A Phenomenological Model of Bulk Force in a Li-Ion Battery Pack and Its Application to State of Charge Estimation

Prismatic Lithium-Nickel-Manganese-Cobalt-Oxide (NMC) batteries encased in a hard aluminum shell were used in this study. Each cell has a nominal capacity of 5Ah and has outside dimensions of 120  ×  85  ×  12.7 mm. Inside the aluminum case is a horizontal flat-wound jelly roll that has 90  ×  77.5  ×  11.7 mm. In this orientation, with the tabs facing upwards as shown in Fig. 2, the bulk of the electrode expansion results in outward displacement of the sides of the aluminum case when unconstrained.¹⁶ The packaging of the jelly roll makes the electrode expand primarily perpendicular to the largest face of the battery due to the wound structure and gaps between the electrode and casing around the top and bottom. This observed cell expansion is, consequently, exerted against the end-plates when the battery pack is assembled as shown in Fig. 1. The space between the batteries is maintained via a plastic spacer with dimples to preserve the airflow channels and still provide a means for compressing the batteries.

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To study the force dynamics in a battery pack, a fixture as shown in Fig. 3 was designed and used in all experiments reported in this article. The fixture consists of three NMC batteries connected in series sandwiched between two 1-inch thick Garolite plates. The flat Garolite surfaces allow for parallel placement and compression of the batteries inside the fixture, and are bolted together using four bolts, one in each corner of the plate; each bolt is instrumented with an Omega LC8150-250-100 sensor to measure the bulk force. The sensor is a 350 ohm strain gauge type load cell with a 450 N full scale range and 2 N accuracy. This arrangement of the fixture preserves a constant compressive distance between the two end Garolite plates which replicates the conditions experienced by cells in the pack.

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The fixture is placed in a Cincinnati Sub-Zero ZPHS16-3.5-SCT/AC environmental chamber to control the ambient/fixture temperature. Current excitation is provided by means of a Bitrode model FTV, and the resulting force and temperature data is acquired via a National Instruments NI SCXI-1520 strain gauge input module and 18-bit data acquisition card. The temperature, current, voltage and force data are sampled at a 1 Hz rate.

Using the described fixture, and performing experiments (E1) and (E2) detailed in Appendix A, this section makes key observations that form the basis for the model. The two experiments, whose outcomes are studied in this section serve different roles. Experiment (E1) is designed to highlight the influence of a change in ambient temperature and the impact that this change has on the measured force. Along the way, the influence of intercalation is also noted by incrementally changing the SOC and allowing force measurements to reach equilibrium. By choosing to draw currents that are small in magnitude, the heat generated in the cell is kept small and hence the cell temperature is almost identical to that of the ambient/ fixture. The second experiment (E2) is designed to investigate the influence of increasing cell temperatures while the temperature of the ambient and hence the fixture are held constant.

Figure 4(a) presents the experimentally observed relation among Force, SOC and temperature, derived from the experiment (E1). At room temperature, the shape of the measured Force-SOC relation bears a striking similarity to the linear surface displacement of a similar unconstrained prismatic cell in Ref. 16 and interlayer spacing in the graphite anode.²⁹ This observation can be explained by noting that in Li-ion cells with graphitic anodes, the volume expansion of the graphite is typically much larger than that of the cathode (Table I) and that the stress-intercalation state of the negative electrode exhibits a similar shape.²¹

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In Ref. 3, the authors assert that the stress-SOC relation is to be expected to be independent of temperature and current. However, contrary to this claim, Fig. 4(a) shows a significant dependence of measured force on the temperature of the pack-fixture. Further, it can also be inferred that a change in temperature does not involve a simple linear superposition of the expected material expansion (contraction) due to temperature increase (decrease). This nonlinear effect can be seen by observing that for the same change in temperature, the increase in measured force at different SOCs is not the same. Figure 4(b) presents approximate curves of the rate change of measured force per-bolt with respect to change in SOC (dF/dz). From this figure, it is noted that as the temperature of the cell decreases, the dF/dz curve starts to get flatter and that the peaks in the curve constantly migrate to the left. This characteristic can perhaps be captured by a temperature dependent shift and scaling function.

