Real-Time Estimation of Lithium-Ion Concentration in Both Electrodes of a Lithium-Ion Battery Cell Utilizing Electrochemical–Thermal Coupling

Different kinds of modeling strategies have been discussed in the literature for Li-ion batteries: (1) data-driven models [1,2], (2) equivalent circuit models [3,4], and (3) electrochemical models. These models are generally used for estimating state-of-charge (SOC) and state-of-health (SOH) of the batteries, which are crucial for real-time monitoring and control. Among these models, the electrochemical models arguably possess the most accurate and physically meaningful information [5]. The pseudo two-dimensional (P2D) model, which involves nonlinear partial differential equations (PDEs), is one of the benchmark models for Li-ion battery cells [6]. However, the P2D model suffers from two disadvantages: (1) too complex mathematical structure for estimator design and (2) high-computational burden for real-time implementation. As a work around, several researchers have proposed reduced-models with less computational and structural complexity [7,8]. The single particle model (SPM) is one of such widely used reduced-models for real-time applications [9,10].

In the SPM, the electrodes are approximated as spherical particles, assuming volume-averaged current rather than distributed current in both electrodes, and also neglecting the charge and electrolyte dynamics. Several SOC estimation approaches based on SPMs have been proposed such as back-stepping PDE estimator [11], extended Kalman filter (EKF) [9,10], and nonlinear observer-based approaches presented by the authors of the present paper [12,13]. Another set of work resorted to adaptive estimation to simultaneously estimate the states and parameters of the model using: particle filter (PF) [14], UKF [15], iterative EKF (IEKF) [16], adaptive PDE observer [17], and nonlinear geometric observer [18]. The authors of the present paper also contributed to this segment by proposing sliding-mode observer [19] and nonlinear adaptive observer [20,21].

From estimation theory viewpoint, the two-electrode SPM suffers a drawback where the Li-ion concentration states are weakly observable from the differential voltage measurement [10]. One of the main reasons for that is the Li-ion concentration states of the electrodes are coupled in the voltage output through their thermodynamic potentials, whereas their dynamics are decoupled. To resolve this observability issue, the existing estimation schemes approximate one electrode's concentration state as a function of the concentration state of other electrode [10–12]. This approximation results in a one-electrode SPM (either anode or cathode) with strong observability of the corresponding states from the voltage measurement. However, this aforementioned algebraic function, and hence the approximation, comes from the underlying assumption of the conservation of the total number of Li-ions in the cell. This assumption may not be true as there are losses of Li-ions due to aging and other phenomena over persistent long-term operation.

In light of the above discussion, the main contribution of this work is: we relaxed the aforementioned assumption (conservation of the total number of Li-ions in the cell) and designed an estimation scheme based on the two-electrode SPM. Essentially, the main difference between previously proposed battery state estimation schemes [9–21] and the current work is that previous schemes estimate the overall cell SOC based on the Li-ion conservation assumption, whereas the current work relaxes such assumption and estimates the SOC (Li-ion concentration) of individual electrodes. Estimates of individual electrode SOC would enable additional crucial benefits/features such as health monitoring of individual electrodes, capacity tracking of the cell, and detection of mechanisms leading to loss of active lithium, e.g., Li plating.

In doing so, we first showed that the observability of both electrode concentration states can be improved by exploiting the electrochemical–thermal coupling. This arises from the fact that the electrochemical parameters of the cell and the thermodynamic potential of the electrodes depend on the temperature. On the other hand, electrode surface concentrations contribute to the heat generation in the thermal model. In the estimation scheme, we used a cascaded two-observer structure that exploits this coupling. Observer I is a sliding-mode observer which is used to estimate the heat-generation term contributed by the electrode concentration states. Then, this estimate of the heat-generation term is fed to observer II as a pseudomeasurement. Observer II, which is based on the two-electrode SPM, uses this pseudomeasurement and the differential voltage measurement to estimate the electrode concentration states. The design of observer II is based on an UKF framework that can account for the model nonlinearities and uncertainties.

