Determination of the Differential Capacity of Lithium-Ion Batteries by the Deconvolution of Electrochemical Impedance Spectra

Abstract

Electrochemical impedance spectroscopy (EIS) is a powerful tool for investigating electrochemical systems, such as lithium-ion batteries or fuel cells, given its high frequency resolution. The distribution of relaxation times (DRT) method offers a model-free approach for a deeper understanding of EIS data. However, in lithium-ion batteries, the differential capacity caused by diffusion processes is non-negligible and cannot be decomposed by the DRT method, which limits the applicability of the DRT method to lithium-ion batteries. In this study, a joint estimation method with Tikhonov regularization is proposed to estimate the differential capacity and the DRT simultaneously. Moreover, the equivalence of the differential capacity and the incremental capacity is proven. Different types of commercial lithium-ion batteries are tested to validate the joint estimation method and to verify the equivalence. The differential capacity is shown to be a promising approach to the evaluation of the state-of-health (SOH) of lithium-ion batteries based on its equivalence with the incremental capacity.

1. Introduction

Electrochemical impedance spectroscopy (EIS) has been proven to be a powerful tool for the diagnosis of complex electrochemical systems, including lithium-ion batteries [1,2,3,4,5,6], fuel cells [7,8], and supercapacitors [9,10]. Electrochemical impedance spectroscopy has been widely used to characterize the polarization processes of lithium-ion batteries [11,12,13,14] and to investigate various prognostics and health management (PHM) methods [15,16,17,18,19]. Electrochemical impedance spectrum is generally analyzed by a carefully chosen equivalent-circuit model (ECM), which requires knowledge about the electrochemical processes that take place at the individual electrodes within the cell [20,21,22,23]. Comparison between EIS-based ECM and incremental capacity has been presented to identify and quantify the effects of degradation modes [24]. However, some non-ideal processes and the overlapping effects lead to a certain level of ambiguity of the ECM during the model identification [25,26,27]. This problem needs to be settled by the deconvolution of the EIS data with respect to the distribution of relaxation times (DRT) [28,29,30,31,32].

Considering the DRT offers an approach that does not rely on any prior knowledge of the investigated electrochemical system [33,34]. Therefore, the use of the DRT is regarded as a model-free approach for system identifications. The DRT method attempts to decompose the impedance of a capacitive electrochemical system into a continuous distribution of resistor-capacitor (RC) elements in the domain of relaxation times [35]. Good practices were reported in the context of the analysis of the impedance of solid oxide fuel cells (SOFCs) and high-frequency impedance of lithium-ion batteries [7,31,32,36]. For low frequencies, however, the differential capacity caused by diffusion processes is non-negligible, and thus, cannot be decomposed by DRT. Consequently, low frequencies limit the application of the DRT method to lithium-ion batteries [5,37,38,39].

For lithium-ion batteries, the differential capacitive tail, as shown in Figure 1, has to be considered at low frequencies. Consequently, the DRT method needs to be modified as it cannot characterize a pure capacitive behavior. Some amending methods have been proposed to estimate the differential capacity, such as the preprocessing method [12,28], the distribution function of differential capacity (DDC) method [39], the distribution of diffusion times (DDT) method [40], and the differential impedance analysis (DIA) method [41,42]. However, the differential capacity and the DRT are estimated separately, resulting in accumulative errors, thereby limiting the applicability of the DRT method to lithium-ion batteries.

In this paper:

(1)

A joint estimation method with Tikhonov regularization is proposed to simultaneously estimate the differential capacity and the DRT with the aim of minimizing the estimation errors and to obtain more information about the diffusion processes by EIS.

(2)

Moreover, the equivalence of the differential capacity C_DC and the incremental capacity C_IC is proven in Section 2.

(3)

Four types of commercial lithium-ion batteries are tested in Section 3 to validate the joint estimation method and to verify the equivalence of the C_DC and C_IC.

(4)

Subsequently, the estimation results of the DRT and the C_DC are discussed in Section 4.

(5)

In addition, an efficient state-of-health (SOH) evaluation method is demonstrated based on the relationship between the C_DC and the cell capacity in Section 4.

(6)

The conclusions of the work are summarized in Section 5.

2. Theoretical

2.1. The Relationship between EIS and ICA

This section derives the relationship between the differential capacity C_DC identified by EIS and the incremental capacity C_IC obtained by ICA.

