1. Model Introduction
In the mid-1990s, M. Doyle, T. F. fuller and J. Newman of the University of California, Berkeley established a pseudo two-dimensional (P2D) model based on the porous electrode and concentrated solution theory, which laid the foundation for the of electrochemical mechanism model. In this model, a series of partial differential equations and algebraic equations are used to describe the diffusion and migration of lithium ions, electrochemical reaction on the surface of active particles and other physical and chemical phenomena. So far, most electrochemical models are derived and developed from this model. Electrochemical model is a first principle model, which can accurately simulate the external characteristics of power battery, such as lithium ion concentration in electrode and electrolyte, reaction overpotential, etc. Compared with other power cell models, the electrochemical model can deeply describe the micro reaction in power cell, and has a clearer physical meaning.
P2D model is universal and extensible, which can be applied to different battery materials, and can be developed and extended to more complex multi field coupling model. Therefore, P2D model plays an irreplaceable role in the process of battery modeling. However, it contains complex partial differential equations and a large number of electrochemical parameters, which puts forward a high demand for the computing power of BMS. At present, numerical methods are mainly used to solve P2D model, such as finite difference method, finite element method and finite volume method.
Figure 1 P2D model structure block diagram of lithium cobalt oxide power battery
Figure 1 shows a schematic diagram of the electrochemical model of a lithium cobalt oxide power battery. The model uses two phases (solid phase and liquid phase) and three regions (positive electrode, negative electrode and diaphragm) to simplify the description of the internal structure of the power battery. The solid phase is modeled using spherical particles. The solid phase diffusion process is described by the diffusion process of lithium ions along the particle radius r. The liquid phase diffusion process is described by the diffusion and migration movement of lithium ions along the thickness of the power battery, that is, the x direction. The charge and discharge reactions of the positive and negative electrodes of the battery are shown in Table 1.
Table 1 Charge and discharge reaction of lithium cobalt oxide battery
During the charging process of the battery, the surface of the active material particles in the positive electrode zone (ie, the solid phase-liquid phase interface) undergoes a chemical reaction in the formula (1) of Table 1 to generate lithium ions and electrons. The electrons flow through the external circuit and the current collector. Return to the positive pole of the external power supply, thereby generating an external current. Lithium ions enter the liquid phase (electrolyte), the concentration of lithium ions on the surface of the active material particles decreases, and a concentration gradient is generated inside the active material, so that the solid phase diffusion of lithium ions from the inside to the outside occurs in the active material in the radial direction. At the same time, because this part of the lithium ions enters the liquid phase, the concentration of lithium ions in the liquid phase in the positive electrode area increases. In the negative electrode, the chemical reaction in the formula (2) of Table 1 occurs on the surface of the active material in the negative electrode area, which absorbs lithium ions and electrons. Lithium ions enter the solid phase particles from the electrolyte, resulting in an increase in the concentration of lithium ions on the surface of the active material, and solid phase diffusion of lithium ions from the outside to the inside of the active material occurs in the radial direction. At the same time, because this part of lithium ions comes from the electrolyte, the concentration of lithium ions in the liquid phase in the negative electrode area decreases. As the concentration of lithium ions in the positive electrode electrolyte increases and the concentration of lithium ions in the negative electrode electrolyte decreases, a concentration gradient is generated in the electrolyte. The lithium ions diffuse and migrate from the positive electrode to the negative electrode through the separator in the electrolyte.
The discharge process of the battery is opposite to the above process. Lithium ions diffuse from the negative active material to the surface, and are released into the electrolyte through the electrochemical reaction that occurs on the surface of the negative active material; then the lithium ions diffuse toward the positive electrode and reach the positive electrode after passing through the separator. And an electrochemical reaction occurs on the surface of the positive electrode active material and then diffuses into the inside of the positive electrode active particles. At the same time, electrons move from the negative electrode current collector to the positive electrode current collector, thereby forming a current in the external circuit.