A preliminary version of this work was presented in Ref. [22]. In the present paper, we extend our preliminary work [22] by including: (1) discussion and results of the application of the scheme to estimate initial capacity of the cell as well as the capacity degradation over time; (2) detection scheme for degradation mechanisms such as lithium plating in cells; (3) additional simulation results under varying operating conditions such as dynamic discharge profiles; and (4) discussions and simulation studies on the effect of the modeling uncertainties and measurement noises on the scheme. It is shown that the proposed estimation scheme is able to estimate Li-ion concentration states in both electrodes from measured cell voltage, temperature, and current. Further, it is shown how the estimation of individual electrode concentrations can be used to compute the real-time capacity of the cell and to help in detecting combinations of degradation mechanisms leading to loss of lithium in the cell.

The organization of the paper is as follows: First, we review the SPM and thermal model for Li-ion cell. Next, a detailed discussion of the system observability is provided. Then, the estimation scheme and its design are presented. This is followed by the application of the proposed scheme for two specific objectives: capacity tracking and lithium plating detection. Then, simulation results are provided for the performance of the scheme along with the effects of modeling, parametric, and measurement uncertainties. Lastly, the paper is concluded with some remarks.

The SPM is derived from the P2D model [6] by assuming volume-averaged current and neglecting electrolyte and charge dynamics. We shall use the schematic given in Fig. 1 and notations given in the Nomenclature section for the SPM equations.

where $Tref$ is the reference temperature at which $k0,ref±$⁠, $Ds,ref±$⁠, and $Rf,ref$ are evaluated.

Note: As noted in the existing literature, the predictive ability of the SPM degrades high charge–discharge rates, especially when the electrolyte dynamics have significant contributions. The extensions of the SPM with electrolyte dynamics and thermal models are also proposed in Refs. [24] and [25] which improve the predictive ability. However, the objective of this paper is to illustrate the observability and estimation concept using the conventional two-electrode SPM along with the averaged thermal model. Extension of our design based on such extended SPM models can be considered in future work. Furthermore, the model inaccuracy of the SPM is compensated to a certain extent by the use of unscented Kalman filter (UKF) algorithm, which will be detailed later.

Next, we approximate the PDEs in Eq. (1) to a set of ODEs (6) and (7), using a finite difference method where the spatial domain is discretized into (M + 1) nodes (see Fig. 1).

where $m=1,…,(M−1)$⁠, $[csP0,csP1,…csPM]$ and $[csN0,csN1,…csNM]$ are Li-ion concentration states, $ΔN=R−/M$⁠, $ΔP=R+/M$⁠, $aN=Ds−(T)/ΔN2$⁠, $aP=Ds+(T)/ΔP2$⁠, $bN=1/as−FΔNAL−$⁠, and $bP=1/as+FΔPAL+$⁠.

Note: In this work, a finite difference method is adopted to reduce the PDE model to ordinary differential equation model. However, several other methods exist that can be used for this model reduction in the proposed scheme [26].

where $xP=[csP1,…csPM]T∈RM$ and $xN=[csN1,…csNM]T∈RM$⁠, $u∈R$ is input current, $yV=V∈R$ is the output voltage, $xPM=csPM∈R$ and $xNM=csNM∈R$ are the surface concentration states, $AP∈RM×M$ and $AN∈RM×M$ are temperature-dependent tridiagonal matrices obtained from the state matrices in Eqs. (6) and (7), respectively, $0M∈RM×M$ is the zero matrix, $BP∈RM×1$ and $BN∈RM×1$ are the input matrices derived from Eqs. (6) and (7), respectively, $UP=U+:R2→R$ and $UN=U−:R2→R$ are the temperature-dependent open-circuit potential maps for the positive and negative electrodes, respectively, $Rf∈R$ is the temperature-dependent scalar film resistance, and $nP:R2→R$ and $nN:R2→R$ are the first two scalar over-potential terms in Eq. (8), respectively.

Around a given nominal operating temperature $T=T*$⁠, the observability of Eq. (10) is studied in Ref. [10] where the states are found to be weakly observable from the voltage output. This unobservability can also be explained by considering the system structure of Eq. (10). There is no coupling between individual electrode concentration states in Eq. (10), i.e., $AP$ and $AN$ have no common terms. However, in the nonlinear output voltage map $yV$⁠, the electrode surface concentrations $xPM$ and $xNM$ are coupled via their thermodynamic potential. This leads to the aforementioned unobservability. Two solutions have been presented in literature to improve on this concern before undertaking estimator design: (1) in Ref. [10], the surface concentration of the negative electrode is approximated as a function of the positive electrode surface concentration using the stoichiometry ratio and (2) in Ref. [11], the opposite approximation is done under the assumption of the conservation of the total number of Li-ions in the cell. However, the conservation assumption may not be true as discussed before. In this paper, we take a different approach where we improve the observability by adding the thermal model to the two-electrode SPM (10).

