An Inverse Method for Estimating the Electrochemical Parameters of Lithium-Ion Batteries II: Implementation

This paper is the second part of a two part study on parameter estimation of Li-ion batteries. The methodology was developed in Part I. In Part II, the methodology is tested for LiCoO 2 , LiMn 2 O 4 and LiFePO 4 positive electrode materials. An inverse method combined to a simpliﬁed version of the Pseudo-two-Dimensional (P2D) model is used to identify the solid diffusion coefﬁcients (D s,n and D s,p ), the intercalation/deintercalation reaction-rate constants (K n and K p ), the initial SOC (SOC n,0 and SOC p,0 ), and the electroactive surface areas (S n and S p ) of Li-ion batteries. Experimental cell potentials for both low and high discharge rates provide the reference data for minimizing the objective function in the best time interval. For all cases simulated, the numerical predictions show excellent agreement with the experimental data.

Lithium-ion (Li-ion) batteries are increasingly employed for energy storage. Their working voltage and energy density are higher than those of similar energy storage technologies. Their service life is longer. They exhibit high energy-to-weight ratios and low selfdischarge. As a result, they have become the preferred energy storage devices in the electronics and the automotive industries.
Mathematical modeling of Li-ion batteries is an essential engineering tool for their design and operation. Two different approaches are usually adopted to predict their behavior. These approaches may be divided, broadly speaking, into empirical models and electrochemical models.
Empirical models are the simplest mathematical models. They are relatively easy to implement and they provide fast responses. This is why they are mostly suited for control systems used in the high tech industry and in the automotive industry. The scope of applications of empirical models is however narrow. Empirical models ignore the physical phenomena that take place in the cell. Consequently, they cannot predict the life and the capacity fading of the battery. Furthermore, they are only valid for the battery for which they have been developed. [1][2][3] Electrochemical models provide, on the other hand, reliable responses of the battery under a wide range of operating conditions and for different applications. They account for the chemical/ electrochemical kinetics and the transport phenomena. Electrochemical models are unequivocally superior to empirical models. But they are also more complex and require longer computation times.
Among the electrochemical models, the Pseudo-two-Dimensional (P2D) model stands out. The P2D model rests on the porous electrode theory, the concentrated solution theory and the use of appropriate kinetics equations. [4][5][6] A simplified and computationally efficient version of the P2D model is the Single Particle Model (SPM). In the SPM, it is assumed that the current distribution along the thickness of the porous electrode remains uniform and that the electrolyte properties are constant. 5,7 Both empirical and electrochemical models need to be calibrated in order to simulate faithfully the Li-ion cells. Moreover, due to their complexity, electrochemical models require extensive data such as the chemical/electrochemical parameters and the physical properties of the battery. Some of these parameters are readily available. They are provided by the manufacturer or can be determined experimentally. Others like the mass transport properties, the charge transfer and the kinetics parameters are more difficult to determine. This is the case of * Electrochemical Society Student Member. z E-mail: barzin.rajabloo@Usherbrooke.ca internal parameters such as the diffusivity of Li + ions in the electrodes (D s,n and D s,p ), the reaction rate constants at the electrodes/electrolyte interface (K n and K p ), the initial State Of Charge of the electrodes (SOC n,0 and SOC p,0 ) and the volume fraction of active material in the electrodes (ε n and ε p ), etc. [8][9][10][11][12][13][14][15] In this case, the collected experimental data must be processed with optimization methods in order to reveal unknown parameters and properties. A literature review of the methods applied to Li-ion batteries, i.e., the Parameter Estimation methods (PE), is provided in the first part of this study. 16 An innovative inverse PE method for identifying the electrochemical parameters of Li-ion batteries was also proposed in Part I. This inverse PE method rests on a simplified version of the P2D model combined with an inverse method and sensitivity curves of the expected parameters. The PE method may predict the solid diffusion coefficients (D s,n and D s,p ), the intercalation/deintercalation reaction-rate constants (K n and K p ), the initial SOC (SOC n,0 and SOC p,0 ), and the electroactive surface areas (S n and S p ) of the Li-ion battery. A Genetic Algorithm (GA) is used to minimize the objective function. The results displayed in Part I show good agreement between the predicted discharge curves and the noisy reference data at both low and high discharge rates. The present paper follows Part I by verifying the proposed methodology for Li-ion cells made of different active cathode materials. The noisy reference data that were generated with the P2D model and used for validating the PE model in Part I are now substituted with actual experimental data. These data come from Li-ion batteries made of different positive electrode materials: LiCoO 2 (LCO), LiMn 2 O 4 (LMO) and LiFePO 4 (LFP).
In the following section, the simplified P2D model, introduced in starting of paper, is reviewed. Next, the Parameter Estimation method is presented. The reference data and the expected parameters are discussed afterward. Sensitivity analysis section focuses on the sensitivity analysis of the Li-ion batteries. The features of the LFP and the challenges of simulating this positive electrode active material are discussed in Results and discussion section. Finally, the PE results are presented in Conclusion section.

