Estimation of State-of-Charge and State-of-Health for Lithium-Ion Degraded Battery Considering Side Reactions

This paper focus on the battery aging effects at current load (≤3C) and temperature range (25°C-45°C). For these conditions it has been shown that the SEI formation is the main cause of battery degradation.²⁶ To further confirm the aging mechanisms on anode and cathode after cycling tests, we opened the cell in the glove box filled with argon and took some samples from both anode and cathode surface for further analysis. Fig. 2 shows photographs of the negative and the positive electrode of aged cell at 3C, 25°C after 180 cycles. The surface of the positive electrode appears to be very smooth. In contrast, the negative shows clear signs of delamination near the edge. The Scanning electron microscope (SEM) analysis was carried out to investigate the surface morphologies. Fig. 3 compares the SEM images of the positive and negative electrode from a fresh cell and a cell cycled at 3C after 180 cycles. The results show that there is no apparent products deposited on the cathode and evident crack formation compared to the anode. Then the fresh and aged cathode samples were sent to X-ray diffraction (XRD) tests. The XRD results in Fig. 4 show that there is no phase change or new phase formation in the cathode active materials of the battery investigated in this paper. That proves that the cathode aging problem is not obvious in this work. These findings could also be seen in Ref. 27. The reason for the phenomenon could be that the reaction of formation of the positive electrode/electrolyte interphase caused by electrolyte oxidation is very slow and the carbon coating of the cathode plays a strong protective role toward formation of surface species on the material.²⁸ To investigate if transition mental dissolved on the negative electrode surface, we conducted EDS (Energy Dispersive Spectrometer) test for the anode samples. The result is depicted in Fig. 5. The elements detected were C, O, Si, F, and P. No transition mental element was detected on the anode surface. We could conclude that metal dissolution is not likely a concern for these cells. This could due to the maximum cycle number and temperature in this paper is not over enough to cause transition metal dissolution.²⁹

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Generally, It is much more difficult to solve for and detect the side reaction effects on the positive electrode than that on the negative electrode.^21,30 Therefore, many papers related to battery aging model descriptions focused on the anode aging phenomena.^19,31

Based on the experiments and aforementioned researches, the following assumption are proposed: the side reactions at cathode would not have a significant impact on the presented aging model for the battery tested in this paper. The side reactions at anode should be regarded as the major cause for battery aging in this paper.

Based on the proposed assumption in Assumption on aging mechanism in this paper section, The model presented here is based on the authors' simplified electrochemical pseudo two-dimensional (P2D) model²⁵ and aging descriptions with modification to side reactions at anode in Ref. 19. The aging model is described briefly as below:

The SEI growth is considered as side reactions at anode. Therefore, a total of two electrochemical reactions occur at the anode, as illustrated in Fig. 6.

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The intercalation current density is , denoted by Eq. 1a. When side reactions occur at the interface between the electrode particles and the electrolyte, the current density of SEI growth is induced as j_side by Eq. 1b.

The total volumetric current density j_total is the sum of the current of the above two reactions:

The activation overpotential of the side reactions η_side, is represented as:

where ϕ_s and ϕ_e are electrical potentials of the solid electrode particle and electrolyte, respectively. R_SEI is the resistance of SEI that is presented only on the anode side. U^θ_{eq, side} is the equilibrium potential of side reactions and assumed to be 0.21V in this paper.

The local current density of side reactions is calculated by the Tafel expression as follows:

where i_{0, side} is the exchange current density of side reactions. n_side is the number of ions involved in the side reactions which is equal to 2 in this study.

The amount of ion loss, C_ionloss can be obtained by integrating the side reaction rate,j_side over the volume of composite anode and operation time:

where δ₋ denotes the thickness of composite anode (m),τ is the total operating time (s), A is the electrode plate area and a_s is the active surface area per electrode unit volume.

The volume fraction of SEI, ε_SEI is defined as:

Loss of active materials is simply described as follows:

where ε_s is the volume fraction of solid active material that is dependent on thickness x and k_iso is a dimensionless coefficient which describes how fast the active materials are isolated from chemical reactions.