where $UPD=(∂U+/∂T):R→R$ and $UND=(∂U−/∂T):R→R$ are functions of $xPM$ and $xNM$⁠, respectively, and $yT=T∈R$ is the measured temperature. It can be seen from Eq. (11) that the thermal model possesses additional information about the electrode surface concentrations in the form of ${UPD(xPM)−UND(xNM)}$⁠. This is the crucial feature behind the improvement of the observability of the concentration state in both electrodes.

where $h=[yVyT]T$ represents the output voltage map, $x=[xPT, xNT]$ represents the state vector, and $u$ represents the input current. The observability of Eq. (11) depends on the functions $UP$,$UN$,$UPD$⁠, and $UND$⁠, which are specific to different Li-ion battery chemistries. Here, we consider the LiCoO₂–graphite chemistry for illustration. The functions $UP$,$UN$,$UPD$⁠, and $UND$ for this chemistry are shown in Fig. 2. The rank of the observability matrix $O$ is checked at different operating points with a four-node discretization for each electrode particle. Therefore, the dimension of the state vector is 7 (three concentration states for each electrode and one thermal state). It is noted that the observability matrix $O$ has rank 7 at all tested operating points (matlab 2013a, 32 bit), except for the operating points with $u*=0$⁠. Therefore, it can be concluded that the system (11) is locally observable for all of the operating points tested except for $u*=0$⁠.

Remark I. In case of $u*=0$⁠, thermal dynamics does not carry any information of the Li-ion concentration states. Hence, the system loses its observability. This is evident from the expression of the thermal dynamics in Eq. (11).

Remark II. Considering Fig. 2, it can be noted that the values of $UPD$ and $UND$ are much smaller than that of $UP$ and $UN$ and are generally in the mV range. However, $UPD$ and $UND$ enter the thermal dynamics equation (4) as $uTUPD$ and $uTUND$⁠, where $T$ is the temperature in K, and $u$ is the current in A. Considering the nominal temperature around 298 K and the nominal current around 10 A, $UPD$ and $UND$ are amplified by the factor of 2980 which makes them comparable to the amplitudes of $UP$ and $UN$⁠. Therefore, the measurements of voltage, current, and temperature should be sufficiently accurate in order to observe the states through these terms. This is one of the limitations of using this model for estimation.

Remark III. Another observation that can be made from Fig. 2 is that the $UND$ function becomes flat after $xNM/xNM−max$ crosses 0.5. This indicates the insensitivity of $UND$ to the negative electrode surface concentration state in the region $0.5xNM−max$ to $xNM−max$⁠. However, this does not lead to the unobservability of the negative electrode concentration as the other three functions, $UPD$,$UP$⁠, and $UN$⁠, are still sensitive to the corresponding states in that region. This makes the overall system still observable.

The estimation scheme is depicted in Fig. 3.

As can be seen from Fig. 3, the scheme has two observers running in cascade.

Observer I: This is designed based on the thermal model of the system. This observer estimates a pseudomeasurement (⁠$ypseudo$⁠) signal using the current and temperature measurement. This $ypseudo$ essentially contains the electrode concentration function that contributes to the heat-generation term.

Observer II: This is designed based on the two-electrode SPM. This observer estimates Li-ion concentration states in both electrodes using the voltage and the pseudomeasurement (⁠$ypseudo$⁠) signal.

will result in $V̇T<0$⁠. For design purposes, the following values can be selected:

$|u|max$⁠: This can be determined a priori based on the operating maximum input current.

$|T|max$⁠: This can be determined a priori based on the maximum possible temperature range for the battery operation.

$|UP(xPM,T)−UN(xNM,T)|max$⁠, $|UPD(xPM)−UND(xNM)|max$⁠: These values can be determined a priori based on the particular electrode chemistries.

where $q$ represents the process noise accounting for the unmodeled dynamics, modeling, and parametric uncertainties; $r¯$ represents the measurement inaccuracies and unmodeled output uncertainties. The corresponding process and measurement noise covariance matrices are $Q$ and $R$⁠. The covariance of the state estimation is $Px$⁠. The UKF implementation follows the steps given in Ref. [30].