Direct Model
Empirical and electrochemical models are the preferred approaches for simulating the State Of Charge and the State Of Health (SOH) of Li-ion batteries. 1,5,17 Empirical models rely on polynomial, exponential, power law, logarithmic, and trigonometric functions to match the experimental data. These models are simple and computationally inexpensive. They are however solely applicable to the battery for which they were calibrated. As a result, the scope of applications of empirical models is limited. 18,19 Electrochemical models are, on the other hand, unquestionably superior to empirical models for predicting the behavior of Li-ion batteries. These models take into account the chemical/electrochemical kinetics and the transport phenomena that take place into the battery. But their complexity may be a drawback. They are also CPU time consuming.
The P2D model is a popular electrochemical model. 6 The P2D model considers both the diffusion and the potential in the solid and in the electrolyte phases. The kinetics is described by the Butler-Volmer expression. The P2D model solves the electrolyte concentration, the electrolyte potential, the solid-state potential, and the solid-state concentration within the porous electrodes. It also predicts the electrolyte concentration and the electrolyte potential within the separator. The transport phenomena, the electrochemistry, and the thermodynamics are portrayed by coupled nonlinear partial differential equations (PDEs) in space and time. 4,7,[20][21][22][23][24][25][26] Most P2D models are, however, elaborate and CPU time consuming. A simplified version of the P2D model is the SPM, which was developed by Atlung et al. 27 and later improved by Haran et al. 28 The SPM accounts for a lumped solution resistance and it ignores the local concentration and potential in the solution phase. It also assumes that the current distribution along the thickness of the porous electrode is uniform. As a conclusion, each electrode is modeled as one spherical particle. Intercalation and de-intercalation phenomena happens through a reaction at the surface and a diffusion inside spherical particle. It should be noted that assuming uniform current distribution and ignoring concentration and potential in the solution phase are not always possible. As a result, the SPM is most suitable for simulating Li-ion batteries with thin and highly conductive electrodes subjected to low current densities. 29 As a compromise between the simplicity of the SPM model and the accuracy of P2D models, a simplified version of the P2D model was introduced in Part I. It rests on the following equations: F ln m 2 n + 4 + m n 2 + I a 0 + a 1 (I t) + a 2 (I t) 2 + ... [1] where SPM differs to simplified P2D model in last term of potential equation (Eq. 1) where a lump solution resistance has been replaced The potential equation is applied in the PE process for which the electrochemical parameters and the unknown variables of the electrolyte potential drop function are estimated.
The proposed simplified P2D model improves the results of the SPM particularly at high C-rate charge/discharge. Moreover, the number of parameters needed for the simplified P2D model is less than that for the P2D model. As a result, the proposed simplified P2D model is befitting online simulation and optimization.

