Design-of-Experiment and Statistical Modeling of a Large Scale Aging Experiment for Two Popular Lithium Ion Cell Chemistries

Pouch cells of two different chemistries were studied: 99 cells which featured iron-phosphate cathodes and graphite anodes (further referred to as LFP cells) and 78 cells which featured nickel-aluminum-cobalt-oxide cathodes and graphite anodes also (further referred to as NCA cells). For reasons of energy consumption and safety of the experiment, a small cell size was selected for the cells. The cells showed initial capacities of 50 mAh (LFP) and 60 mAh (NCA) and consist of three square electrode sheets of the sequence anode (single layered) – cathode (double layered) – anode (single layered). During the whole experiment the cells were pressurized by using plastic clamps sitting on the outside of the pouch foil, in analogy to the straps commonly used in automotive applications, to hold the electrodes firmly in place. A common carbonate electrolyte (EC/EMC) and PP separator was used. The cells were especially designed and built for this experiment, using base electrode materials from Südchemie AG and cell building know-how from our project partner GAIA Akkumulatorenwerke GmbH.

Parallel to building cells, a Design-of-Experiment (DoE) was conducted. The DoE is a procedure which helps to plan and conduct an experiment such as to obtain statistically valid and objective conclusions.²⁸ It consists of 5 steps, of which the first four will be explained in this subsection, for the last step see the explanations in sections Results and Discussion.

Statement of Experimental Target

Screening and Test Plan Conception

The second step of the DoE procedure is the screening of the tested cells as well as the conception of the full factorial test plan. Screening means defining the range limits of the settings of the external influence factors (i.e. −15, 20, 50°C are settings for the factor "temperature") and testing of these limits regarding the cells' sensitivity in voltage response to setting combinations (i.e. are high c-rates at low temperatures possible?). The screening was mainly done by literature research, where many single-factorial variation experiments are on display.^{12,15,16,18,30} The fishbone-diagram in Fig. 1 was created out of the testing constraints and the results found in literature, wherein the "controllable design factors" are the influence factors to be actively varied in the test plan. From the number and assumed influences of "uncontrollable factors" the number of repetitions of each test point was determined to be at least three times, in order to obtain statistically valid results. On the bottom side of the effect arrow in Fig. 1 the factors are listed, which require special attention during the planning phase. It must be ensured that the "held-constant factors" are truly constant for the whole duration of the experiment, whereas the "nuisance factors" are points where very accurate calibration or additional measurement devices are supplied.

The four influence factors for this experiment are the temperature, the current (I, always given in c-rate), the average state-of-charge (SoC) and the amplitude or delta state-of-charge (dSoC). The temperature is given in degrees Celsius [°C], the range limits were set to −15°C and 70°C. For the current (expressed as c-rate) the generally accepted definition was applied, that the "c-rate is the current which corresponds to the cell capacity".²² The range of test for the c-rate is 0C (pure calendar aging) to 5C for charge as well as discharge. The average SoC is given in a range of [0...1]. The test will not exploit the full SoC range but will stay within 0.1 and 0.9, because it is not expected that a real-life application relies on the outer limits of the cell operation window. The dSoC is the SoC swing which is run through by a single constant-current pulse, which means that the SoC of the cell oscillates around the required average SoC by half of the dSoC value. The range limits of the dSoC will be 0 (calendar aging) to 0.8.

The full-factorial test plan is constructed out of each combination of the controllable factors in each of its factor levels. Factor levels are the numeric values in which the influence factor range is discretized, meaning for example the current factor range is 0 to 5 C, in which the levels are for example 0, 0.5, 1, 2 or 5 C. When considering the four identified controllable factors with five setting levels each, the full test plan has 4⁵ = 1024 possible sample points and should be repeated at least 3 times in each point to eliminate the risk of testing an outlier with >90% confidentiality.

