Lithium-ion batteries (LIBs) have become one of the popular energy storage devices in various high-power applications, such as power grid systems and electric vehicles. The advancement of simulation techniques is the important task related to the design, performance, and safety of such systems. Currently, simulations of the battery performance using the multiple resistor-capacitor branches equivalent circuit model (Foster-type model) have been carried out. Although this model is easy to calculate the battery voltage and the current through the cell, it is a behavioral model, deviating from the actual electrode structure and the electrochemical reaction. Therefore, it is unlikely that the equivalent circuit represents the electrochemical physics of the battery performance.

Our objective here is to develop a unique equivalent circuit model working in conjunction with numerical calculations according to electrochemical theories. Authors demonstrate that details of the electrochemical reaction in the porous electrode, such as distributions of the overvoltage, the reaction current, and the ion concentration, can be calculated on a circuit simulator by using this model.

In a simulation technique presented below, both currents through each electric elements and terminal voltages on the equivalent circuit are calculated by the circuit simulator running on MATLAB/Simulink, and these are used as boundary conditions to calculate the Butler-Volmer equation combined with diffusion equations. Thereby a circuit simulation and electrochemical calculations are coupled and the battery performance is calculated according to both theories of the electric circuit and the electrochemistry. The Butler-Volmer equation and diffusion equations are solved by using numerical calculation function implemented in the Simulink.

In developing LIBs model, it is very important to express the electrochemical reaction of the porous electrode. This is because the reactive sites spread in the electrode thickness direction, so reaction conditions vary depending on the position of the electrode depth. Therefore, in a equivalent circuit model of LIBs, the porous electrode structure must be reflected. In this simulation technique, a equivalent circuit of a transmission line model (TLM) of the porous electrode is adopted (see Figure 1). The electrolyte resistance of pore inside (Re’), the reaction resistance (Rf’), and the electric double layer capacitance (Cdl’) are given for each segment of electrode pores divided into 100. By using a TLM, it is expected that distributions of the potential and the current in the actual porous electrode can be expressed.

Figure 2 shows one segment in a TLM and the method of estimating Rf ' is described below. First, the electric double layer voltage (= overvoltage) is calculated by a circuit simulation, and then both the reaction current and the Li-flux of the intercalation/deintercalation through the electrolyte-electrode interface are calculated by using the Butler-Volmer equation as a function of the overvoltage. Second, the Li ion concentration [Li+] in the electrolyte, the Li concentration [Li] and the vacant site concentration [θ] in the electrode are estimated by solving diffusion equations under the boundary condition of the Li-flux. Moreover, [Li+], [Li], and [θ] are fed back to the Butler-Volmer equation. Third, value of the Rf’ is determined by dividing the overvoltage by the reaction current. Thus, Rf’ is modeled as a variable resistor which changes with overvoltage. Forth, the calculated Rf’ is transferred to the equivalent circuit. Thereby the equivalent circuit is updated from the initial state and the battery current, the electric double layer charging current, the reaction current, and the overvoltage are recalculated on a circuit simulator. The above is the calculation sequence in one calculation step, and this calculation sequence is repeated for each calculation step.

The above calculation is carried out for the variable Rf’ element of all segments in Figure 1. Reaction conditions (ex. overvoltage) for each segment is automatically calculated according to a TLM on a circuit simulator. Therefore, Rf’ is a variable resistance which depends on the electrode thickness direction. Electrolyte resistances of the pore outside (Rsol’), Re’, and Cdl’ are estimated from the electrochemical impedance measurement. Diffusion coefficients etc. are determined from literature values.

Figure 3 shows the transient current curve at the potential step of 0.5 V obtained with the positive electrode of LIBs. The experimental and the simulated currents were found to be in excellent agreement. In this simulation technique, the behavior of the electric double layer charging current immediately after the potential step and the steady reaction current can be calculated. Moreover, distributions of reaction current and overvoltage in the electrode thickness direction can be estimated (see Figure 4).