The Parameter Estimation Process
Inverse problems belong to a class of ill-posed mathematical problems. Their solution is strongly dependent on the initial conditions, on the boundary conditions and on the measured signals. This makes the solution of inverse problems more challenging than that of direct problems.
In PE problems, the unknown parameters of a system can be determined with an inverse method. Of course, the parameters should be measurable and identifiable. Fig. 1 depicts a schematic of the PE procedure. 16 The objective function for the identification of the Li-ion parameters is defined as the difference between the experimental data for the time-varying cell potential and the numerical predictions generated by the direct model.
The vector of experimental data (V * cell ) is comprised of N time intervals between zero and the cutoff time (0 < t ≤ t c ). It is expressed as The objective function (S) is defined as the ordinary least-square function of the experimental data (V * cell ) and the numerical predictions (V cell ) 30 for one charge or discharge cycle: For M charge/discharge processes: 8 The aim of the PE method is (1) to minimize the objective function with a mathematical optimizer and (2) to determine the resulting unknown parameters (P): min S = S (P) subject to P j,low ≤ P j,1 ≤ P j,high [8] P j,low and P j,high are the minimum and the maximum possible values of P j respectively. Stochastic techniques are well suited for inverse problems with many parameters in the objective function. These techniques are also recommended for functions that exhibit local minimums in the vicinity of a global minimum. This is why a Genetic Algorithm (GA) was adopted in Part I for estimating the large number of electrochemical parameters of the Li-ion battery.

The Reference Data and the Parameters
The reference data.-The calculation procedure exemplified in Part I of this study will now be tested for three Li-ion batteries whose positive electrode is made of LiCoO 2 (LCO), LiMn 2 O 4 (LMO) and LiFePO 4 (LFP) materials.
Experimental data for Li x C 6 /Li y CoO 2 and Li x C 6 /Li y Mn 2 O 4 materials were retrieved from the literature. The data for the Li x C 6 /Li y FePO 4 cell were generated by standard galvanostatic charge/discharge tests on a commercial cell.
The experimental data of a pouch cell reported by Santhanagopalan et al. 8 were used as the reference data for the C/LCO cell. Charge and discharge tests were conducted for a carbon (MCMB)/LCO cell for rates of C/5, C/2, 1C, and 2C.
Experimental data for C/LMO cell are provided by the work of Doyle et al. 21 Potential versus capacity curves are reported for discharge rates ranging from 0.1C to 4C.
Experimental data from a cylindrical 18650 graphite/LFP cell were employed for the validation of the PE procedure on LFP cells galvanostatic discharge curves at both low and high C-rates were used to find the physical and the electrochemical parameters.
The equations used for calculation open circuit potential (OCP) as a function of state of charge for graphite, LCO and LMO are given in Appendix. The open circuit potential of LFP positive electrode was calculated by interpolating the experimental data at low discharge current. Note that the open circuit potential is an intrinsic characteristic of each positive electrode material. The open circuit potentials are dependent upon the ionic intercalation/deintercalation mechanisms. As a result, the range of influence of the battery's parameters may change, hence the need to perform a sensitivity analysis for validating the PE procedure for various active materials.
The parameters.-Numerous physical and chemical parameters are needed for simulating the behavior of Li-ion batteries and for predicting their performance. Some parameters such as the physical dimensions or the chemistry of materials are readily available from the manufacturer. Other parameters are more difficult to determine. These hard-to-get parameters include design parameters (porosity, particle size, etc.), electrode specific parameters (diffusion coefficients, electrical conductivity, contact resistance, etc.), and kinetics parameters (transfer coefficients, reaction-rate constants, etc.). Parameter estimation methods allow the determination of these parameters from experimental charge/discharge curves. For the present study, eight electrochemical parameters were determined with the parameter estimation method. These parameters are the solid diffusion coefficients (D s,n and D s,p ), the intercalation/deintercalation reaction-rate constants (K n and K p ), the initial SOC (SOC n,0 and SOC p,0 ), and the electroactive surface areas (S n and S p ). In spite of the fact that all these parameters are identifiable with the Li-ion direct model, their magnitude is de-pendent on the positive and negative materials. Table I  On the other hand, different active materials exhibit different behavior during charge/discharge processes. The chemical/electrochemical properties are influenced by the intercalation/deintercalation mechanisms and by the structure of the active materials. This behavior will be elucidated in the sensitivity analysis.