Note that aging and other phenomenon of the cell will result in loss of cycle-able Li-ions over persistent long-term operations, and hence the value of the parameter $n̂Li$ will decrease with time.

In this part, we will illustrate lithium plating detection as another application of the estimated Li-ion concentrations in positive and negative electrodes. Lithium plating happens during charging the Li-ion cell and essentially results in deposition of metallic lithium on the graphite anode. Several factors such as electrolyte characteristics, ratio between negative and positive electrode capacities, and charging rates affect the negative electrode kinetics and Li-ion diffusion rates in such a way that lithium is deposited on the negative electrode instead of intercalating the negative electrode [31,32]. This mechanism may degrade the Li-ion cell performance and safety.

To detect lithium plating, we adopt the approach in Ref. [33] where the Li-ions transport rates in and out of the electrodes were used as indicators of the same. However, the approach given in Ref. [33] was based on two-electrode models with only output voltage, which suffers the previously mentioned unobservability. In this paper, we overcome this limitation by using our extended model and cascaded observers.

The plating phenomenon is detected when the extraction rate $EP$ exceeds the maximum possible extraction rate (⁠$EP−max$⁠) and $IN$ drops below the minimum possible insertion rate (⁠$IN−min$⁠). These limits can be obtained in the following way: Collect $EP$ and $IN$ data from the healthy battery either by: (1) Monte Carlo simulation studies on the battery model on varying different operating conditions, different levels of measurement noise, and different levels of modeling uncertainties or (2) experimental studies on a physical battery under different operating conditions [35]. Then, choose the maximum and minimum of $EP$ and $IN$ as the limits.

To validate the proposed estimation scheme, we perform simulation studies in this section. For simulation purpose, battery cell model parameters of a 6.8 A h LiCoO₂–graphite cell are taken from Ref. [36]. The plant-model used for this purpose is a two-electrode SPM with averaged thermal dynamics. However, to verify the convergence, observer II is initialized with incorrect initial conditions. As the temperature is measured, we initialize observer I with the correct initial condition. In the simulation scenarios, we start with verifying the performance with smaller measurement noise level. Then, gradually we inject higher levels of measurement noise and parametric uncertainties to verify the effectiveness of the scheme.

In this section, we present simulation scenarios where the measured variables from the plant-model are injected with zero mean Gaussian noise 1 mV, 1 mA, and 0.1 °C.

The estimation performance under a constant 1C current discharge is shown in Figs. 4–7. In Fig. 4, the output estimation performance is shown, which validates that the cascade of observers I and II is able to track the temperature and voltage with sufficient accuracy. Figures 5 and 6 show the reasonable performance in estimation of the surface and bulk SOC for both electrodes. Bulk SOC represents the volume-averaged combination of the Li-ion concentration state variables. In Fig. 7, the performance of observer I is shown in estimating the pseudomeasurement. Moreover, Fig. 7 also shows the estimation of capacity indicator $nLi$⁠, which is accurately estimated in a few minutes.

In the next study, we show the estimation performance under a dynamic discharge profile in Fig. 8. The current profile is generated from a scaled-down version of the urban dynamometer driving schedule velocity profile. Moreover, a bias component is added to avoid the zero-current limitation mentioned earlier. It is evident that the observer-based scheme estimates the bulk SOC and the capacity indicator with reasonable accuracy.

In this study, we compare the estimation performance of the scheme using the two-electrode formulation proposed in this work against a similar estimation scheme that only uses a single-electrode formulation using the assumption of conservation of total number of Li-ions in the cell. The second estimation scheme is an UKF designed based on only the positive electrode concentration dynamics and voltage output. In the simulation plant-model, we have injected a side reaction current component which leads to capacity fade with time. The result of the estimation performance is shown in Fig. 9. It is evident that the estimate from the scheme with conservation assumption diverges from the actual value with time, while the proposed scheme is able to track with reasonable accuracy.

Here, we illustrate the observer's performance in tracking the capacity fade of the battery cell under 1C discharge scenario. The capacity fade is simulated by adding a side reaction current component into the plant-model which is not known to the estimators. The performance of the capacity fade tracking is shown in Fig. 10.

