The global push towards “Carbon Peak” and “Carbon Neutrality” has catalyzed an unprecedented transformation in the energy sector, driving the rapid, high-quality development of renewable energy. The vision is to construct a new type of power system where new energy sources play a dominant role. Within this evolving landscape, lithium-ion battery energy storage systems have emerged as a cornerstone technology. Compared to alternatives like nickel-cadmium or nickel-metal hydride batteries, li ion batteries offer superior advantages, including higher cell voltage, longer cycle life, and faster charging capabilities. Consequently, large-scale li ion battery storage power stations are being widely deployed to provide critical grid services such as frequency regulation, peak shaving, and renewable energy integration.
At the core of any effective Battery Management System (BMS) lies a precise and reliable model of the li ion battery. The accuracy of this model fundamentally determines the performance of state estimation (like State of Charge, SOC) and the reliability of safety monitoring. Furthermore, a well-defined model is indispensable for the optimal tuning of control parameters in energy storage systems. Fine-tuned controls enhance the system’s ability to support grid voltage and frequency, thereby improving the stability and efficiency of the new power system. For researchers and engineers, a model that balances accuracy with practicality enables more effective simulations, leading to better insights into the integration challenges of large-scale storage systems with the power grid.
Existing modeling approaches for li ion batteries can be broadly classified into three categories: black-box models, empirical Equivalent Circuit Models (ECMs), and electrochemical mechanism models. Black-box models, often constructed using machine learning algorithms, create a mapping function between external variables like voltage, current, and SOC. While flexible, they lack physical interpretability, require massive amounts of training data, and their performance is highly sensitive to data quality, limiting their widespread application in core BMS functions.
Empirical Equivalent Circuit Models, such as the Rint, RC, or Thevenin models, have gained significant popularity. Their strength lies in their simplicity and intuitive electrical topology. With relatively few parameters, they can be easily described by state-space equations or transfer functions, making them highly convenient for system-level simulation and control parameter tuning. A key advantage is their straightforward scalability from a single cell to a full battery pack, which is crucial for modeling large-scale energy storage stations. However, empirical ECMs suffer from two major drawbacks. First, they are essentially curve-fitting exercises that ignore the internal physicochemical processes of the li ion battery. They can approximate external terminal behavior under specific conditions but cannot explain the internal causes of performance changes or safety issues. Second, their parameters are typically identified through tests like the Hybrid Pulse Power Characteristic (HPPC) test. These parameters are often valid only for a narrow range of operating conditions, necessitating extensive and time-consuming re-calibration across different scenarios to ensure model robustness. This lack of generalizability and physical basis can lead to inefficient management or even safety risks.
On the other end of the spectrum are detailed electrochemical models. The Pseudo-Two-Dimensional (P2D) model is the most comprehensive, describing lithium diffusion in solid particles, ion transport in the electrolyte, and electrochemical kinetics at the interfaces. While highly accurate, its complexity—involving coupled partial differential equations (PDEs)—makes it computationally prohibitive for real-time BMS applications. A powerful simplification is the Single Particle Model (SPM). Under low to moderate charge/discharge rates (typically around 1C and below), the SPM offers an excellent balance between accuracy and computational load. It assumes each electrode can be represented by a single spherical particle, homogenizing the lithium concentration and potential distributions within each electrode. This simplification transforms the complex PDEs into more manageable ordinary differential equations (ODEs) while retaining crucial physical insights into internal states like solid-phase surface concentration. The SPM does not require extensive empirical fitting; its parameters have direct physical meanings and can often be obtained from material datasheets or simple tests.
Despite its merits, the SPM lacks the immediate, intuitive appeal of an electrical circuit representation. It is not naturally suited for direct use in control law derivation or for the straightforward series-parallel aggregation needed for large battery pack simulation. Therefore, a significant research gap exists: a model that marries the physical accuracy and parameter stability of an electrochemical model with the circuit-based convenience and control-friendly formulation of an empirical ECM. This article addresses this gap by deriving a novel Mechanism-Informed Electrical Equivalent Model for li ion batteries. This model is founded on the electrochemical principles of the SPM but expresses them through an electrical circuit topology, complete with clearly defined components whose parameters are explicitly linked to the fundamental electrochemical properties of the li ion battery.