Sensitivity Analysis
A sensitivity analysis may be conducted for delineating the time interval domain for which the output of the system, i.e., the cell potential, is most sensitive to the input parameters and the properties. The sensitivity analysis improves the accuracy of the inverse method and the parameter estimation process.
The sensitivity or the Jacobian matrix (J) is defined as the first order partial derivative of the calculated cell potential (V cell ( P)) with respect to the unknown parameters (P j ), that is: As explained in Part I, the governing equations for the SPM are employed to calculate the dimensionless sensitivity coefficients for each parameter. The sensitivity coefficients depend on time and on the discharge rate.
The discharge curves may be divided into three regions: 1. the early stages of the discharge process; 2. the intermediate region of the discharge process when the potential varies gradually; 3. the end of the discharge process characterized by a sharp decline in the potential. Farkhondeh et al. 49 fitted experimental data for Li/LFP half-cells for these three regions. The authors were able to determine the regions where each of the cell parameters is the most influential on the battery's behavior.
In the present study, the best time domain for the PE of different electrochemical parameters is determined from a sensitivity analysis of the electrode chemistry.
The best time domains for estimating the parameters of a graphite/LiCoO 2 cell are shown in Fig. 2 (see also Figure 5f in Part I). high discharge rates, respectively. Therefore, the best time domain for estimating D s,p at low discharge rates is region 2, and region 3 for high discharge rates. It was concluded, in Part I, that D s,n , K n and K p can be determined from region 2, for both low and high discharge rates. The best time domain for estimating D s,p at low discharge rates is region 2, and region 3 for high discharge rates. For all discharge rates, region 3 is best for determining S n and SOC n,0 . Finally, regions 1 and 3 are the best time intervals to find S p and SOC p,0 , at both low and high discharge rates. The best time domains for estimating the parameters of the graphite/LiMn 2 O 4 cell were obtained from a sensitivity analysis. The results are depicted in Fig. 3. In this case, the discharge curve was divided into 2 regions. The main conclusions to be drawn from this figure are:

Green and red colored parameters can be estimated better in low and
r For the parameters SOC n,0 , S n , D s,n , K n and K p : Region 1 appears to be the best time domain for both low and high discharge rates.
r For the parameter D s,p : Region 1 is recommended for low discharge rates. Region 2 is better for high discharge rates.
r For the parameters S p and SOC p,0 : Region 2 is best for both low and high discharge rates.
The sensitivity analysis also provides the order of sensitivity of the parameter, which is S p , SOC p,0 , SOC n,0 , S n , K p , K n , D s,p , D s,n . This order results from different Jacobian value for each parameter. For instance, it was found that the output cell potential is more sensitive to S p than to D s,n .
A similar sensitivity analysis was conducted for graphite/ LiFePO 4 . The results are illustrated in Fig. 4. The conclusions concerning the best time domains for estimating the parameters of this battery may be summarized as follows:  r For the parameters D s,n , K n and K p : region 2 appears to be the best time domain for both low and high discharge rates.
r For the parameter D s,p : Region 1 is recommended only for low discharge rates (C/10). The dimensionless sensitivity coefficient for D s,p is close to zero at other discharge curves.
r For the parameters ε n and SOC n,0 : Region 3 is suitable at all discharge rates.
r For the parameters ε p and SOC p,0 : Region 1 is prescribed for both low and high discharge rates.
The parameter sensitivity resulting from the PE analysis may be stated in the following decreasing order: SOC n,0 , ε n , SOC p,0 , K n , D s,n , ε p , K p , D s,p . This order reveals that the PE process for graphite material is more sensitive than that for LFP. This behavior is shown in Fig. 5. The open circuit potential of the electrodes and the cell potential are compared. Fig. 5 reveals that the cell potential follows the open circuit potential of the LFP in region 1 only. In regions 2 and 3, the LFP has no influence. Graphite is the dominant electrode. The LFP electrode is made of fine powder compared to that of graphite electrode. As a result, the capacity of the positive electrode is higher than that of the negative electrode when the thickness and the porosity of two electrodes are almost same. This is the case for graphite/LFP cells designed for high power applications. Here, the LFP electrode is almost always operated in the plateau region. Since graphite material dictates the overall potential, the effect of the LFP electrode on the performance of the graphite/LFP cell is insignificant.
Fortunately, for future studies such as introducing an aging model considering the changes in dominant parameters, PE methodology is still valid and helpful. Due to the fact that graphite is recognized as the most dominant electrode in losing capacity mechanisms. In addition, it is worth noting that the methodology is anticipated to be well capable to estimate the influential parameters of LFP positive electrode material when LFP is the dominant electrode with capacity less than graphite. In fact, in this case, the order of sensitivity of the potential to the parameters would change and parameters related to the capacity of the positive electrode such as diffusion coefficient and porosity would possess higher sensitivities.
In order to simulate graphite/LFP cell, a Mosaic model was developed. 50 The Mosaic model proposed by Andersson et al. was adopted. 50 This model assumes a particle-radius dependency on the current density. Small radius particles are involved in the lithiation/delithiation process at high current densities. Large radius particles are, on the other hand, influential at lower current densities. Maheshwari et al. 43 adjusted the radius of the positive electrode particles for each C-rate in order to match the simulation results with the experimental data. Maheshwari et al. concluded that employing a current dependent radius is equivalent to simulating the actual particle size distribution (PSD). Prada et al. 40 observed a similar behavior for the evolution of the particle radius with the current density. The higher the current density, the smaller is the effective particle radius. Prada et al. have developed a Mosaic model in which the radii of both positive and negative electrodes were adjusted according to the experimental data. In this study, a modified version of the Mosaic model has been used. The adjusted radii related to each particle in different discharge currents are calculated based on this model. The model will be reported in our future paper.
Here, the Mosaic model was used to simulate the discharge curves of graphite/LFP cells of the 18650 battery type. The model was applied to the negative electrode (the graphite electrode) so as to obtain a good fit between the numerical predictions and the experimental data.