In this section, we illustrate the observer's performance in lithium plating detection. The simulation scenario used for this case study is C/10 charging at ambient temperature −20 °C. To illustrate the lithium plating condition in simulation scenario, we have injected a steplike additive C/5 charging input current component to the positive electrode of the plant-model. This additive input current causes a sudden change in current density in the positive electrode, while the current density in the negative electrode is kept constant. This essentially represents a loss in active lithium-ions, as the Li-ion extraction rate from the positive electrode would be higher than the insertion rate in the negative electrode. In this case study, the positive electrode current density increases at t = 600 s. The bulk SOC estimation performance is shown in Fig. 11 where the observer-based scheme is able to track the bulk SOC even after the occurrence of the plating. Further, the plating detection signals $EP$ and $IN$ are shown in Fig. 12. It can be seen that $EP$ crosses the maximum limit after the plating has occurred, and hence the plating is successfully detected.

Note: The case study of lithium plating is used to illustrate the capability of the scheme for detecting any mechanism that leads to loss of lithium in the cell which in turn results in capacity loss. Loss of lithium can be interpreted as: the total amount of Li-ion leaving one electrode is not reaching the other electrode; some of them are lost due to undesirable reactions. Lithium plating is one of such scenario. Therefore, any such mechanism that results in loss of lithium can be detected by the estimation scheme. So, the reversible plating cannot be detected by the proposed scheme as it does not cause capacity loss. On the other hand, reversible plating which causes capacity loss will be detected by the scheme. Furthermore, the detection accuracy depends on the modeling and measurement uncertainties. The loss of lithium due to plating must be sufficiently high such that the lithium extraction rate goes beyond the predefined limit $EPmax$⁠.

Here, we study the robustness properties of the estimation scheme under different uncertainties. These studies are conducted under 1C discharge scenario.

It was mentioned in the design section that this scheme is sensitive to measurement noise. In this section, we study the effect of measurement noise via simulation by injecting different noise levels in the all three measurements (voltage, temperature, and current). It is found that the scheme generates reasonable estimates approximately up to 25 mV voltage measurement noise and 50 mA current measurement noise. Above those ranges, the steady-state estimation error goes significantly higher reducing the effectiveness of the scheme. Other than measurement noise, measurement bias is also considered. For voltage and current measurements, it is noted that the scheme performs satisfactorily up to 5 mV and 12 mA measurement bias. Beyond these values, the performance of the scheme degrades.

Another interesting point is noted in simulation studies regarding the temperature measurement noise. Above 0.5 °C noise level in the temperature measurement, the estimates tend to diverge after a certain point when the signal-to-noise ratio gets smaller. One such scenario is shown in Fig. 13. This is because of the reason that in high-temperature measurement noise, the sliding surface deviates significantly from $T̃=0$ and settles to $sT=T̃=η$⁠, where $η$ is the lumped effect of temperature measurement noise. This creates a significant bias in pseudomeasurement which the UKF is unable to handle. Similar effects have been noted under temperature measurement bias where the estimation performance degrades beyond 0.1 °C bias.

In this section, we study the estimation performance under parametric uncertainties. The plant-model observer is initialized with incorrect parameter values. We chose the following parameters for this study: $Ds,ref−$,$Ds,ref+$,$as−$,$as+$,$Rf,ref$,$h$⁠, and $Cp$⁠. The observer is initialized with 50% error in one of these parameters in each case. It is noted that the scheme performs well with reasonable accuracy in case of uncertainties in $Ds,ref−$⁠, $Ds,ref+$⁠, $as−$⁠, and $as+$⁠. The steady-state errors for these parametric uncertainties are shown in Table 1. Note that the capacities of individual electrodes can be written as: $Q±=ε±AL±Fcs,max±|y100%−y0%|$⁠, where $y0%$ and $y100%$ are the stoichiometry points of the individual electrode [37]. Further, the definition $as±=3ε±/R±$ (see after Eq. (1)) indicates that 50% uncertainty injection in $as±$ is essentially equivalent to 50% uncertainty injection in $ε±$ which in turn injects uncertainty in the individual electrode capacities $Q±$⁠. Therefore, it can be concluded that the scheme performs reasonably under the deviation of individual electrode capacities due aging.