Theoretical Foundation: From Electrochemistry to Circuit Elements
The terminal voltage of a li ion battery, according to electrochemical principles, can be expressed as the sum of equilibrium potentials and various overpotentials. In the framework of the Single Particle Model, this is given by:
$$V(t) = E(t) + \eta_{\text{SEI,p}}(t) – \eta_{\text{SEI,n}}(t) + \eta_{\text{act,p}}(t) – \eta_{\text{act,n}}(t) + \eta_e(t)$$
where:
- $V(t)$ is the terminal voltage.
- $E(t)$ is the open-circuit voltage (OCV), a thermodynamic property.
- $\eta_{\text{SEI,p}}, \eta_{\text{SEI,n}}$ are the overpotentials due to the Solid-Electrolyte Interphase (SEI) film resistance on the positive and negative electrodes.
- $\eta_{\text{act,p}}, \eta_{\text{act,n}}$ are the activation overpotentials from the electrochemical reaction kinetics at the electrode surfaces.
- $\eta_e(t)$ is the ohmic overpotential due to ion transport in the electrolyte.
The core innovation of the proposed model lies in re-interpreting these electrochemical terms as distinct electrical circuit segments.
1. The Open-Circuit Voltage Sub-Circuit: Capturing Solid-Phase Diffusion
The OCV, $E(t)$, is not a constant voltage source. It varies with the lithium concentration at the surface of the active material particles in both electrodes, a process governed by solid-phase diffusion. The surface concentration ($c_{s,\text{surf}}$) dynamics relative to the average concentration ($c_{s,\text{avg}}$) can be effectively approximated in the frequency domain using a two-state model for each electrode (i=p for positive, n for negative):
$$
\frac{\hat{c}_{s,\text{surf},i}(s)}{\hat{j}_n,i(s)} = -\frac{R_{s,i}}{3} \left( \frac{\lambda_1}{5D_{s,i}T_{1,i}s + 1} + \frac{\lambda_2}{5D_{s,i}T_{2,i}s + 1} \right)
$$
$$
\hat{c}_{s,\text{avg},i}(s) = -\frac{1}{R_{s,i}s} \hat{j}_{n,i}(s)
$$
where $j_n$ is the pore-wall flux, $R_s$ is the particle radius, $D_s$ is the solid-phase diffusion coefficient, and $T_1, T_2, \lambda_1, \lambda_2$ are time constants and weighting factors derived from the diffusion dynamics.
The OCV is a nonlinear function of the electrode’s state of charge, defined as $\theta_i = c_{s,\text{surf},i} / c_{s,\text{max},i}$. By linearizing the OCV curve $E_i(\theta_i)$ around an operating point, we get $E_i \approx U_{b,i} + k_{\text{ref},i} \theta_i$, where $U_{b,i}$ is a voltage source and $k_{\text{ref},i}$ is a proportional coefficient. Combining this linearization with the diffusion dynamics equations and the relationship between flux and external current $I(t)$, we can derive an equivalent electrical circuit for the OCV. This circuit physically represents how the slow diffusion process modulates the available equilibrium voltage.
The derived OCV sub-circuit for the full cell consists of two such electrode circuits in series opposition (positive minus negative). The final topology for this segment includes:
- Two integrating capacitors ($C_{1,p}$, $C_{1,n}$): These represent the main storage of charge, related to the total lithium inventory in each electrode. Their voltage corresponds to the integrated effect of current on the average lithium concentration.
- Four RC parallel branches (Two per electrode: $R_{2,i}C_{2,i}$ and $R_{3,i}C_{3,i}$): These represent the dynamic voltage drops associated with the relaxation of the lithium concentration gradient within the solid particles. They capture the transient voltage response when current changes.
- Two linearized voltage sources ($U_{b,p}$, $U_{b,n}$): These provide the baseline voltage from the linearized OCV relationship.
The parameters of these electrical components are not arbitrary; they are explicit functions of the li ion battery’s electrochemical and geometric properties:
$$
C_{1,i} = \frac{3 \varepsilon_{s,i} A L_i F c_{s,\text{max},i}}{k_{\text{ref},i}}, \quad R_{2,i} = \frac{5 k_{\text{ref},i} R_{s,i}}{\varepsilon_{s,i} A L_i F c_{s,\text{max},i} D_{s,i} \lambda_1}, \quad C_{2,i} = T_{1,i} / R_{2,i}
$$
$$
R_{3,i} = \frac{5 k_{\text{ref},i} R_{s,i}}{\varepsilon_{s,i} A L_i F c_{s,\text{max},i} D_{s,i} \lambda_2}, \quad C_{3,i} = T_{2,i} / R_{3,i}
$$
where $A$ is electrode area, $L_i$ is electrode thickness, $F$ is Faraday’s constant, and $\varepsilon_{s,i}$ is solid-phase volume fraction. The time constants $T_{1,i}$ and $T_{2,i}$ are intrinsic properties determined solely by $R_{s,i}$ and $D_{s,i}$.