If the factor levels are chosen each within a physically possible range, the interaction with other factors' levels can be limited by the cell physics or the measurement equipment accuracy. The range of each factor, as well as the interaction limits found in literature, are marked as full (LFP) and dotted (NCA) lines in Fig. 2. In this figure the influence factors are given in columns and rows, so each plot shows the test space spanned by the two influence factors marked on the outside of the corresponding column and row.

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While considering imposable test limits, it is important to keep the test space convex,²⁷ because after cutting away all the restricted spaces from the rectangular test space, a space must remain in which any two points can be connected by a straight line without crossing a limit or range border. Four such physically necessary limits were identified:

The first limit is the combination of dSoC with SoC, giving values larger than the SoC range, when added 1. The limit leads to the triangle shaped test space in the SoC vs. dSoC plot of Fig. 2 (center plot).

Second, the interaction of low temperatures and high currents was formulated as a limit, since cell cycling in this area could not be maintained within the operating voltage window. The current is empirically limited to 1C at −15°C, from 10°C upwards the full 5C is available 2 and top right plot in Fig. 2).

In 2 the temperature (T) is given in degrees Celsius [°C] and the current (I) in c-rate [h⁻¹]. These numeric values depend strongly on cell micro and macro geometry as well as cell chemistry. Therefore this limit is not generally applicable with these values, contrary to the first limit.

The third and fourth limits concern the dSoC and current interaction (top center plot in Fig. 2). Both limits counteract the problem, that at zero current (I = 0) the dSoC is also always zero (and vice versa). When approaching this point with very small currents and/or very small dSoCs, the cycling equipment's accuracy will be the limiting factor. The limits were defined to not restrict the space more than up to the next factor level in current direction (1C) and in dSoC direction (0.1).

Since both limits are dependent of the cell chemistry and cycling equipment accuracy used, they are not applicable with exactly these values to every experiment, but the formulas are generally valid.

For the full-factorial test plan, the limits reduce the 1024 possible sample points by 230 not feasible points. So summing up for the cell count of the full-factorial experiment, a total of 3 * (4⁵ - 230) = 2382 cells in 794 different sample points should be tested.

Test Plan Reduction

In the third step of the DoE process, the full factorial test plan is reduced, by using a modular algorithm for the design of large scale D-optimal experiments. The procedure and its implementation is thoroughly explained by Haselgruber in²⁷ and used unaltered. The reduction is based on a criterion describing the usefulness of a single experimental point for the parameterization of an expected model for the target parameters. The score for the usefulness is called D-optimality of a design plan in statistical terms.

The algorithm selects a set of points in the corners of the test space as well as some points randomly distributed within the space. In each following optimization step it takes another random point out of the full factorial test plan, exchanges it with one point of, or adds it to, the initially selected points and calculates the change in D-optimality.²⁸ If the D-optimality increases, the changes are kept and another point is exchanged. The optimization is finished if no more increase in D-optimality can be found. Such a coded test point matrix is displayed in,²⁶ Table 2 for a visual reference.

After the screening process, it was assumed that a second-order statistical model would be the best way to express the expectations,³¹ without constraining the experiment too much. The target values (capacity decrease and impedance increase) are either linear or quadratic functions of a single factor or linearly dependent of the multiplication of two influence factors. The model was defined such as to incorporate offsetting the center of a quadratic term into the experimental range, an example of this model for capacity decrease is shown in 5.

This model definition accounts for all effects found in literature at the time of setting up the experiment, describing the capacity decrease and impedance increase depending on the chosen external influence factors.^7,9,30 Table I gives an overview of influence factor transformations, which can be deduced from literature^{7–9,11,16,30} and which were used as a basis for the target parameter model of the DoE.

Table I. Overview of the influence factors and their transformations, deduced from literature, as basis for the target parameter model. The variables x and y stand for any numeric values within the ranges of the influence factors and introduce an offset from zero for the changes of influencing behavior. Sqrt(I) ... square root of I.