2. Mathematical Equation
In order to establish the mathematical equations of the electrochemical model, the P2D model makes the following assumptions:
Hypothesis 1: Only lithium ions inside the battery participate in the chemical reaction;
Hypothesis 2: In the solid phase, the lithium ions are transferred along the radial direction of the active particles, and in the liquid phase, the lithium ions are transferred along the thickness direction of the battery (that is, the x-axis direction in Figure 1);
Hypothesis 3: The positive and negative active materials are spherical particles with the same radius;
Hypothesis 4: The electrochemical reaction on the surface of the active material conforms to the Butler-Volmer equation;
Hypothesis 5: The transfer mode of lithium ions in the solid and liquid phases is diffusion and Migration;
Hypothesis 6: The liquid phase volume fraction of the positive and negative poles of the battery remain unchanged during use;
Hypothesis 7: The heat generated inside the battery is negligible during the use of the battery;
Hypothesis 8: During the use of the battery, the battery volume does not change.
According to the working principle of lithium-ion power battery and model assumptions, the P2D model can establish the following six sets of mathematical equations:
① The diffusion equation of lithium ions in the liquid phase. The transfer mode of lithium ions in the electrolyte only includes diffusion and migration. The transfer process is described by Fick's second law, and the description area includes the positive electrode, the negative electrode and the diaphragm.
② The diffusion equation of lithium ions in the solid phase. The diffusion of lithium ions in the positive and negative electrode active materials is also described by Fick's second law. Different from the liquid phase, since the P2D model assumes that the positive and negative active particles are spherical particles with the same radius, a spherical coordinate system needs to be used to establish the diffusion equation of lithium ions in the solid particles, and the description area includes the positive electrode and the negative electrode.
③ Liquid phase Ohm's law equation. Since the transfer mode of lithium ions in the electrolyte is diffusion and migration, the change law of the liquid phase potential inside the battery is described by the modified Ohm's law, and the description area includes the positive electrode, the negative electrode and the diaphragm.
④ The solid phase Ohm's law equation. The transfer mode of lithium ions in the solid-phase particles is only diffusion, and the internal solid-phase potential of the battery can be described by Ohm's law. The description area includes the positive electrode and the negative electrode.
⑤ The charge conservation equation. According to the law of conservation of charge, the sum of the liquid phase current density and the solid phase current density at any position inside the power battery is the charge and discharge current density of the power battery. The description area includes the positive electrode, the negative electrode and the diaphragm.
⑥ Butler-Volmer reaction kinetic equation. When an electrochemical reaction occurs on the surface of the electrode active particles, an over-potential will be generated on the surface. This potential is described by the Butler-Volmer equation, and the description area is the solid-liquid interface of the electrode active particles.
The mathematical equations and boundary conditions of the P2D model are summarized in Table 2.
Table 2 Mathematical equations and boundary conditions of the P2D model
Refer to Table 3 for the meaning of each physical quantity in the above table.
Table 3 The meaning of P2D model parameters
3. Main Results
Figure 2a shows the comparison between the simulation results of a cycle and the experimental values under DST conditions, and Figure 2b shows the simulation error of the terminal voltage in the range of 100% to 20% SOC.
Figure 2 (a) Comparison of terminal voltage simulation value and experimental value (b) Simulation error
It can be seen from Figure 2 that in the middle and high SOC section, the electrochemical model has good simulation accuracy, the simulation error is usually kept within 20 mV, and the accuracy is reduced when the current is discharged at a high rate, but it is still kept within 50 mV. The above results show that the simplified P2D model can accurately simulate the characteristics of the power battery.
 R. Xiong. Core Algorithms of Battery Management System. Beijing：China Machine Press，2018. (Chinese) (Chapter Three Section One)
 R. Xiong, L. Li, Z. Li, Q. Yu and H. Mu, "An electrochemical model based degradation state identification method of Lithium-ion battery for all-climate electric vehicles application", Applied Energy, vol. 219, pp. 264-275, June 2018. (Download)
 R. Xiong, L. Li, Q. Y, “Improved Single Particle Model Based State of Charge and Capacity Monitoring of Lithium-Ion Batteries”, 2019 IEEE 89th Vehicular Technology Conference (VTC2019-Spring), 2019. (Download)
 M. Doyle, T.F. Fuller, J. Newman, “Modeling of Galvanostatic Charge and Discharge of the Lithium / Polymer / Insertion Cell”, Journal of the Electrochemical Society, 140(6): 1526–1533, 1993. (Download)
 S.J. Moura, F.B. Argomedo, R. Klein, et al, “Battery State Estimation for a Single Particle Model with Electrolyte Dynamics”, IEEE Transactions on Control Systems Technology, 25(2): 453–468, 2017. (Download)
5. Available Resources
(1) Electrochemical model data: click to download (monograph chapter)