In school, we learn that the voltage across circuit components in parallel is the same, and the current is split between them according to their resistances. For components in series, the current through each is equal and the voltage drops off. In a simple model, the total capacity of a battery pack with cells in series and parallel is the complement to this. If cells have capacity Q, and they are arranged in a simple layout with np cells connected in parallel, then ns times in series (see the figure below), we have:
But the real picture is complicated by the presence of cell-to-cell variation. Such variations can arise during the manufacturing process—electrode thickness, electrode density (or porosity), the weight fraction of active material [1,2,3], and the particle size distribution [4,5] have been identified as key parameters that impact cell-to-cell capacity variation in lithium-ion cells. Resistance variation also occurs [6]. Cell-to-cell variations resulting from the manufacturing process are then compounded by other factors that arise during battery pack operation, such as temperature variations [7].
In this blog post, we’re just going to look at how cell-to-cell variation affects the discharge capacity of an assembled battery pack. In this model, each cell in the battery has a nominal capacity Q, and an actual capacity Qij which is a random variable:
Which means that the (random) capacities of each cell are independent from each other, and that they are each drawn from an identical probability distribution which is a normal distribution. This normal distribution has a mean equal to the nominal capacity Q and a standard deviation 𝜎Q (this value quantifies the cell-to-cell variation). The subscripts i and j label the position of the cell in the battery pack. i labels the position of each cell in series (from top to bottom in the figure above) and j labels its position from left to right in the parallel arrangement.
This post has been built based on the support and sponsorship from: Thermo Fisher Scientific, Eatron Technologies, About:Energy, Quarto Technical Services and Alvatek.
If probability theory isn’t your cup of tea, you can think about what this means in a different way—in a sense, the other way round. Produce a batch of cells with nominal capacity Q, but which in fact show some cell-to-cell variation—not due to a systematic issue in the manufacturing process, but instead due to random processes which have been controlled as much as possible. Plot a histogram of the actual measured capacities of each cell, and if the batch size is large enough, it should have the shape of a bell curve centred at Q and with a standard deviation of 𝜎Q. To give an example of actual numbers, one paper reports the ratio 𝜎Q/Q to be 0.33% [6]. Small, but as we will see, this compounds to have a bigger effect on the total capacity of the battery pack.
To complete the battery pack model, we need to know how different cell capacities combine to give the overall capacity Q. Going back to our analogy at the start of the post, we can see that the capacity of each cell arrangement in parallel will sum up. But how about those arrangements in series? The discharge capacity of a module will be effectively limited to the capacity of the weakest (lowest capacity) cell or group of cells in parallel [1,8]. This is because the lower discharge limit will be determined by a voltage measurement across each parallel group of cells, the first parallel group of cells (smallest group capacity) to hit this lower voltage limit stops the discharge.
To summarise in a single formula:
With this information, we can put together our statistical model.
Statistical model
Due to properties of probability distributions, we can break this problem down into two steps. Find the probability distribution of the cells in parallel, then find the probability distribution of the parallel cells arranged in series (i.e. of the overall battery pack). It turns out the first step is easy: because we’re assuming the capacity of each cell is normally distributed, and the statistical distribution of a sum of normally distributed variables is another normal distribution:
The second step is trickier. So let’s simplify things a bit: because the distribution above looks the same (normal) no matter how many cells are in parallel, let’s take the simplest case of np=1. That’s the same as having a string of cells arranged in series. And for the sake of readability, let’s simplify our notation a bit. I’ll use n in place of ns for the number of cells in series, and 𝜎 instead of 𝜎Q for the standard deviation of cell capacities. We will also drop the j subscript from our cell capacities (in this simplified configuration, cells only need to be labelled by their position in series, not in parallel).
So, to be completely clear, we’re now trying to work out the probability distribution of Qtot as it is defined in the formula above. To work this out we’re going to make use of a neat little trick: if Qtot is larger than some other arbitrary value, q, that means the capacity of every cell in series also has to be larger than q. And correspondingly, in terms of probabilities, the probability that Qtot is larger than some other value q, is equal to the probability that every capacity in the module is larger than q. We can write this as:
We can now work out the cumulative distribution function (CDF) of Qtot, making use of this equation:
We can see that this results in an equation for the CDF of Qtot that in turn contains a CDF for the random variable Qi which we know is normally distributed. A probability density function (PDF) can be calculated from the corresponding CDF by differentiation, which gives us:
which can also be expressed in the standard PDF / CDF notation using lowercase f / capital F:
On the right hand side, we now have the CDF and the PDF of the normal distribution with mean Q and standard deviation 𝜎Q. This is straightforwardly handled in any statistical library, and can be plotted exactly as well as calculating its mean and variance. But we can also analyse the distribution a little more closely and learn something about the effect cell-to-cell variation has on battery pack capacity.
