Single-particle models (SPM) are a class of electrochemical models used to simulate the behavior of lithium-ion batteries by simplifying the complex physicochemical processes occurring within the cell. These models are widely adopted in battery management systems (BMS) and real-time applications due to their computational efficiency, making them suitable for embedded systems where resources are limited. The core idea behind SPM is to reduce the multi-scale, multi-physics problem of battery dynamics into a more tractable form by focusing on the electrode particles while making strategic simplifications.

The primary simplification in SPM is the assumption that each electrode can be represented by a single spherical particle, hence the name. This means that the lithium concentration gradients within the active material of the electrodes are modeled as radial diffusion processes in a single representative particle for the anode and cathode. By doing so, the model ignores the spatial distribution of particles across the electrode thickness, which is a key feature of more complex models like the pseudo-two-dimensional (P2D) model. Additionally, SPM neglects the dynamics of the electrolyte, assuming that the electrolyte potential and concentration remain uniform or vary insignificantly during operation. This simplification drastically reduces the number of equations to solve, enabling faster computation.

Another simplification in SPM is the treatment of the solid-electrolyte interphase (SEI) and charge transfer kinetics. While more detailed models account for the potential drop across the SEI layer and the nonlinear Butler-Volmer kinetics, SPM often uses a linearized approximation for the reaction rates. This linearization is valid under small current perturbations but may introduce errors under high C-rate conditions where nonlinear effects become significant. The ohmic losses in the current collectors and external circuit are also typically ignored or lumped into an effective internal resistance.

The trade-offs between computational speed and accuracy are a defining characteristic of SPM. By neglecting electrolyte dynamics and spatial variations in electrode properties, SPM achieves a computational cost that is orders of magnitude lower than that of the P2D model. For example, while a P2D model may require solving partial differential equations (PDEs) in multiple dimensions, SPM reduces the problem to ordinary differential equations (ODEs) or even algebraic equations in some cases. This makes SPM particularly attractive for real-time applications such as state-of-charge (SOC) estimation, state-of-health (SOH) monitoring, and power prediction in BMS. However, the accuracy of SPM deteriorates under conditions where the simplifications break down, such as high current loads, low temperatures, or when electrolyte depletion or polarization effects become significant.

In comparison to the P2D model, SPM lacks the ability to capture phenomena like electrolyte concentration gradients, which can be critical for predicting cell performance under extreme operating conditions. The P2D model accounts for lithium transport in both the solid phase (particles) and the liquid phase (electrolyte), as well as the potential distribution across the cell. This makes it more accurate for detailed design and analysis but computationally prohibitive for real-time control. SPM, on the other hand, provides a middle ground where reasonable accuracy is maintained for moderate operating conditions while keeping computational demands low.

Use cases for SPM are predominantly found in BMS and onboard vehicle applications. For SOC estimation, SPM can be coupled with filtering techniques like the Kalman filter to provide real-time updates on the battery's charge state. The model's simplicity allows it to run on microcontrollers with limited processing power, making it feasible for deployment in electric vehicles and portable electronics. Similarly, SPM is used for power capability prediction, where the battery's maximum charge and discharge currents are estimated to prevent overloading or damage. In these applications, the model's fast execution enables frequent updates, ensuring timely responses to changing load conditions.

Another application of SPM is in battery control algorithms for hybrid and electric vehicles. By predicting the voltage response under different current profiles, the model helps optimize energy usage and prolong battery life. For example, SPM can be used to design charge protocols that minimize stress on the electrodes while maintaining acceptable charging times. The model's efficiency also makes it suitable for large-scale simulations, such as those used in fleet management or grid storage systems, where thousands of cells must be monitored simultaneously.

Despite its advantages, SPM is not a one-size-fits-all solution. Its limitations become apparent in scenarios requiring high precision, such as detailed cell design or failure analysis. In these cases, the P2D model or even more refined approaches are necessary. However, for many real-world applications where speed and simplicity are prioritized, SPM remains a valuable tool. Ongoing research aims to improve SPM by incorporating selective enhancements, such as empirical corrections for electrolyte effects or hybrid models that combine SPM with data-driven techniques. These advancements seek to extend the model's applicability without sacrificing its computational benefits.

In summary, single-particle models offer a pragmatic balance between accuracy and computational efficiency for lithium-ion battery simulation. By focusing on the essential dynamics of electrode particles and omitting less critical details, SPM enables real-time applications that would be infeasible with more complex models. While it cannot replace high-fidelity models in all scenarios, its role in BMS and control systems underscores its importance in the practical deployment of battery technologies. As computational resources and modeling techniques evolve, SPM continues to adapt, maintaining its relevance in an increasingly demanding field.