However, the scheme is found to be sensitive to the parametric variations in $Rf,ref$,$h$⁠, and $Cp$⁠. Estimation error for the bulk SOC is shown in Fig. 14. It can be seen that the negative electrode bulk SOC estimation is degraded in case of any of these parameter variations. However, for positive electrode case, it is still reasonable in case of uncertainties in $Rf,ref$ and $h$ but degrades significantly in case of $Cp$ uncertainty. The reason behind this is that in the voltage measurement expression, the contribution of the positive electrode concentration is significantly higher than that of the negative electrode concentration. Therefore, any smaller uncertainties in the voltage expression such as deviation in $Rf,ref$ make the negative electrode estimation poor. However, the uncertainty in $Cp$ affects the pseudomeasurement estimation significantly and therefore affects estimation of the concentration in both electrodes.

In the final simulation, we show the performance of the scheme under zero current. As discussed and showed in the observability analysis, the model loses observability under zero input current due to the absence of the heat-generation term. It is confirmed in the result shown in Fig. 15 where the input current is made to zero at 1000 s and the estimation errors diverge to a different nonzero value. So, it is important to engage the proposed estimation scheme under some nonzero current.

Note: The limitations of this approach are: (1) addition of the temperature measurement for each cell increases the hardware cost to some extent and (2) a one-state thermal dynamic model needs to be solved in observer I. However, the gain from the additions of the model and measurement is the availability of the estimates of each electrode states, which can be used for battery health monitoring (e.g., capacity tracking and plating detection) as detailed in the Applications section.

In this paper, we present an estimation scheme to estimate Li-ion concentrations in the negative and positive electrodes of the Li-ion cell. To capture the battery internal physics, we adopt the two-electrode electrochemical SPM along with averaged thermal dynamics for the observer design. First, the improvement of the observability of the Li-ion concentration states in both electrodes is given measurement of temperature and voltage. Next, the details of the cascaded observer structure are outlined. The first observer estimates a function containing Li-ion surface concentration information from the thermal dynamics and the measured temperature. The second observer uses this information along with the measured voltage to estimate the Li-ion concentration states of both electrodes. Two important applications of the proposed estimation scheme are presented: cell-capacity estimation and lithium plating detection. The effectiveness of the scheme is verified by simulation studies. Furthermore, the effect of measurement noise and parametric uncertainties is evaluated.

However, there are some aspects that should be explored as future work of this study. First, the scheme is illustrated using particular battery chemistry (LiCoO₂–graphite). The effectiveness of the scheme should be explored for other chemistries. Next, solutions should be investigated to deal with the sensitivity to high-noise levels in the measurements. Finally, the proposed scheme should be validated using experimental data to verify its effectiveness.

This research was supported, in part, by the U.S. Department of Energy GATE program under Grant No. DE-EE0005571 and NSF under Grant No. CMMI-1055254.

$as±$ =

electrode surface area, specific (cm²/cm³)

$A$ =

current collector area (cm²)

$As$ =

surface area of the cell (cm²)

$ce$ =

Li-ion concentration, electrolyte phase (mol/cm³)

$cs±$ =

Li-ion concentration, solid-phase (mol/cm³)

$cs,e±$ =

Li-ion surface-concentration, solid-phase (mol/cm³)

$cs,max±$ =

Li-ion max. concentration, solid-phase (mol/cm³)

$Cp$ =

specific heat capacity (J/g K)

$Ds±$ =

solid-phase diffusion coefficient (cm²/s)

$Ds,ref±$ =

solid-phase diffusion coefficient at $Tref$ (cm²/s)

$ER$ =

film resistance activation energy (J/mol)

$EK±$ =

diffusion coefficient activation energy (J/mol)

$EDs±$ =

reaction rate constant activation energy (J/mol)

$F$ =

Faraday's constant (C/mol)

$h$ =

cell heat transfer coefficient (W/cm² K)

$I$ =

current (A)

$K±$ =

reaction rate constant (cm^2.5/mol^0.5/s)

$Kref±$ =

reaction rate constant at $Tref$ (cm^2.5/mol^0.5/s)

$L±$ =

cell length (cm)

$r$ =

radial coordinate (cm)

$R$ =

solid active particle radius (cm)

$R¯$ =

universal gas constant (J/mol K)

$Rf,ref$ =

contact film resistance at $Tref$ (Ω)

$T$ =

temperature (K)

$Tref$ =

reference temperature (K)

$T∞$ =

ambient temperature (K)

$U±$ =

open-circuit voltage (V)

$α±$ =

charge transfer coefficient

$v$ =

cell volume (cm³)

$ρ$ =

cell density (g/cm³)

Superscript

± =

positive/negative electrode