Determination of the Di ﬀ erential Capacity of Lithium-Ion Batteries by the Deconvolution of Electrochemical Impedance Spectra

: 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 o ﬀ ers a model-free approach for a deeper understanding of EIS data. However, in lithium-ion batteries, the di ﬀ erential capacity caused by di ﬀ usion 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 di ﬀ erential capacity and the DRT simultaneously. Moreover, the equivalence of the di ﬀ erential capacity and the incremental capacity is proven. Di ﬀ erent types of commercial lithium-ion batteries are tested to validate the joint estimation method and to verify the equivalence. The di ﬀ erential 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.


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 modelfree 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.

Figure 1.
Electrochemical impedance spectrum of a lithium-ion battery with a differential capacitive tail, measured from 5 kHz to 50 μHz.
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 CDC and the incremental capacity CIC 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 CDC and CIC. (4) Subsequently, the estimation results of the DRT and the CDC are discussed in Section 4. (5) In addition, an efficient state-of-health (SOH) evaluation method is demonstrated based on the relationship between the CDC and the cell capacity in Section 4. (6) The conclusions of the work are summarized in Section 5.

The Relationship between EIS and ICA
This section derives the relationship between the differential capacity CDC identified by EIS and the incremental capacity CIC obtained by ICA. Electrochemical impedance spectrum of a lithium-ion battery with a differential capacitive tail, measured from 5 kHz to 50 µHz.
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.

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: where ω is the angular frequency, U(ω) is the excitation voltage, I(ω) is the current response, j is the imaginary unit, R ohm is the ohmic resistance, and R pol (ω) is the polarization resistance. The differential capacity C DC can be extracted theoretically by processing the limit at extremely low frequencies: where dQ dU 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.
Energies 2020, 13, x FOR PEER REVIEW 3 of 14 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: where is the angular frequency, ( ) is the excitation voltage, ( ) is the current response, j is the imaginary unit, ohm is the ohmic resistance, and pol ( ) is the polarization resistance. The differential capacity CDC can be extracted theoretically by processing the limit at extremely low frequencies: where d d is the incremental capacity [43,44], denoted as CIC. The equivalence of the CDC and CIC can be proved by Equation (2). The detailed derivation is given in Appendix A. Figure 2 gives a graphical interpretation of the relationship between CDC and CIC for a better understanding of the equivalence. This equivalence relationship expands the applicability of EIS to lithium-ion batteries, given that CDC and CIC are equal. This is highly beneficial, since estimating CDC by EIS in certain cases is more straightforward and time-efficient compared with the use of ICA to measure CIC.

The Joint Estimation Method with Tikhonov Regularization
The value of the CDC 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 exp measured at several sweeping frequencies were fitted by a model DRT as follows [12,28,29,32]: where represents the characteristic time constants and g( ) 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 DRT was modified to obtain the following expression:

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 DRT as follows [12,28,29,32]: where τ represents the characteristic time constants and g(τ) 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 DRT was modified to obtain the following expression: where Z C DRT 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: where ω is a column vector with n entries equal to the sweeping frequencies, A represents the approximation matrix of the DRT of the real part of the EIS data, A " 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 C DRT (ω), which implies the minimization of the following sum of squares: where Ω and Ω represent the frequency matrices of the DRT, 1 is a column vector with n entries all equal to 1, Z Re exp is the real part of the experimental data, and Z Im exp 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: 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 ohm , the differential capacity C DC , and the parameter x of the DRT can be simultaneously estimated by minimizing J(x) in Equation (7). Table 1 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 x Co y Mn z O 2 (NCM) cathodes, one had a LiFePO 4 (LFP) cathode, and one had a mixed cathode consisting of NCM and LiMn 2 O 4 (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.

The Test Conditions
Energies 2020, 13, x FOR PEER REVIEW 5 of 14 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.

The Test Profiles
Two test profiles were designed to verify the equivalence of the CDC and CIC. The profile for the EIS is described in Table 2 and the profile for the ICA is described in Table 3. 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]. Table 2. Test profile 1 for the EIS measurements.
Step No.
Step  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

Aging Characterization of the Cells
Several D-type cells (denoted as D1-D7) were subject to cycling at 45 °C with a charge/discharge

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 and the profile for the ICA is described in Table 3. 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]. Table 2. Test profile 1 for the EIS measurements.
Step No.
Step Name Duration Current Cycle No.  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

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]: where Q act is the actual capacity in the cell's present condition and Q nom represents the nominal capacity of the cell. Detailed information about the aging cells and their testing procedures are listed in Tables 4 and 5, 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).  Table 5. Characterization procedures of the aging D-type cells.
Step No.
Step Name Duration Current

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.  are provided, and a comparison of the CDC and the CIC is conducted. CDC values of the cells with different capacities were estimated, and their relationship with the SOH of the cells was evaluated. 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. The CDC 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 CDC 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 CDC, 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 CDC compared with the traditional DRT method.

SOH Evaluation Based on the Relationship between the CDC and the Cell Capacity
The CDC values of the aging D-type cells were estimated ( Table 6). The red crosses shown in Figure 8 are the estimated CDC values at different cell capacities. The relationship between CDC and the cell capacity can be described by Equation (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 for the D-type cells. The blue line shown in Figure 8 is the

SOH Evaluation Based on the Relationship between the CDC and the Cell Capacity
The CDC values of the aging D-type cells were estimated ( Table 6). The red crosses shown in Figure 8 are the estimated CDC values at different cell capacities. The relationship between CDC and the cell capacity can be described by Equation (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 for the D-type cells. The blue line shown in Figure 8 is the

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). 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): Energies 2020, 13, 915 where Q is the capacity of the cell; a 1 , a 2 , b 1 , and b 2 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 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): fitting curve of the CDC values described by Equation (9), which provides an accurate fit. Subsequently, the SOH can be evaluated by combining Equations (8) and (9):  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. 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  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. 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 a 1 , a 2 , b 1 and b 2 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.

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.

Conflicts of Interest:
The authors declare no conflict of interest.

Nomenclature
A approximation matrix of the DRT for the real part of the EIS A" approximation matrix of the DRT for the imaginary part of the EIS a, b fitting coefficient of C DC and Q C DC differential capacity

Appendix A
The theoretical solution of the C DC is: In conclusion, Equation (A2) can be approximated as: which is equivalent to the incremental capacity C IC .