Experimental data obtained from (E2) are plotted in Fig. 5 wherein the first two subplots are of primary interest. In this experiment, a 50 A, 1 Hz current was applied to the cell to effect a temperature increase in the cell through Joule heating; the gray section in subplot four indicates the period of pulse excitation. Notice that during this period, the temperature of the battery increases by ∼ 13 °C and the resulting per-bolt force is close to 400 N at 50% SOC. The value of force as the battery reaches thermal equilibrium is close to the force that is measured when the cell is fully charged (100% SOC) at 25 deg C. This observation further affirms our belief that thermal effects should not be ignored when modeling the bulk mechanical stress/ force in prismatic battery packs.

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Although, the exact physical origin of this nonlinear temperature effect is not understood, we will attempt in this paper to lump it to a temperature dependent effective Young's modulus. Hence we assume that the modulus of the cell changes with temperature and attempt to find a physical relationship that describes the shift and scaling that temperature variability imposes to the measured SOC-only dependent force at one reference temperature.

The fixture presented schematically in Fig. 3, is modeled equivalently as a bar in series with a spring as shown in Fig. 6; the bar and spring are representative of the lumped collective of cells and spacers in the fixture. In this equivalent mechanical model, the bar of length L is attached to a spring with spring constant k_sp, and both are constrained and held in place by plates H apart (inner distance). The restorative force in the spring is the same as the sum of force measured by the load sensors and is computed by solving the force balance equation

where ΔL and ΔH are the changes in length of the bar and the fixture respectively. The stress σ is related to the force f by the surface area A of the battery which is under compression.

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In deriving the mathematical relations in remainder of this section, the following assumptions pertaining to the thermal characteristics of the elements in the representative structure of Fig. 6 are made.

• It is a common observation that prismatic battery packs are held together with some minimal pre-stress. This initial pre-loading ensures that cells do not shift and also improve the conductivity of materials inside the cell. We assume that the pack which is being modeled is under some pre-stress.
• The bolts that hold the fixture together, much like the bar, are assumed to undergo thermal expansion. That is, suppose the temperature of the ambient changes by ΔT_F ( = T_F − T_ref), then the bolts increase in length by ΔH. That is,

  

  where α_F is the coefficient of thermal expansion.
• The dependance of the Young's modulus of the bar on its temperature is assumed to be locally approximated by a quadratic function. That is,

  
• The length of the bar, aside from being impacted by the battery's temperature, is also affected by the SOC of the cell. The SOC dependent strain experienced by the bar is denoted as .

From the above assumptions, the change in length of the bar resulting from a ΔT_b change in battery temperature and/ or SOC can be computed according to Eq. 4.

Suppose that, when the fixture and the cells in the pack are in thermal equilibrium with the ambient whose temperature is equal to T_ref, the bulk force is measured over the entire rage of SOC. Denote by , the function that relates SOC to measured force at T°_refC. That is,

From the above, and by representing the longitudinal cross-sectional area of the cell by A, the restorative force in the spring at any temperature of fixture and battery can be computed as follows.

Substituting for from Eqs. 5 into Eq. 6 and solving for f(ΔT, z) = σA, the measured force at any value of fixture and battery temperature, and at any SOC is expressed as

where ΔT = [ΔT_b, ΔT_F]'. Parameters of the static model in Eq. 7 are estimated using experimentally collected data using the methodology described Parameterizing the static model in Appendix B and their estimates are tabulated in Table II. In deriving the estimates, the reference temperature, T_ref, is assumed to take the value 25°C.

The first three parameters in Table II correspond to physical dimensions which have been measured. The thermal expansion coefficient for steel is assumed for α_F corresponding to the bolts. The estimated thermal expansion of the battery α_L is reasonable for the given materials comparing to the values provided in Ref. 22. The estimated value for the spring constant k_sp corresponds to the plastic separator, given the contact area and width of the separators this corresponds to an effective modulus of E_sp = k_sp*L_sp/A_sp = 0.75 GPa, which is also reasonable for injection molded plastic.

Remark1. In the above derivation, the function deserves special attention. The functional form of is empirically derived and is valid only for conditions identical to that when the function was initially parameterized. Particularly, if the pre-stress of the pack changes, be it owing to a change in the fixture or irreversible deformation of the batteries themselves, the function may not be valid as originally parameterized.

In the previous subsection, a static model accounting for the influence of temperature (both ambient and fixture) and intercalation on the measured bulk force was modeled; however, the static model does not entirely capture the response of the actual system. As an example consider the measurements derived from experiment (E1); in particular, force measurements from the end of the Constant Current (CC) phase of every CCCV step (Step (S1.5) of experiment (E1) Parameterizing the static model in Appendix B). At the end of the CC phase, the SOC of the cell is typically ∼ 99% and does not change considerably as the cell is trickle charged. Figures 7(a) and 7(b) sketch the trajectories of the difference between measured force and the static model immediately following the end of the CC phase of the charging operation at different temperatures and for different initial conditions respectively.