An in-depth look at the module capacity PDF
Let’s look at that last equation and the simplest case of n = 2 (n = 1 is trivial, it’s just the normal distribution for a single cell). In fact, this case corresponds to the skew-normal distribution. As its name suggests, this is a modification of the normal distribution that incorporates skew—it introduces an extra parameter that solely controls the distribution’s skewness, and is scale-independent. In our equation, that parameter has a value of -1, a low value meaning the distribution has low skewness. But introducing this parameter also has an effect on the mean and variance of the distribution, which will be of more interest to us as those correspond to the expected value of the module’s total capacity and its standard deviation
In the figure above I’ve plotted how the distribution varies as n, the number of cells in series, increases (with consistent values of Q and 𝜎). We can see that there is a continual decrease in the mean value—meaning, the expected value for total pack capacity—as n increases. The distribution also becomes narrower: that means, if we assemble lots of packs with n cells in series, we expect to see a smaller spread of measured total capacities within the packs as we increase n. The x-axis is given in units of standard deviations: using our value of 𝜎/Q = 0.33% we can see that for large modules the drop in expected total capacity due to cell-to-cell variation amounts to about 1% of the nominal capacity.
But recall that there is another component to the problem—which is that cells can be assembled in parallel then in series. And the parallel subassemblies also have a modified distribution (see the formula at the top of the previous section), with a mean total capacity proportional to np and a standard deviation proportional to √np (note: this is the same statistical phenomena that gives rise to combining errors in quadrature). This decreases the expected drop in total capacity from its nominal value by a factor of √np. The formula for this is:
where α is the number of standard deviations that the mean shifts by, as a function of ns (refer to the figure above).
One illustrative case is to consider two battery pack configurations with the same nominal total pack capacity (230Ah). The first pack configuration has np=46 cells arranged in parallel, which are then arranged in series with ns=96. Each cell has a (mean) capacity of 5Ah. The second configuration has np=2 and ns=96, with 115Ah (mean) capacity cells. Clearly, the nominal total pack capacity (and voltage) is the same in each configuration. I’ve plotted the distributions below: the first configuration with 5Ah cells is in blue, and the second with 115Ah cells is in green. Like the plot above, I’ve plotted both the true (solid line) and approximate (dashed line) distributions.
In a future post, I will take a look at some subtler details of the total pack capacity distribution and how it relates to pack configuration, as well as how we have implemented the calculation in the Battery Calculations Workbook.
References
- Ben Kenney, Ken Darcovich, Dean D. MacNeil, Isobel J. Davidson, Modelling the impact of variations in electrode manufacturing on lithium-ion battery modules, Journal of Power Sources, 213 (2012), pp.391-401, https://doi.org/10.1016/j.jpowsour.2012.03.065
- Simon F. Schuster, Martin J. Brand, Philipp Berg, Markus Gleissenberger, Andreas Jossen, Lithium-ion cell-to-cell variation during battery electric vehicle operation, Journal of Power Sources, 297 (2015), pp.242-251, https://doi.org/10.1016/j.jpowsour.2015.08.001
- Shriram Santhanagopalan, Ralph E. White, Quantifying Cell-to-Cell Variations in Lithium Ion Batteries, International Journal of Electrochemistry (2012), https://doi.org/10.1155/2012/395838
- Louisa Hoffmann, Jan-Kirsten Grathwol, Wolfgang Haselrieder, Ruben Leithoff, Tobias Jansen, Klaus Dilger, Klaus Dröder, Arno Kwade, Michael Kurrat, Capacity Distribution of Large Lithium-Ion Battery Pouch Cells in Context with Pilot Production Processes, Energy Technol., 8 (2020), https://doi.org/10.1002/ente.201900196
- Lars Bläubaum, Fridolin Röder, Christine Nowak, Hoon Seng Chan, Arno Kwade, Ulrike Krewer, Impact of Particle Size Distribution on Performance of Lithium-Ion Batteries, ChemElectroChem, 7 (2020), https://doi.org/10.1002/celc.202001249
- Katharina Rumpf, Maik Naumann, Andreas Jossen, Experimental investigation of parametric cell-to-cell variation and correlation based on 1100 commercial lithium-ion cells, Journal of Energy Storage, 14 (2017), pp.224-243, https://doi.org/10.1016/j.est.2017.09.010
- Xinhua Liu, Weilong Ai, Max Naylor Marlow, Yatish Patel, Billy Wu, The effect of cell-to-cell variations and thermal gradients on the performance and degradation of lithium-ion battery packs, Applied Energy, 248 (2019), pp.489-499, https://doi.org/10.1016/j.apenergy.2019.04.108
- Mina Naguib, Phillip Kollmeyer, Ali Emadi, Lithium-Ion Battery Pack Robust State of Charge Estimation, Cell Inconsistency, and Balancing: Review, IEEE Access, 9 (2021), pp. 50570-50582, https://doi.org/10.1109/ACCESS.2021.3068776