Influence Factor
Temperature	Current	State-of-Charge	Delta State-of-Charge
T	I	SoC	dSoC
T2	Sqrt(I)	SoC2	dSoC2
(T - y)	I2	(SoC - x)
(T – y)2		(SoC - x)2

The exact model formulation, meaning whether every influence factor transformation has enough significance to remain in the resulting model, as well as the values for a_i are not known until the analysis of the experiment.

Using the algorithm of²⁷ the test plan was successfully reduced to 46 sample points, now fitting to the restrictions of the equipment available (channel number). These points are indicated in Fig. 2 for LFP cells (boxes) and for NCA cells (stars).

The fourth step of the design-of-experiment procedure is the conduction of the experiment itself, which is explained in the following subsections. Step 5 is the data analysis, which is presented in the sections Results and Discussion.

The experimental planning was conducted under the premise that the measurement data acquisition is intermittent. This was necessary, because the capacity of the cell cannot be measured during 10% dSoC swings. So an experimental time schedule was drawn up, pictured in Fig. 3.

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The cells were built with the electrolyte in a squashable plastic bag, so they could be activated whenever the cell was required for the experiment. The first step after being connected to the cycling equipment is the formation procedure. After the initial cycles a reference test follows, which is described in detail in 2.4, conducted at reference temperature (30°C). Following this initial test, in order to make the cells comparable at start, the reference test procedure is applied a second time at the design-point temperature of the cell. The capacity measured in this second test forms the basis for the current rate to obtain the c-rate, at which the load cycling will be conducted. It was decided not to use the nominal c-rate for aging cycling, because otherwise the current density-dependent aging effects would be increasingly accentuated.

The cycling load as well as calendar aging (at a c-rate of 0 and a dSoC of 0), is periodically interrupted by reference tests, conducted at design temperature and at reference temperature. After reaching an end-of-life criterion in a reference test at reference temperature, the cell is taken out of the experimental setting.

The reference test procedure (RTP) was designed to be a measurement sequence which allows for a calculation of capacity decrease, impedance increase and their respective estimation errors as well as additional analysis data sets, such as differential capacity and differential voltage. It consists of an initial segment in which the cell is preconditioned to a full SoC (constant voltage charge at 3.8 V (LFP) / 4.2 V (NCA) until current < 0.05C). Then, a slow discharge at 0.2C is conducted for measuring the actual discharge capacity. This capacity is used as base for the following c-rate scan with 0.5, 1, 2, and 5 C (charge and discharge), each including constant voltage full charges in between (same as above). The third section is a galvanostatic intermittent pulse test, which has 5 pulse and rest periods at 0.5C in discharge and in charge direction (at SoC = 0.9, 0.7, 0.5, 0.3 and 0.1). This section is the base for calculation of the actual impedance³² of the cell. In the last part of the RTP a charge step followed by a controlled 50% discharge step set the cell to be ready to start a new load cycle. The RTP initialization and the load cycling restart are done by hand, because the new capacity has to be manually decreased in the control program of the cycling equipment.

The cycling equipment is an Arbin BT2000 with 96 channels ± 5 A in between 0 and 5 V with a measurement accuracy of ± 0.002 V and ± 40 uA in the used current range of ± 100 mA. Under the assumption that the measurement error follows a Gaussian distribution, the error for the capacity measurement is < 0.25% and the error for the impedance calculation is < 15%, for worst case settings with the given cells.

The status of aging is always calculated from a single RTP step, including the cell capacity, the cell impedance, the differential voltage and differential capacity:

• The cell capacity is calculated as being the value of the coulomb counter after the 0.2C discharge, which has been reset in end of the preconditioning charge preceding the discharge.
• The cell impedance is calculated from the pulse test of the RTP. For each current pulse of 0.5C a pulse impedance is calculated while switching on and switching off.³² The current rise time of the cycling equipment is <30 ms, so a time delta of 500 ms was chosen for the highest measurement frequency of voltage and current. Since the pulse test was done in charge and discharge direction, this equals to four calculated values per SoC level. A mean value for the impedance is calculated from all values at the same SoC level, since the expected difference between charge and discharge impedance is smaller than the measurement error. A linear regression is then applied to the mean values for the SoC levels to receive the median impedance for the cell.
• The differential voltage and differential capacity data is calculated from the same 0.2C discharge as the capacity. A constant current discharge is not the optimal base for this transformation, but a viable one according to.³³