A typical EIS involves sweeping the excitation frequency with a sinusoidal voltage or current. In the EIS data, the complex impedance can be described by a frequency-dependent function:

$$Z\left(\omega \right)=\frac{U\left(\omega \right)}{I\left(\omega \right)}=R_{\mathrm{ohm}}+R_{\mathrm{pol}}\left(\omega \right)+\frac{1}{j\omega C_{\mathrm{DC}}}$$

(1)

where $\omega$ is the angular frequency, $U\left(\omega \right)$ is the excitation voltage, $I\left(\omega \right)$ is the current response, j is the imaginary unit, $R_{\mathrm{ohm}}$ is the ohmic resistance, and $R_{\mathrm{pol}}\left(\omega \right)$ is the polarization resistance. The differential capacity C_DC can be extracted theoretically by processing the limit at extremely low frequencies:

$$C_{\mathrm{DC}}=\underset{\omega \to 0}{\lim }\frac{1}{j\omega \left(\frac{U\left(\omega \right)}{I\left(\omega \right)}-R_{\mathrm{ohm}}-R_{\mathrm{pol}}\left(\omega \right)\right)}\approx \underset{\omega \to 0}{{\displaystyle \lim }}\frac{Q\left(\omega \right)}{U\left(\omega \right)}=\frac{\mathrm{d}Q}{\mathrm{d}U}=C_{\mathrm{IC}}$$

(2)

where $\frac{\mathrm{d}Q}{\mathrm{d}U}$ is the incremental capacity [43,44], denoted as C_IC. The equivalence of the C_DC and C_IC can be proved by Equation (2). The detailed derivation is given in Appendix A. Figure 2 gives a graphical interpretation of the relationship between C_DC and C_IC for a better understanding of the equivalence. This equivalence relationship expands the applicability of EIS to lithium-ion batteries, given that C_DC and C_IC are equal. This is highly beneficial, since estimating C_DC by EIS in certain cases is more straightforward and time-efficient compared with the use of ICA to measure C_IC.

2.2. The Joint Estimation Method with Tikhonov Regularization

The value of the C_DC cannot be directly calculated by Equation (2) because the sweeping frequencies are discrete and have a lower limit. Therefore, the DRT method must be modified. The experimental data $Z_{\exp }$ measured at several sweeping frequencies were fitted by a model $Z_{\mathrm{DRT}}$ as follows [12,28,29,32]:

$$Z_{\mathrm{DRT}}\left(\omega \right)=R_{\mathrm{ohm}}+{\displaystyle {\int }_0^{\infty }\frac{g\left(\tau \right)}{1+j\omega \tau }}\mathrm{d}\tau$$

(3)

where $\tau$ represents the characteristic time constants and g$\left(\tau \right)$ represents the distribution of the polarization resistance. Furthermore, the differential capacity is non-negligible for lithium-ion batteries as it contains information about the diffusion processes. Consequently, considering the differential capacity of lithium-ion batteries, the model $Z_{\mathrm{DRT}}$ was modified to obtain the following expression:

$$Z_{\mathrm{DRT}}^{\mathrm{C}}\left(\omega \right)=R_{\mathrm{ohm}}+{\displaystyle {\int }_0^{\infty }\frac{g\left(\tau \right)}{1+j\omega \tau }}\mathrm{d}\tau +\frac{1}{j\omega C_{\mathrm{DC}}}$$

(4)

where $Z_{\mathrm{DRT}}^{\mathrm{C}}$ represents the DRT model considering the differential capacity C_DC. Subsequently, the discretized DRT model derived for Equation (3) in Ref. [29] can be reformulated into:

$$Z_{\mathrm{DRT}}^{\mathrm{C}}\left(\mathit{\omega }\right)=R_{\mathrm{ohm}}+{\left({\mathit{A}}^{\mathbf{\prime }}\mathit{x}\right)}_{\mathrm{n}}+{\left({\mathit{A}}^{\mathbf{″}}\mathit{x}\right)}_{\mathrm{n}}+\frac{1}{j\mathbf{\omega }{C}_{\mathrm{DC}}}$$