Energy conservation of the whole cell (lumped thermal model) is expressed below:

Detailed descriptions of these governing equations can be found in the literature.³²

OCV_fresh is defined as the difference between the cathode equilibrium potential and the anode potential, expressed as:

End of charge (EOC) and end of discharge (EOD) correspond to OCV_max and OCV_min, respectively. Capacity is calculated based on the amount of charges transferred from EOC to EOD as follows:¹⁸

It is supposed that p and n are used to denote the electrodes stoichiometry of positive and negative electrodes, respectively. ε_s is the volume fraction of the active material. δ denotes the thickness of composite anode (m). A is the electrode plate area. F is the Faraday constant. Electrodes stoichiometry i ∈ {p, n} are defined as the ratio of the surface Li-ion concentration C_{s, surface}, over the maximal Li-ion concentration C_{s, max}.

For a fresh cell, n_EOC,n_EOD p_EOC and p_EOD equal to  n_max,n_min p_min, and p_max, respectively, so the capacity of a fresh cell is

A relationship between p and n for a fresh cell can be then deduced:

The values of n_max,p_min n_min and p_max are provided by the manufacturer in the beginning. A linear dependence of one of the OCV-SOC model parameters on temperature is expressed as below:

where T is a reference temperature (e.g., 25°C), and φ(T_ref) is the value of the OCV-SOC model parameters at T_ref. Δφ is the sensitivity to temperature determined for each parameter φ(T).³³ The OCV-SOC characterization curve almost remains the same regardless of the temperature of the battery.³⁴ Thus, the OCV curve of a fresh cell under room temperature, 25°C is chosen as the reference. The OCV curve of a fresh cell are plotted versus stoichiometric number in Fig. 7a.The actual stoichiometric number on the anode side, n, and volume fraction of the active material, ε_{s −}, could decrease due to loss of ions and active material, respectively. Thus, the variation of anode stoichiometric number, n_shift and the decrement of volume fraction of active material, Δε_{s −} are included in the ultimate correlation as below:

Δε_{s −} is expressed in Eq. 7 and n_shift is graphically shown in Fig. 7 as the amount of shift in n axis, which is illustrated by the amount of ion loss:

where C_ionloss is calculated based on Eq. 5. Δε_{s −} is obtained from Eq. 7, which is negative.

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Based on Eq. 13, 15, 16, the relationship between the aged anode stoichiometric number n_agedand the fresh anode stoichiometric number n can be deduced as:

Finally, the aged cell OCV is given as:

The mathematical relationship for U₊(p) and U₋(n) of fresh cells is listed in Table I. The values are from the battery manufacturer.

When an aged cell is charged to EOC, the OCV_max is determined by two equilibrium potentials.

The two variables, p_EOC and n_EOC are solved using Eq. 15 and Eq. 16c. Similarly, when an aged cell is discharged to EOD, the OCV_min is denoted as:

p_EOD and n_EODbecome less than p_min and n_max, respectively. It means that the cathode electrode is unable to be fully charged and the anode electrode can be over-discharged at the same OCV_min, as depicted in Fig. 7b.

When a certain amount of the active material is lost (Δε_{s −}), n changes faster with respect to p, according to Eq. 15. This can be analyzed by shrinking the n axis, as shown in Fig. 7c. By updating n in Eq. 16c, the anode OCV curve is obtained considering both loss of ions and loss of active materials. From Fig. 7, we can see that when side reactions are considered, the following relationship for aged cells can be deduced:

where n_EOC, p_EOC, n_EOD and p_EOD are numerically solved using Eqs. 16–17 and plotted in Fig. 7. OCV_max and OCV_min are determined by the cell's cutoff voltage which remains constant during cycling. Furthermore, the capacity is calculated by the following equation:

SOH defined by Eq. 20 is also obtained.

The 2^nd order RC ECM is selected because the 2^nd order ECM is an optimum choice for implementation of most battery energy management strategies.³⁵ As depicted in Fig. 8, R₁ and C₁ is the polarization resistance and capacitance, respectively. R₂ and C₂ represents the diffusion resistance and capacitance, respectively. V_OCV denotes the open circuit voltage which is mapped to SOC based on the proposed OCV-SOC relationship. V is the terminal voltage.

