phoenix and Boston).

I tried to work backward to see where the empirically derived battery aging factors that make my model work could have come from. It turns out that I can get the factors quite close by assuming that the speed of chemical reactions doubles for every 15.2 degrees C., rather than every 10 degrees C. Again, not saying this is right, but I did find it interesting. More info to come as I discover (or invent?) it. :lol:

Stoaty said:
Fascinating, please keep up the good work. Wish I had time to participate more actively in this thread.

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