(5)

where ω is a column vector with n entries equal to the sweeping frequencies, ${\mathit{A}}^{\mathbf{\prime }}$ represents the approximation matrix of the DRT of the real part of the EIS data, ${\mathit{A}}^{\mathbf{″}}$ represents the approximation matrix of the DRT of the imaginary part of the EIS, and x represents the parameter vector for the DRT approximation. Then, the joint estimation function can be obtained by fitting the data with the improved discretized DRT model $Z_{\mathrm{DRT}}^{\mathrm{C}}\left(\mathit{\omega }\right)$, which implies the minimization of the following sum of squares:

$$J\left(\mathit{x}\right)={\Vert {\mathbf{\Omega }}^{\mathbf{\prime }}\left(R_{\mathrm{ohm}}\cdot \mathbf{1}+{\mathit{A}}^{\mathbf{\prime }}\mathit{x}-{\mathit{Z}}_{\exp }^{\mathrm{Re}}\right)\Vert }^2+{\Vert {\mathbf{\Omega }}^{\mathbf{″}}\left(\frac{1}{\mathit{\omega }{C}_{\mathrm{DC}}}+{\mathit{A}}^{\mathbf{″}}\mathit{x}-{\mathit{Z}}_{\exp }^{\mathrm{Im}}\right)\Vert }^2$$

(6)

where ${\mathbf{\Omega }}^{\prime }$ and ${\mathbf{\Omega }}^{″}$ represent the frequency matrices of the DRT, 1 is a column vector with n entries all equal to 1, ${\mathit{Z}}_{\exp }^{\mathrm{Re}}$ is the real part of the experimental data, and ${\mathit{Z}}_{\exp }^{\mathrm{Im}}$ is the imaginary part of the experimental data. Implementation of the traditional DRT method is well established in the literature [7,12,28,29,38]. Hence, in the present work, we extend the traditional DRT to cover the C_DC part of the curve and perform a joint estimation. So, we only provide the modified optimization function to account for the C_DC based on Equation (6) as follows:

$$\min \left\{J\left(\mathit{x}\right)={\Vert {\mathbf{\Omega }}^{\prime }\left(R_{\mathrm{ohm}}\cdot 1+{\mathit{A}}^{\prime }\mathit{x}-{\mathit{Z}}_{\exp }^{\mathrm{Re}}\right)\Vert }^2+{\Vert {\mathbf{\Omega }}^{″}\left(\frac{1}{\mathit{\omega }{C}_{\mathrm{DC}}}+{\mathit{A}}^{″}\mathit{x}-{\mathit{Z}}_{\exp }^{\mathrm{Im}}\right)\Vert }^2+\lambda {\mathit{x}}^{\mathrm{T}}\mathit{M}\mathit{x}\right\}$$

(7)

where λ is the regularization coefficient and M is the regularization matrix, which is derived in Ref. [29]. The problem stated in Equation (7) is the well-known Tikhonov regularization problem whose solution can be obtained by various numeric algorithms [29,38,45,46]. Then, the ohmic resistance $R_{\mathrm{ohm}}$, the differential capacity C_DC, and the parameter x of the DRT can be simultaneously estimated by minimizing J(x) in Equation (7).

3. Experimental

3.1. The Test Conditions

Table 1. Specifications of the tested lithium-ion batteries.

Battery	Cathode	Anode	Capacity (Ah)	Voltage Range (V)
A	NCM	G	3.2	2.5–4.2
B	NCM	G	4.8	2.5–4.2
C	LFP	G	20	2.0–3.65
D	NCM + LMO	G	24	2.5–4.2

lists the specifications of the four types of commercial lithium-ion batteries that were tested. The batteries will be henceforth referred to by the capitals A, B, C, and D for convenience. Two batteries had LiNi_xCo_yMn_zO₂ (NCM) cathodes, one had a LiFePO₄ (LFP) cathode, and one had a mixed cathode consisting of NCM and LiMn₂O₄ (LMO). Each battery had graphite anodes, marked as G in Table 1.

The battery test platform is shown in Figure 3. The test platform consisted of a CT-4008-5V100A-NTFA tester (Neware, Shenzhen, China) a BTH-150C thermal chamber (DGBELL, Dongguan, China) an Autolab PGSTAT302N electrochemical workstation (Metrohm AG, Herisau, Switzerland) and a host computer. The Neware tester was used to charge and discharge the tested cells. The sampling frequency of the Neware tester was 1 Hz and its measurement accuracy was ±0.05% of its full scale. The thermal chamber provided the required ambient temperature with an accuracy of ±0.5 °C. The electrochemical workstation was used for EIS tests with a sampling frequency of 10 MHz. The host computer was used to control the tests and for data storage.

3.2. The Test Profiles

Two test profiles were designed to verify the equivalence of the C_DC and C_IC. The profile for the EIS is described in

Table 2. Test profile 1 for the EIS measurements.