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There are two principle observations to made by inspecting Fig. 7(a); the first observation is that the difference appears to exhibit some dynamics as the current is reduced to zero and during the ensuing relaxation period. One could contend that this observation is related to the internal temperature of the cell lagging behind the surface and undergoing some dynamics. Spatial distribution of temperature (along the thickness) inside the cell would result in a volume averaged expansion that is different from that computed using the parameters of the static model using the measured surface temperature, and could account for the difference presented in Fig. 7(a).

To test the above theory, the thermal model of the battery cell used in this study, presented in Ref. 1, was used to simulate the temperature in a prismatic cell soaked at different temperatures, in response to a 1C constant charge starting from ∼ 0% SOC. Fig. 8 presents the trajectory of the difference in temperature between the hottest and coolest spots in the battery as it was charged and allowed to rest, and Table III tabulates the max norm of the difference. The maximum difference in temperature is computed to not exceed 0.25°C, this difference is found to be unable to account for the difference in force presented in Fig. 7(a).

Table III. L_∞ norm of the temperature differential and the change in static force prediction for this difference in temperature.

Tamb (°C)	(°C)	Δf (N)
−5	0.24	6.1
10	0.127	3.8
25	0.08	0.5

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Figure 7(a) also provides reason to believe that the dynamics during relaxation is dependent on the temperature of the cell. As the cell's temperature increases, the effective error decreases while the time constant for the relaxation appears to not change noticeably.

To ascertain if the value of the difference between the measured and the static model's prediction is dependent on the initial SOC at which the charging operation (CCCV protocol) was initiated, Figure 7(b) traces two separate curves. The dashed curve shows the evolution of force for the case when the charging operation was initiated with the cell close to 5% SOC; the dash-dotted curve presents the case when the initial SOC was 50%. Since in both cases, the battery was charged at the same rate, 1 C, the time to reach the end of CC phase for the first case is almost twice that of the second. From Fig. 7(b), by comparing the values of the peaks of the curves, it is immediately apparent that the duration of charge appears to have an impact on the maximum value of the difference, suggesting that the state that affects bulk force, might have a rather large time constant.

Based on the above observations and inspired by the dynamics of dashpot, the dynamic equations of δf is written as

In a bid to reduce the model complexity and inspired by the current rate dependency of measured expansion results presented in Ref. 16, as a first attempt, we employ the following first order dynamic equation to describe the difference between the measured data and the static model.

where δf is the deviation from the static bulk force, I is the battery current; ζ₁ and ζ₂ are tunable parameters and assumed to be temperature dependent. These parameters are estimated by solving a least square optimization problem described in Appendix B and tabulated in Table IV. The identified parameters appear to support our observation derived deductions about the dynamics of the difference; the time constant of the additive state is about 10000s and the input gain is temperature dependent. The values of ζ₁ appears to not be significantly influenced by temperature and hence could be made a constant.

The final form of the bulk force model, with an aim of describing the influence of intercalation, temperature and current on the measured force, is expressed in terms of nominal force-SOC characteristic, and a temperature dependent Young's modulus, E(T_b), as follows

where T_F(t), T_b(t) are the trajectories of ambient and cell temperatures respectively; Q is the capacity of the cell in Ampere hours (Ah) and I is the battery current in Amperes; z is the state of charge (SOC).

Using parameter estimates computed Model parameterization in Appendix B, Figures 9 and 10 compare the measured bulk force in experiments (E1) and (E2) Experiments A to the output of the model in Eq. 10. From these figures, it can be observed that the parameterized model is able to reasonably describe the steady state force at different SOCs and temperatures. The overall shape of the cell response and the model output is the same; however, when using just the static component of the model, there are significant errors during charging and relaxation periods. Simulated bulk force of the combined static and dynamic models over the entire data-set are compared to measurements in Fig. 10. As evidenced by the second subplot, the inclusion of the dynamic model, on an average, helps reduce the error across the entire trajectory. In particular, root mean square (rms) error of model output is decreased by 36.3%.