The experimental results for the capacity decrease are shown in Fig. 4 (LFP cells) and in Fig. 5 (NCA cells). There is a strong dependency of the capacity decrease on the test temperature for both chemistries. Although the test space for the two chemistries was slightly different (range limits for LFP cells: −15°C to 70°C, for NCA: 10°C to 50°C), the tests at 10°C were the ones resulting in the longest cell life time for both chemistries. For the LFP cells it is remarkable that cells being cycled at 1C do not age much faster than calendar aged cells, whereas cells cycled at 5C age significantly faster. For the NCA cells any cycling load ages the cells faster than calendar aging samples. For these cells the influence of the dSoC is stronger than the c-rate influence. These effects and additional, not directly visible effects of the other influence factors on the two cell chemistries will be highlighted during model built.

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The experimental results for the impedance increase are shown in Fig. 6 (LFP cells) and in Fig. 7 (NCA cells). The plots are structured the same way and show the data the same way the capacity decay plots do. The impedance and the time are again normalized to the initial measurement respectively the longest cell endurance.

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The impedance ticks on the y-axes are in logarithmic scales, due to the fact that the impedance is depending rather strongly on temperature. When temperatures below 10°C are applied to new cells, the impedance rises approximately 2.5 times per decade Celsius. Rising temperatures above 30°C do not affect the new cells as strongly, an impedance decrease of a factor of about 1.5 times per decade Celsius is observed. For aged LFP and NCA cells alike, these dependencies become less significant. Influences of factors other than temperature are not directly visible within the impedance data.

Thompson³³ and more recently Dubarry²¹ described how to read the changes of differential capacity over aging. Therefore, differential capacity plots were generated for the cells, as can be seen in Fig. 8 for a LFP cell (a and b) and a NCA cell (c and d). However, no other changes than a vanishing peak (upmost plot of b) to the right) for LFP and a shift to lower potentials for NCA could be observed. As both of these features are well visible in the discharge curves already, the peak shift data from differential capacity is not included in statistical analysis.

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The fifth and last step in the DoE process (section 2.2) is modeling and analysis. There is a wide selection of statistical model methods available, which can be applied to the generated experimental data. Yet for the analysis of the DoE, the model which has been used for calculating the optimized design plan will be used for modeling. The basic model type is already shown in 5 and is a linear statistical model.

A linear model is not a time-dependent modeling method reproducing the data as seen in Figs. 4–7. The actually modeled data is represented by a single point in time per cell measurement. For the presented models this time is defined as the time needed for the cell to reach the end-of-life criterion for the modeled target value (see also Fig. 9).

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Each of the two target values, capacity decrease and impedance increase, will be treated in a separate model. Additionally separate models per chemistry were fitted. So a total of four parameter sets are presented in Table II (capacity decrease) and Table III (impedance increase).

The tables show three columns of values for each of the parameter sets. The values in the first columns are the estimates for the parameters of the model, received by least-square fitting. These estimates and the influence factor combinations for each term (found in the leftmost index column) form the model formula. For an increased accordance of the capacity decrease to a linear model (Fig. 9), the target variable t_CD was transformed to a logarithmic scale. This step was not necessary for the target variable of the impedance increase t_II. As examples, the final model formulations of the capacity decrease and impedance increase models for LFP from Tables II & III are presented in 7 and 8.

The center column displays the quadratic standard error for each estimated value coming from the least squares estimation. This value helps in deciding if a model term and the estimate are reliable and useful parameters or not, which is then displayed in the third column. Significance is given by a star rating, as common in many statistics tools such as R, more stars mean better reliability of the estimation (for more information see the R online help manual³⁴), the formula for calculating significance is displayed in 9. The hypothesis, which the p-test is testing against, is that the estimate is actually zero. If true, the term is insignificant for the data description model.