Step No.	Step Name	Duration	Current	Cycle No.
1	Rest	180 min
2	EIS test
3	Discharge	18 min	1/3 C
4	Cycle, step 1–3			10
5	Rest	180 min
6	EIS test
7	End

and the profile for the ICA is described in

Table 3. Test profile 2 for the ICA measurements.

Step No.	Step Name	Duration	Current	Condition
1	Rest	180 min
2	Discharge		I = 1/20 C	V = upper limit
3	Rest	180 min
4	Charge		I = 1/20 C	V = lower limit
5	End

. EIS tests were conducted at 10% SOC intervals ranging from 100% to 0% SOC. The amplitude of the applied voltage in the EIS tests was 5 mV, and the frequency range is 2 kHz–2 mHz (60 points). For the ICA, the charging data of 1/20 C was adopted and processed by the probability density function (PDF) method [44].

3.3. Aging Characterization of the Cells

Several D-type cells (denoted as D1–D7) were subject to cycling at 45 °C with a charge/discharge rate of 1 °C. D1 was a fresh cell, while D2 to D7 have been exposed to varying cycling, and hence, possessed varying SOH. In this paper, SOH is defined by assessing the actual capacity divided by the nominal capacity as follows [47,48,49,50,51]:

$$SOH=\frac{Q_{\mathrm{act}}}{Q_{\mathrm{nom}}}$$

(8)

where $Q_{\mathrm{act}}$ is the actual capacity in the cell’s present condition and $Q_{\mathrm{nom}}$ represents the nominal capacity of the cell.

Detailed information about the aging cells and their testing procedures are listed in

Table 4. Capacity and SOH of aging D-type cells.

Cell Number	D1	D2	D3	D4	D5	D6	D7
Capacity (Ah)	24.2	22.6	22.0	21.1	20.1	19.3	15.5
SOH (%)	100	93.4	90.6	87.2	82.8	79.5	63.9

and

Table 5. Characterization procedures of the aging D-type cells.

Step No.	Step Name	Duration	Current
1	Rest	180 min
2	Discharge		I = 1/3 C
3	Rest	180 min
4	Charge to 3.68 V		I = 1/20 C
5	Rest	180 min
6	EIS test
7	End

, respectively. The SOH of the batteries ranged from 100% to 63.9% (Table 4), which covers the whole life cycle of commercially available lithium-ion batteries. The characterization procedures given in Table 5 mainly consist of EIS tests at a certain open-circuit voltage (OCV).

4. Results and Discussion

In this section, the test results are given, the estimation results using the joint estimation method are provided, and a comparison of the C_DC and the C_IC is conducted. C_DC values of the cells with different capacities were estimated, and their relationship with the SOH of the cells was evaluated.

4.1. The Estimation Results of the DRT and the C_DC

Figure 4 shows the EIS results of Cells A, B, C and D. The EIS results are shifted in the y-direction for better visualization.

The C_DC and the DRT were simultaneously estimated by the joint estimation method based on the shown EIS results. A comparison of the traditional DRT method and the proposed joint estimation method is shown in Figure 5, where the blue line shows the results obtained by the traditional DRT method, and the red one shows the results obtained by the proposed joint estimation method. The solid and dotted lines are the EIS results in different frequency bands. The frequency range of the solid lines was 2 kHz–2 mHz while the frequency range of the dotted lines was 2 kHz–20 mHz, which means that the solid line contains the low frequency (LF) and the dotted line does not. The C_DC cannot be accurately determined by the traditional DRT method, which can be seen from the highest peak in Figure 5. This peak is caused by the C_DC, and its height and position are affected by the frequency range of the EIS. The results of the proposed joint estimation method are hardly affected by the frequency range of the EIS. Therefore, the proposed joint estimation method can effectively solve the problem of determining the C_DC compared with the traditional DRT method.

Figure 6 shows the results of the C_DC estimated by the joint estimation method (red crosses) and the ICA curves (blue lines). Here, the C_DC values estimated by the joint estimation method are compared with C_IC values obtained by ICA at the corresponding voltage according to Equation (2), as the traditional DRT method cannot provide the C_DC. The estimation C_DC values are close to the C_IC values at the corresponding voltage, indicating that the joint estimation method exhibits adequate accuracy. The relative errors of the C_DC and C_IC values are shown in Figure 7. The relative errors are below 10% except for the individual points, indicating the method’s sufficient accuracy.