Results and Discussion
The inverse methodology introduced in Part I was implemented to find each parameter in its best time domain. The PE was carried out for different Li-ion batteries. Table II presents the estimated parameters for the graphite/LiCoO 2 cell. Each parameter was calculated for its best time domain. Fig. 6 compares the predicted and the experimental discharge curves for different currents (C/5, C/2, 1C and 2C). The agreement between the simulation and the experiment is striking.
The PE methodology developed in Part I was next tested for the spinel LiMn 2 O 4 positive electrode. The estimated parameters are summarized in Table III. Estimated parameters introduced in Table III were used as the input of simplified P2D model. Fig. 7 confronts the predicted and the measured discharge curves. Once again, the numerical predictions match the experimental data. Table IV provides the estimated parameters for the graphite/LFP. The simulations were conducted using a current dependent radius   for the negative electrode (based on the Mosaic model). In this case, the electroactive area is a dependent parameter (Eq. 2 and 3). The porosities of the electrodes (ε p and ε n ) were chosen as the estimated parameters instead of the total electroactive area of the electrodes. The Mosaic model was applied where the apparent radius of the graphite active material needed to be calculated from the discharge current. Fig. 8 shows the decrease of particle radius as current increases. Fig. 9 depicts the simulated and the experimental discharge potentials of the cell. The accuracy of the mathematical model is validated at both high and low discharge rates.
It is common to check the risk of over-fitting in PE studies. This issue may happen when number of parameters in PE is large. In this case, inverse method finds the parameters that are more representatives of the noise of the system than the general trend. These set of parameters simulates the behavior of the cell perfectly, however they are unable to find the results in other conditions. To prevent this issue   it is always better to find solutions that are more general. A strategy to examine over-fitting is to check the predictability of the model in a new condition other than conditions that are used for PE. In Fig. 9, simulated potential for discharge rate of 10 C is provided by parameters estimated from lower discharge rates (mentioned in Table IV). It is obvious that the model is able to predict the performance of cell outside the conditions used for PE. By using Eq. 10, the specific error values for each discharge curve is calculated for each cathode materials and presented in Table V.

Conclusions
A methodology for the electrochemical Parameter Estimation (PE) of Li-ion batteries was developed. The methodology rests on an in-verse method combined to a simplified version of the Pseudo-two-Dimensional (P2D) model. It is designed to identify the solid diffusion coefficients (D s,n and D s,p ), the intercalation/deintercalation reactionrate constants (K n and K p ), the initial SOC (SOC n,0 and SOC p,0 ), and the electroactive surface areas (S n and S p ) of the Li-ion battery. The methodology was tested for LiCoO 2 , LiMn 2 O 4 and LiFePO 4 positive electrode materials. The numerical predictions showed excellent agreement with the experimental data in all cases.

Acknowledgments
The authors are very grateful to Hydro-Québec and to the Natural Sciences and Engineering Council of Canada (NSERC) for their financial supports. Where the surface state of charge of the negative electrode (SOC n ) equals to the stoichiometric value x in Li x C 6 .