2. The Reaction and Ohmic Impedance: Equivalent Series Resistance
The remaining voltage terms in the equation—activation, SEI, and electrolyte overpotentials—are all linearly or approximately linearly proportional to the instantaneous current $I(t)$ under low-rate conditions. They can be aggregated into a single equivalent series resistance $R_{\Sigma}$.
Activation Overpotential & Equivalent Reaction Resistance: The Butler-Volmer equation describes the reaction kinetics. Under the low-polarization assumption typical for SPM validity, it simplifies, and the activation overpotential becomes approximately linear with current:
$$
\eta_{\text{act},i} \approx \pm \frac{RT}{F \alpha_{a,i}} \cdot \frac{I(t)}{a_i A L_i j_{0,i}} = r_{\eta,i} \cdot I(t)
$$
where $j_{0,i}$ is the exchange current density, and $a_i$ is the specific surface area. The term $r_{\eta,i}$ is defined as the electrochemical reaction equivalent resistance, directly calculable from kinetic parameters:
$$
r_{\eta,i} = \pm \frac{RT}{F \alpha_{a,i} a_i A L_i j_{0,i}} = \pm \frac{RT}{2F \alpha_{a,i} a_i A L_i \cdot r_{k,i} (c_e)^{\alpha_{a,i}} (c_{s,\text{max},i} \theta_i)^{\alpha_{a,i}} (c_{s,\text{max},i}(1-\theta_i))^{\alpha_{a,i}}}
$$
Total Equivalent Series Resistance $R_{\Sigma}$: The total resistance in the model’s circuit is the sum of the reaction resistances for both electrodes, the SEI film resistances, and the electrolyte ohmic resistance:
$$
R_{\Sigma} = r_{\eta,p} – r_{\eta,n} + R_{\text{SEI,equiv}} + R_{e}
$$
where $R_{e}$ is calculated from the ionic conductivity and geometry of the electrolyte-filled pores in the electrodes and separator.
The Complete Mechanism-Informed Electrical Equivalent Model
Synthesizing the OCV sub-circuit and the series resistance, we arrive at the complete circuit topology for the proposed li ion battery model.
The model’s state-space representation is highly convenient for control and simulation. The state vector $\mathbf{x}$ consists of the voltages across the six capacitors ($V_{1,p}, V_{1,n}, V_{2,p}, V_{2,n}, V_{3,p}, V_{3,n}$). The state-space equations are:
$$
\dot{\mathbf{x}} = \mathbf{A} \mathbf{x} + \mathbf{B} I(t)
$$
$$
V(t) = \mathbf{C} \mathbf{x} + U_{b} – R_{\Sigma} I(t)
$$
where the system matrix $\mathbf{A}$ is a diagonal matrix with entries $0, 0, -1/\tau_{2,p}, -1/\tau_{2,n}, -1/\tau_{3,p}, -1/\tau_{3,n}$ ($\tau = R \cdot C$), input matrix $\mathbf{B}$ contains the inverses of the capacitances, and output matrix $\mathbf{C}$ is $[1, -1, 1, -1, 1, -1]$. The model output clearly separates the dynamic OCV component ($\mathbf{C}\mathbf{x} + U_b$) from the instantaneous ohmic drop ($R_{\Sigma}I(t)$).
Model Analysis, Application, and Advantages
The proposed model provides a direct bridge between li ion battery design, control, and monitoring.
Design Insights: The circuit components offer immediate design guidance. For higher capacity (linked to $C_{1,i}$), designers should increase electrode volume ($A L_i$), active material loading ($c_{s,\text{max},i}$), or porosity ($\varepsilon_{s,i}$). To reduce energy loss and voltage sag (linked to $R_{\Sigma}$), designers should aim for higher porosity, smaller particle size ($R_{s,i}$), and electrolytes with higher ionic conductivity. The transient response, governed by the RC time constants $\tau_{2,i}, \tau_{3,i}$, is fixed by material properties ($R_{s,i}, D_{s,i}$), guiding material selection for power performance.