A rating of three stars means that the term is highly significant and should not be left unconsidered. Two and one star are a little less significant. A point means the user should decide based on his knowledge of the subject, if the term should be considered.

Not all the influence factor combinations mentioned possible in Table I can actually be found in Tables II and III, because the models have already been reduced by all insignificant combinations. This process starts with the graphical interpretation of a factor plot as seen in Fig. 10. It exemplarily displays one model target parameter (t_CD) for the LFP cells in dependence of a single influence factor (x-axis) averaged over all combinations with the remaining three factors. This plot tells which of the influence factor terms will very likely be significant and need to be taken into account for the model. In the upper right hand plot of Fig. 10 the maximum at 10°C visualizes the need for a center value different from zero for the quadratic temperature influence term for LFP, since this term would otherwise always be centered at zero. Alike, plots exist for the first order interactions, as shown in Fig. 11 for the data of the LFP cells. It tells which interactions are to be retained in the model for t_CD of LFP. In the rightmost graph a weak interaction of SoC and dSoC is indicated, which showed to be insignificant during model reduction. It is a repeated process of taking a model, starting out with all possibilities of Table I, and reducing the insignificant ones, then refitting the model. This process is repeated three times per target value and chemistry, always reducing the model by all parameters which had no or very low significance (< 10% in the p-test). The improvement in model quality during the process is monitored in the four model graphs (Residual vs. Scaled, QQ-Normal, Scale Location and Leverage, see ²⁸) given by the plot model command in R and the decreasing AIC value (Akaike Information Criterion).^31,35 These steps are not explained in detail here, but can be found in the R help files.³⁴ The models stayed very much the same across the chemistries during the first repetition, but differences arose in the following iterations. All models acquired good model quality during the fitting process.

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When reviewing the aging measurements of the capacity measurements it is notable that the cells do have some variance in their response to the influence factor combinations in a lot of the sample points. Since all cells were hand-made and the cells are only covered by a single layer of pouch bag foil, this was expectable to happen (Fig. 1). A countermeasure taken was to steady the active material parts by plastic clamps before the start of the experiment. As a second measure, a larger number of cells than required by the DoE plan was tested in some sample points to characterize the real-cell variance well. These sample points are enlarged in Fig. 12 for LFP (left) and NCA (right). The sample point selected was 30°C, 1C, 50% SoC and 80% dSoC for reasons of space for additional cells and channels being available in the 30°C test chamber and the electric load being a medium test point except for dSoC. The resulting variance in this test point is rather large, but it is also visible that most of the cells had the same characteristic path in their capacity decay. Among cells of the same chemistry the path is very common, but not shared by the other chemistry. LFP containing cells seem to age in a double dip behavior with an elongated plateau in between, while NCA cells experience a capacity rise in the first stage of their life and decrease their capacity continually down without plateaus later on. For both chemistries the different stages of aging (Fig. 13) are characteristic. Both cell types use the same anode, which indicates that at least the difference in aging behavior within LFP stage I & II versus NCA stage I can be attributed to cathode causes. The length in time of single stages is dependent on the influence factors, but the strength of influence factors is different between the stages in each sample point.

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Additionally having a build variance, some of the cells showed electrolyte leaks at the electrode bushings. This was an effect which was not initially expected and no specific countermeasures were taken. During model fitting it showed that there was no significant alteration of the aging path for both chemistries due to these leaks presumably due to an electrolyte excess, though it did accelerate the capacity decrease of heavily affected cells. The data of those cells was taken out of the data base before the statistical analysis.

The aging measurements of the impedance were expected to show visibly more variance than capacity measurements due to the measurement accuracy of just less than 15%. This expectation cannot be directly supported by the data presented in Figs. 6 & 7, because the measurements show a more articulate dependence on the actual temperature than on the aging influence factors or the measurement accuracy. The expectation of a characteristic shape of impedance increase has therefore to be reduced to a linear increase, fitting very well to the linear modeling. Still this linear increase has a slope depending significantly on all influence factors, not only temperature. For future experiments all reference tests should be at reference temperature to get less disturbed results. The found impedance increase ranges between factor 2 and 4,5 during the lifetime of the cell. At low temperatures there exist values exceeding the EoLc (factor 3) which was set for the impedance, but only measurements at reference temperature (30°C) were taken for the activation of EoLc.

