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Penn State team develops mathematical formula to predict factors influencing Li-ion battery aging

A team of Penn State researchers has developed a mathematical formula to predict what factors most influence lithium-ion battery aging. Volvo Group Trucks Technology supported this work.

Led by Christopher Rahn, professor of mechanical engineering, the team started out developing models for the specific chemistry of batteries used by Volvo Trucks. After showing the models matched experimentally, the researchers focused on simplifying the aging model—a nonlinear, electrolyte-enhanced, single particle model (NESPM) that includes aging due to solid electrolyte interphase layer growth—and have now brought it down to a formula, said Rahn.

The model is validated with experimental full charge, discharge, HEV cycle, and aging data from 4.5 Ah graphite/LiFePO4 cells. The NESPM is capable of operating up to 3C constant charge–discharge cycles and up to 25C and 10 s charge–discharge pulses within 35–65% state of charge (SOC) with less than 2% error. The NESPM aging model is then simplified to obtain explicit formulas for capacity fade and impedance rise that depend on the battery parameters and current input history. The formulas show that aging increases with SOC, operating temperature, time, and root mean square (RMS) current. The formula predicts that HEV current profiles with the (i) same average SOC, (ii) small SOC swing, (iii) same operating temperature, (iv) same cycle length, and (v) same RMS current, will have the same cell capacity fade.

—Tanim and Rahn (2015)

Schematic diagram of an SPM (single particle models without electrolyte dynamics) Li-ion cell model. The cell consists of three domains: anode with spherical graphite particles; separator; and cathode with spherical active material particles. The NESPM model developed by Tanim and Rahn includes aging due to solid electrolyte interphase (SEI) layer growth. Source: Tanim and Rahn (2015) Click to enlarge.

According to Rahn, a battery ages, or degrades, whether it is sitting on a shelf or used. The main cause of lithium-ion battery aging is the continuous formation of the solid electrolyte interphase (SEI) layer in the battery.

The SEI layer must form for the battery to work because it controls the amount of chemical reactions that occur in the battery. As the battery is continually used, however, small-scale side reactions build up at the SEI layer, which decreases battery capacity. Models allow researchers to understand how different factors affect this degradation process so that longer-lasting, more cost-efficient batteries can be made.

According to the researchers, this new simple aging formula takes into account only the factors shown to most influence lithium-ion battery aging by affecting growth of the SEI layer, which include state of charge, how often the battery charges/discharges completely, operating temperature, and current.

Car companies can use this formula to quantify which factors are contributing the most in aging and focus more on them and less on all of the other factors that don’t play as much of a role.

—Tanvir Tanim, graduate student in mechanical engineering, Penn State

As part of the study, Tanim and Rahn compared the accuracy of their formula to that of more complex models using commercially available batteries. They found that their simple formula works just as well.

Whenever you simplify a model, there are some things lost. We have complicated models because they are very accurate. As you simplify, you have to justify every assumption that you make. I wasn't sure we could simplify the model down to a formula. It’s pretty amazing to explicitly see how things depend on one another.

—Christopher Rahn

Rahn and Tanim have already seen the benefits of having a simple formula to model battery aging by using it to show that increasing the temperature of lithium-ion batteries in hybrid electric vehicles extends the life of the battery, which is contrary to what most researchers think. This effect was something that Volvo had previously seen with their batteries, and using this aging formula, Rahn and Tanim could explain why.


  • Tanvir R. Tanim, Christopher D. Rahn (2015) “Aging formula for lithium ion batteries with solid electrolyte interphase layer growth,” Journal of Power Sources, Volume 294, Pages 239-247, doi: 10.1016/j.jpowsour.2015.06.014

  • Githin K. Prasad, Christopher D. Rahn (2013) “Model based identification of aging parameters in lithium ion batteries,” Journal of Power Sources, Volume 232, Pages 79-85, doi: 10.1016/j.jpowsour.2013.01.041



Sounds really useful.
I wonder how many types of batteries it covers and what the implications for real car (and domestic storage) designs are - i.e. should we add supercaps to keep the peak currents down, etc.


GM did years of extensive testing on packs, I never saw any stories about how Tesla plowed through the same rigor with their packs.

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