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In this section, a phenomenological dynamic model for the relation among bulk Force, SOC, Temperature and current was developed. The developed model uses measured data at a reference temperature (T_ref), taken to equal 25 °C, to describe the evolution of force as the current, and battery and ambient temperature of the fixture change. The next section studies the simple case when temperature effects are not significant in using the developed model to check the feasibility of estimating the SOC of Li-ion cells, specifically for the chemistry used in this study.

The aim of this paper is to study the influence of temperature and intercalation on the measured bulk mechanical stress. To that end, the battery pack is subject to the following experiment, henceforth referred to as (E1), involving changes in ambient temperature and SOC of the pack. Figure 12 depicts the profile of current and ambient temperature in (E1). At every temperature ∈ { − 5, 10, 25, 45}°C, the sequence of steps followed is listed below.

• (S1.1) The cell is first charged using a constant current, 1C-rate¹ (5A) constant voltage (4.2 V) charging profile using a CCCV protocol, until the current reaches C/20 (0.25 A) at a fixed ambient temperature of 25°C as regulated by the thermal chamber. Follow charge by a three hour rest.
• (S1.2) Pulse discharge using 0.5C current pulse to reduce the SOC and rest for two hours.^⋆
• (S1.3) Repeat (S1.2) until SOC reaches five percent.
• (S1.4) Rest for two hours.
• (S1.5) Charge to 100% using the CCCV protocol with 1C constant current and C/20 cutoff, and rest for three hours.
• (S1.6) Discharge at 0.5C-rate until SOC reaches 50% and rest for two hours
• (S1.7) Set ambient temperature to 25°C.^▲
• (S1.8) Charge to 100% using 1C current and rest for three hours.
• (S1.9) Discharge to 50% SOC using 0.5C current.
• (S1.10) Rest for two hours and repeat (S1.1) at a new temperature.

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Notes 1.

• ^⋆The State of Charge (SOC) of the cell is computed by coulomb counting. The pulses applied, as the pack is discharged until 5% SOC, are not identical. The SOCs at the end of the various pulses are, in order of occurrence

  

  This discharge profile, pulsing and resting, is the same at every temperature.
• ^▲As a protocol, we have chosen to change the temperature of the pack only at 50% SOC. In addition, before every change in ambient temperature, an intermediary step wherein the ambient is raised/ lowered to 25°C is performed. This decision is motivated by a need to ensure that the prestress in the pack and or the sensors does not drift; changing prestress influences the dynamics and drifting sensors render the experimental data unreliable. Further, this step could also reduce the impact of any path dependence of measured force on the trajectory of temperature and SOC.
• ^▼Note that in the designed experiment, each change in temperature and SOC is followed by a relaxation period to allow for equilibration of terminal voltage, cell temperature and force measurements. Steps (S1.8) and (S1.9) are performed to reduce any ambient temperature dependent stress accumulation.

Based on measurements derived from the experiment (EI) described above, we create two different data-sets:

• (E1.1) A quasi-steady-state map of the force measured at various SOCs and at various temperatures (Steps (S1.2) and (S1.3)).
• (E1.2) Transient data in which the measured force undergoes some dynamics, either owing to a change in SOC or temperature.

Experiment 2 (E2)

The second experiment performed on the fixture, attempts to isolate the influence of a persistent temperature differential between the ambient and the battery on the measurements of bulk force. To create this difference in temperature, the fixture is allowed to soak at room temperature inside a temperature controlled chamber until measurements of force and temperature reach equilibrium. Thereafter, the test profile can be broken into the following

• (S2.1) The pack is charged until the SOC reaches 100% using a CCCV protocol – the constant current employed is at 1C and the discharge cutoff is C/20 A.
• (S2.2) Following a rest of 1.5 hrs, the pack is discharged using a 1C current until the pack's SOC reaches 50%.
• (S2.3) Upon resting for a further 1.5 hours, the cell is excited with bi-directional rectangular pulse current for a duration of 2.5 hours. The amplitude of the current is 50 A and its frequency is 1 Hz.

Figure 5 depicts the trajectory of total force, temperatures and SOC that are a consequence of applying the above steps pertaining to experiment (E2).

Parameterizing the static model

This section describes the approach adopted in parameterizing the model. the parameters of that we seek to estimate can br expressed as Λ = [α_L, k_sp, k₁, k₂, k₃]'.

The force at any SOC and any value of fixture and battery temperature was derived in Eq. 7 and is reproduced, for convenience, below.