The results from the impedance increase measurements are especially interesting for the design and sizing of cooling systems and power inverters. It implies that a system able to cope with all temperature levels of a new cell, can handle an aged cell with very little additional capacities needed.

Single factor variation experiments are rather easy to analyze, since one can tell the difference in cell aging by looking at the graphs of the different sample points. In this study a total of 46 sample points are tested and the factor level combinations are not linearly varied, so one has a hard time to see differences in sample points intuitively. Hence, a discussion of the model parameters of the presented statistical models is needed. In these parameter sets one can see the influence strength, the differences between sample points and chemistries in a more compact way. Fig. 14 and Fig. 15 show the capacity decrease model and the impedance increase model respectively in a telling way. These plots show within the top right triangle the LFP model and in the lower left part the NCA model. In both halves of the plot are box graphs bounded by the limits of the design space as introduced in Fig. 2. The axes of the design spaces are given in the main diagonal, the labeled contour lines within the design space boxes are the model responses (the survival time until EoLc). So a higher contour label encircles an area of an exponentially longer survival area. Each box graph gives the variation in survival deviating from the standard point of 30°C, a current of 1C and SoC as well as dSoC equal 0.5. If not varied for the graph, these values rest constant.

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As for the influence parameters in both models it is remarkable – although in different powers and with different center values – how basic influence factor combinations are reduced per model and chemistry to a maximum of three out of four parameters tested. A stronger reduction is present in the interactions, which were reduced from 6 possible to about 3 significant interactions per chemistry only.

Temperature

There were five measurement settings in temperature for LFP cells, covering a range of −15°C to 70°C, and only three for NCA cells, ranging from 10°C to 50°C. The measurements suggest a maximum survival time at around 10°C to 15°C (Fig. 10), which was modeled by offsetting the temperature influence to +15°C. The effects leading to the maximum in survival time are suggested by literature^8,36 also, underlined by the fact that the cells, which operated at low temperatures, showed lithium deposition on the anode upon visual inspection after experiment ending. The strength of the aging influence due to temperature as postulated from the measurements is schematically depicted in Fig. 16. The finding is that a cell operation temperature of around 10°C to 15°C would keep capacity decrease and the impedance increase low, although this temperature shows a higher power derating due to increased impedance compared to standard 30°C operation. So the compromise between life and operation conditions would be an operation in the window of 15°C to 20°C.

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Current

dSoC

All four models favor smaller dSoCs for the reduction of aging effects. The lowest tested dSoC was equal to zero, meaning the cell was under calendar aging only, which also was the most favorable point.

The dSoC was found to especially enhance NCA aging when being high. SEM pictures, taken after disassembling aged cells, point toward a mechanism of particle destruction (Fig. 17).

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SoC

Interaction of SoC and dSoC

Interaction of current and dSoC

Interaction of current and SoC

Interaction of temperature and current

The model not only provides a good overview of the experimental results, but also has a real value for someone trying to calculate the survival time a cell under a load profile. Any of the applications mentioned in the introduction demand loads which fit well within the test space and can therefore be safely evaluated with this model. Some current publications^39,40 use models which only rely on single data sets of dSoC variation, which could profit very much by applying models like the one presented here.

When an expected profile is at hand, calculation of the average values for the four influence factors is needed. Taking those as input of the model it will return a survival time for the cells used in this experiment. Since the model can very quickly be evaluated for many different load profiles versus battery pack combinations, this can give a very good hint on how to tune the load or size of the pack in order to reach the desired lifetime. The model presented holds true for the cells tested and needs reparameterization every time a new cell is regarded. General trends on cell aging depending on the cell chemistry can, however, be derived.