Application Workflow:
- Parameterization: Obtain fundamental electrochemical parameters (e.g., from datasheets or characterization tests).
- Initialization: Calculate initial SOC ($\theta_i$) and set initial capacitor voltages (e.g., $V_{1,i,0} = k_{\text{ref},i}(\theta_{i,0})$).
- Online Operation: At each time step:
- Update $\theta_i$ using a coulomb counting approach informed by the model’s capacity parameters ($C_{1,i}$).
- Update all circuit parameters ($R$’s, $C$’s, $k_{\text{ref},i}$, $U_{b,i}$, $R_{\Sigma}$) as functions of the new $\theta_i$.
- Solve the state-space equations with the updated matrices to get the terminal voltage $V(t)$.
This workflow seamlessly integrates the physical state (lithium concentration) into the electrical model parameters, eliminating the need for separate, extensive HPPC-based lookup tables required by empirical ECMs.
Simulation Validation and Performance
The model’s performance was validated against a high-fidelity numerical solution (COMSOL) and a conventional 2nd-order RC empirical ECM. A standard Li-ion battery with parameters from the literature was used for simulation under various constant current (0.2C, 0.5C, 1C, 2C) and dynamic 1C profiles.
| Discharge Rate (C-rate) | Mechanism-Informed ECM Error | 2nd-Order Empirical ECM Error |
|---|---|---|
| 0.2C | < 1 mV | ~5 mV |
| 0.5C | < 1 mV | ~10 mV |
| 1C | ~5 mV | ~20 mV |
| 2C | ~20 mV | ~30 mV (with peaks >300 mV) |
The results demonstrate the superior accuracy of the mechanism-informed model, especially at 1C and below, aligning perfectly with the SPM’s domain of validity. The empirical ECM shows larger errors and significant deviation during the dynamic test, exhibiting “lag” due to its fixed parameters. In contrast, the proposed model’s parameters adapt based on the internal state ($\theta_i$), allowing it to accurately track the voltage even during transients. The increase in error at 2C for the proposed model is expected, as the underlying SPM assumptions start to break down at higher rates where electrolyte transport limitations become significant.
Conclusion
This article has presented a novel Mechanism-Informed Electrical Equivalent Model for li ion batteries. By deriving electrical circuit elements directly from the governing equations of the electrochemically-accurate Single Particle Model, this approach successfully bridges a critical gap in energy storage modeling. The proposed model retains a clear, intuitive circuit topology amenable to state-space formulation, control analysis, and scalable pack modeling—the key strengths of empirical ECMs. Simultaneously, it incorporates the physical fidelity, parameter stability, and internal state observability of electrochemical models. The parameters of its resistors and capacitors are not arbitrary fitted values but are explicit functions of the li ion battery’s fundamental material properties and geometric design.
The model is particularly well-suited for applications within the new power system paradigm. It enables more accurate and reliable simulations of large-scale li ion battery storage stations, providing a better tool for studying grid integration challenges. For system operators and BMS designers, it offers a physically-grounded framework for precise control parameter tuning, enhancing the support functions that storage provides to the grid. Furthermore, by linking external electrical behavior to internal electrochemical states like electrode-level SOC ($\theta_i$), it provides a more profound basis for advanced state estimation and proactive safety monitoring, moving beyond the limitations of purely empirical modeling. This work contributes a valuable tool towards the safer, more efficient, and more predictable integration of li ion battery technology into our sustainable energy future.
| Electrochemical Parameter | Physical Meaning | Primary Influence on Electrical Model |
|---|---|---|
| $c_{s,\text{max},i}$, $\varepsilon_{s,i}$, $A$, $L_i$ | Total active lithium inventory, electrode size | Main storage capacitance $C_{1,i}$ (capacity) |
| $R_{s,i}$, $D_{s,i}$ | Particle size, solid diffusion coefficient | RC branch time constants $\tau_{2,i}, \tau_{3,i}$ (transient response) |
| $r_{k,i}$, $\alpha_{a,i}$, $c_e$ | Reaction rate, charge transfer symmetry, electrolyte concentration | Reaction equivalent resistance $r_{\eta,i}$ |
| $\kappa_i^{\text{eff}}$, $L_{\text{sep}}$ | Effective electrolyte conductivity, separator thickness | Electrolyte ohmic resistance $R_e$ |
| $R_{\text{SEI},i}$ | SEI film resistance | Series resistance component |