4.2. SOH Evaluation Based on the Relationship between the C_DC and the Cell Capacity

The C_DC values of the aging D-type cells were estimated (

Table 6. Differential capacity C_DC of the aging D-type cells.

Cell Number	D1	D2	D3	D4	D5	D6	D7
CDC (105 F)	1.80	1.33	1.12	0.928	0.857	0.785	0.673

). The red crosses shown in Figure 8 are the estimated C_DC values at different cell capacities. The relationship between C_DC and the cell capacity can be described by Equation (9):

$$Q=a_1\exp \left(b_1\cdot C_{\mathrm{DC}}\right)+a_2\exp \left(b_2\cdot C_{\mathrm{DC}}\right)$$

(9)

where Q is the capacity of the cell; a₁, a₂, b₁, and b₂ are fitting coefficients. In this paper, cftool provided by MATLAB is utilized for the fast parameter identification of Equation (9) as it can realize many types of linear and nonlinear curve fitting. The nonlinear least square (NLR) method is set, and the Levenberg-Marquardt algorithm is adopted to identify the fitting coefficients. The values of the fitting coefficients are listed in

Table 7. Fitting coefficients of the aging D-type cells.

Fitting Coefficient	a1	a2	b1	b2
Value	18.86	−2550	1.386 × 10−6	−9.197 × 10−5

for the D-type cells. The blue line shown in Figure 8 is the fitting curve of the C_DC values described by Equation (9), which provides an accurate fit. Subsequently, the SOH can be evaluated by combining Equations (8) and (9):

$$SOH=\frac{Q_{\mathrm{act}}}{Q_{\mathrm{nom}}}=\frac{a_1}{Q_{\mathrm{nom}}}\exp \left(b_1\cdot C_{\mathrm{DC}}\right)+\frac{a_2}{Q_{\mathrm{nom}}}\exp \left(b_2\cdot C_{\mathrm{DC}}\right)$$

(10)

The estimated SOH of the aging D-type cells given by Equation (10) and the relative errors between the real SOH and the estimated SOH are given in

Table 8. SOH evaluation results of the aging D-type cells.

Cell Number	D1	D2	D3	D4	D5	D6	D7
Real SOH (%)	100	93.4	90.6	87.2	82.8	79.5	63.9
Estimated SOH (%)	100	93.6	90.7	86.5	83.8	79.1	64.0
Relative error (%)	0	0.23	0.02	0.80	−1.13	−0.51	0.05

. These results show that the demonstrated SOH evaluation method exhibits adequate accuracy along the whole life cycle of the cells.

Hence, the SOH of batteries of the same type can be evaluated only by EIS at a specific potential when a₁, a₂, b₁ and b₂ are obtained. It is worth noting that this study chose 3.68 V as the measured equilibrium potential for EIS, simply because the highest peak of the ICA is located at approximately 3.68 V. One can select this equilibrium potential arbitrarily as long as its corresponding ICA value changes with the aging of the battery. If the measured equilibrium potential for EIS changes, the fitting coefficients of Equation (9) will change, but the structure of Equation (9) will not.

The EIS based SOH evaluation method is more efficient compared with the ICA based SOH evaluation method, which only needs impedance spectra rather than charging/discharging data of the battery.

5. Conclusions

The present work proposed a joint estimation method to estimate the differential capacity C_DC and the DRT simultaneously based on the EIS of lithium-ion batteries. Four types of commercially available lithium-ion batteries were tested to evaluate the joint estimation method and to verify the equivalence. Experimental data showed that the proposed joint estimation method outperforms the traditional method in the estimation of the C_DC. Moreover, the estimated C_DC values are consistent with the C_IC values obtained by ICA. Key points of this study are summarized as follows:

(1)

A joint estimation method with Tikhonov regularization is proposed to simultaneously estimate the differential capacity C_DC and the DRT with the aim of minimizing the estimation errors and to obtain more information about the diffusion processes by EIS.

(2)

The equivalence of the differential capacity C_DC and the incremental capacity C_IC was shown.

(3)

An efficient state-of-health (SOH) evaluation method is demonstrated based on the relationship between the C_DC and the cell capacity.

The proposed joint estimation method can provide an intuitive understanding of the capacitive characteristics of lithium-ion batteries and can be used for SOH evaluation. Further research is being conducted to estimate the differential capacity C_DC from time-domain data for online applications.