• (1) An online Recursive Least Squares (RLS) algorithm is chosen to identify ECM parameters (R₁, C₁, R₂, and C₂). The identifiable form is derived by Eqs. 21–22. k₁, k₂, k₃, k₄ and

  

  k₅ are parameters to be identified by RLS.

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The initial values of the parameter estimate θ(0) and its error covariance matrix P(0) need to be firstly prescribed. The constant λ is the forgetting factor and is very important to obtain an accurate estimated parameter set with small error, typically 0.95 < λ < 1.

• (2) After τ₁, τ₂, R₁, R₂ are identified in step (1). The third-order Extended Kalman Filter (EKF) is taken to estimate SOC. The state vector is X = [SOC, U₁, U₂]^T. The state space function is indicated as below:

  

  

where T is the sample time, 1s. Q is the capacity. Considering different timescales of SOC and SOH, the OCV-SOC correlation is updated every 20 cycles based on Eq. 16c and then is utilized in C(k). The entire SOC estimation process is shown in Fig. 9.

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Before testing the SOC estimator with our proposed OCV-SOC relationship, it is necessary to validate the effectiveness of the simplified battery model integrated with the updated OCV-SOC correlation after different cycles. The discharge curves under 1C, 2C and 3C for simulation and experiment after different cycles are listed in Fig. 10. As it can be seen in Fig. 10, the simulated voltage almost superposes the experimental value regardless of the different C-rates. Hence, the battery aging model is proved to be effective.

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To better evaluate the robustness of the SOC estimator in an environment close to a realistic one, artificial Gaussian noise is added to the current collected by the tester. The mean and standard deviation of the noise are −1A and 1.6A (around 2% of the full range), respectively.

In addition, we assume that the absolute measurement error of the cell OCV is 8 mV.³⁶ The 1C, 2C and 3C cycling datasets for different cycles are used in this subsection. The initial error of the EKF is assumed to be 10%. The initial values of the EKF prediction and correction gain matrices are fixed for all cases. For fresh cells, the SOC estimator is tested under 1C, 2C and 3C charging current, respectively. Fig. 11a shows the SOC estimation results for fresh cells. It can be seen that the presented estimator efficiently compensates for the initial error and has a satisfactory accuracy of 5%, indicating a good robustness.

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For aged cells, the OCV-SOC curve is updated based on the proposed model and then applied to the SOC and SOH estimator. In addition, to investigate the effectiveness of the updated OCV-SOC curve, the OCV-SOC relationship for fresh cells is considered as constant for all cases and applied to the SOC estimator under different aging states for comparison. The RMS errors of the SOC estimation for aged cells under different cycling conditions after varying cycles are shown in Fig. 11b. It can be seen that if the OCV-SOC correlations are not updated, the estimator performance deteriorates significantly. The RMS Error at 3C is slightly larger than that at 1C and 2C. One reason for this phenomenon may be that high C-rate significantly and instantaneously accelerates the side reaction rate and leads to some unknown reactions such as lithium plating or changes in porosity.³⁷ Obviously, the RMS errors of the SOC estimator without OCV-SOC parameter updated are clearly larger than those updated. Updating the OCV-SOC relationship can improve the SOC estimation in most cases. Moreover, the RMS estimation errors in the validation datasets under three loading current profiles are around 2%, verifying a satisfactory robustness and effectiveness against different charging C-rates.

Fig. 12 highlights that the SOH estimation method introduced in Eq. 19–20 shows good convergence toward experimental values and is capable of monitoring capacity fade over the battery lifespan. The SOH estimator is intentionally operated offline. For cycles at 1C, 2C and 3C, the maximum error is mostly below 2%. At the final stage of cycling, the noticeable fluctuations in error(some>0.05) around 180 cycles for 3C or 360 cycles for 2C are likely caused not only by degradation effects on model accuracy but also by excessive noise from experimental data.

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