For convenience, let the modulus (E_b) in the above static model can be re-written as

By substituting the expression for temperature dependent modulus E(T_b), and by defining the following terms,

the bulk force is expressed as

Using the above equation, model parameterization is performed in two stages. In the first stage, the values of γ_i are estimated using data from experiment (E1.1); these estimates are in-turn used in the second phase to identify the individual parameters in Λ by utilizing the fact that α_F corresponds to the thermal expansion coefficient of steel. In the ensuing discussion, the following notations are adopted — vectors are denoted using capitalized variables (X); individual elements of vectors are denoted using small letters and are indexed in their subscript (x_i); and matrices are denoted in block capitals (A).

The four unknowns in Eq. B-4 do not all appear linearly and hence a nonlinear optimization problem is to be solved to estimate their values. Using data-set (E1.1), we solve the following global optimization problem to arrive at estimates of γ_{1, ..., 4}

where the measured force data, stacked into a vector, is denoted by Y; Z is the vector of SOCs, ΔT is a vector of battery temperatures along the trajectory, stacked; and g( · ) is as defined in Eq. B-4. The constraint in Eq. B-5 is motivated by physics that the modulus of the material be positive. The estimation is performed by repeated application of Genetic Algorithm (GA) and by refining the solution of GA using a local nonlinear optimizer based on Sequential Quadratic Programming (SQP).⁷

The second phase of the parameterization procedure is aimed at distinguishing between the values of the two terms that constitute γ₁; specifically, to identify θ₁ and θ₂. This is performed by minimizing the Least Squares minimization of Eq. B-6 using the data collected in experiment (E2).

where Θ = [θ₁, θ₂]'. The measurements of force along the trajectory are stacked into a vector F, as are the deviations of fixture and battery temperatures about the reference T_ref, which are themselves denoted by ΔT_F and ΔT_b.

Using the solution to the problem in Eq. B-6 and relations in Eqs. B-3, the individual values of the parameters in the model are computed; these values are tabulated in Table II.

Computing variances of parameter estimates

For a general model of the form

where ε is an additive noise assumed to be Normally distributed, the estimate of error variance is given by

The Cramer-Rao bound of the least estimator is given by

The variance of θ_i in Θ, assuming the estimator is unbiased, is then the a_{i, i} entry of the CRB matrix.

For the model under study, using the identified parameters, the CRB matrix is computed to equal

Then the variances of γ₁, γ₂, γ₃ and γ₄ are given by 5.56 · 10^{− 1}, 1.65 · 10^{− 6}, 1.07 · 10^{− 9} and 6.58 · 10^{− 3} respectively.

Remark 2. There are interesting observations to be made concerning the problem of estimating Λ using data collected in experiments (E1) and (E2). Using information from experiment (E1.1), none of the parameters in Λ can be individually identified; instead, variables γ_{1, ..., 4}, combinations of parameters in Λ are estimated; γ_{1, ..., 4} are random variables with a mean and variance as computed above. Using γ_{1, ..., 4} to estimate θ₁ and θ₂ from the problem in Eq. B-6 is plagued by a potential issue — the variables C and d in Eq. B-6 are random variables (them being functions of random variables γ_{1, ..., 4}). While it is desirable that a generalized least squares technique such as Total Least Squares (TLS)¹⁰ be applied, the fact that computing the variance of C is a nontrivial problem, has driven us to employ the Normal Least Squares (NLS) method in this paper. Having employed NLS, variances of parameters Λ, do not convey any meaningful information on the uncertainty in estimates and have been omitted.

Parameterizing the Dynamic Model

The dynamic model of δf is, for convenience, re-stated below.

where δf is the deviation from the static bulk force, I is the battery current; ζ₁ and ζ₂ are tunable parameters and assumed to be temperature dependent.

Parameters of the dynamic model, ϑ = [ζ₁, ζ₂]', are identified by minimizing the Euclidean norm of the difference between measurement and model prediction, the resulting parameters are ϑ*. That is, we solve the following optimization problem

where N_f is the number of measurement points, t_s is the duration of the sampling period and z_i is the SOC at the data point. The first and the second equality constraints denote the dynamics and initial condition, respectively. In particular, the first equality constraint is constructed by applying the forward Euler to Eq. B-11. The model is trained using data-set (E1.2) at various ambient temperatures. The minimization problem is solved by using Sequential Quadratic Programming (SQP) and the identified parameters are provided in Table IV.