Venkat R. Subramanian

Solving the solid-state diffusion equation in a typical pseudo-two-dimensional (P2D) battery model using conventional finite difference methods (FDM) substantially increases the model complexity (in terms of number of equations) as well as the computational time. In our previous work, we approximated the solid-state diffusion in spherical particles using Galerkin weak form method based on symmetric polynomials, which significantly improved the robustness of modeling, achieving a tenfold reduction in computational time, even for non-linear diffusion problems. However, we found that the equations represented in non-dimensional form for mathematical compactness are relatively difficult to interpret and apply in P2D and other battery models. Therefore, as a follow-up to previous work, we present the weak form formulations of solid-state diffusion in dimensional form to provide better physical interpretation and easier adaptability to battery modeling and other applications. Further, considering diverse particle morphologies in modern battery materials, we extend these formulations for cylindrical and rectangular coordinates, in addition to the spherical coordinate. We also present simulations using various experimentally reported concentration-dependent diffusivities and find that the solution convergence is strongly influenced by the degree and order of non-linearity in diffusion. The study enables the proposed formulation to be widely applicable facilitating accurate and fast simulations.

Dynamic operating conditions significantly impact the lifetime of lithium-ion batteries in electric vehicles. While battery lifetime can be extended by optimizing charging profiles to reduce degradation, many existing charging optimization approaches are developed based on fixed, idealized full charge-discharge cycles. These differ from the random and partial charge-discharge behavior in real-world operation and do not consider the influence of battery degradation on charging current profile optimization. To address these limitations, this study designs two groups of battery aging tests to study charging optimization in real-world operation: one subjected to four fixed charging scenarios based on typical daily commuting patterns, and the other to dynamically changing charging scenarios based on state of health change. Capacity degradation, internal resistance increase, and charging time of all cells are analyzed and compared. Degradation modes such as loss of active materials and lithium inventory are examined through incremental capacity and differential voltage analyses. A novel tanks-in-series thermal-aging model is proposed to rapidly simulate battery behavior under dynamic charging, enabling rapid exploration of more charging scenarios constrained by experimental channels or costly to perform. Results demonstrate that dynamically switching charging strategies based on state of health can effectively extend battery lifetime while reducing overall charging time. Moreover, the model proves efficient in identifying optimal charging strategies. These findings offer valuable insights into charging optimization considering practical use scenarios, and present a promising tool for charging optimization.

Developing resilient aqueous energy storage systems, such as Zn batteries, is essential for advancing energy sustainability. A key challenge in these systems is the dendritic growth of metals, which causes poor rechargeability in battery electrodes. Classical diffusion-limited dendritic growth is predicted to occur when the charging rate exceeds the limiting current density J lim determined by electrolyte transport properties. Understanding the electrochemical behavior of dendrites is critical for designing fast-charging metal electrodes, where dendritic growth is not only likely but sometimes unavoidable due to uneven current distribution. Conventional wisdom suggests that high-aspect-ratio fractal dendrites are susceptible to bottom-initiated dissolution, leading to mechanical break-off from the current collector and the formation of “dead” metal. Surprisingly, our results show that near-unity charge–discharge reversibility can be achieved even with highly ramified classical diffusion-limited dendritic Zn metal structures. In particular, the reversibility improves with an increasing discharge rate, demonstrating a strong positive correlation. A two-orders-of-magnitude (∼200×) enhancement in cycle life is observed when Zn electrodes charged identically fast, but discharged at higher versus lower rates. Operando visualization reveals that dendrite fragmentation is significantly suppressed at higher discharge rates. Complementary post-mortem microstructural analysis shows that, consistent with predictions based on the Wagner number (Wa), high-rate discharges promote preferential tip-initiated stable retraction, whereas low-rate discharges induce “pitting” corrosion that mechanically weakens the dendrite backbone and promotes fragmentation. These findings challenge the prevailing assumption that dendritic growth necessarily limits the reversibility and offer new design principles for metal battery electrodes operating at unprecedented high rates approaching the diffusion limit.

Fast charging is being heavily researched for the widespread implementation of lithium-ion batteries for electric vehicles. However, charging at high currents accelerates several parasitic reactions that lead to the degradation of the cell, affecting its lifetime. It is possible to study material degradation mechanisms and predict their impact on capacity loss under several operating conditions using physics-based multi-scale battery models. These models can be integrated with battery management systems (BMSs) to control the cell’s performance and to design novel charging protocols that enable safe and optimal cell performance and suppress cell degradation. Our group has successfully applied BMS 2 based on a physics-based battery model to improve life and reduce charging time for different batteries (as shown in Figure 1). This seminar will present some results from our group for cells, modules, and packs. The talk will also include the theoretical development of pulse profiles as predicted by optimal control of phase-field models. 3 Model-based BMS algorithms require fast and efficient production codes that can predict and estimate battery parameters in real-time and control the battery’s performance under different loads. The theory is reasonably well developed and defined for one or all of (a) ordinary differential equations (ODEs), (b) elliptic partial differential equations (PDEs) with homogenous boundary conditions, and (c) well-conditioned linear equations . Solvers, optimizers, and software have been well developed for the same by optimizing/utilizing one or all of (a) high-performance computing, (b) GPU acceleration, (c) adaptive time stepping, (d) adaptive mesh refining, (e) estimation, (f) parallel computing, (g) efficient Jacobian/Hessian/adjoint calculation, and (h) sparse linear algebra . When physics-based models are taken to BMS – solvers and optimizers can (and often) fail. A Google search (aided by Generative AI) on “where solvers fail” returns four common scenarios (not all of them are accurate or relevant). Numerical issues & convergence issues – Inaccurate/inconsistent models, stiff problems, inconsistent problems, tolerances, nonunique solutions, mesh quality. Model complexity & structure – High nonlinearity, rigid body motion, unconstrained bodies, missing contacts/fixtures. Solver-related issues – Limitations, bugs/errors, infeasible solutions. Other-potential issues – Results/folder permissions, corrupt study, installation issues. Our efforts to alleviate some of these issues for battery models will be presented. Challenges, benefits, and pitfalls of disguising Differential Algebraic Equations (DAEs) 4 as ODEs, advective phase-field/level set PDEs as elliptic PDEs, 5 unconstrained optimization problems as constrained optimization problems, immersed-interface approach to solving moving boundaries, multi-phase problems will be presented. A digital twin is a virtual representation of an object or system designed to reflect a physical object accurately. 6 Demonstrated fast computational capabilities from our group enable the development of robust and efficient Functional Mock-up Unit (FMU) based digital twin for batteries. 7 This enables scaleup and adaption to different chemistries from cell, pack and module level. Preliminary results from the same indicate more than 40% reduction in development costs and time. Disclaimer & References https://wheresolversfail.blogspot.com/. Venkat hopes to write (update) and maintain a blog on “Where Solvers Fail.” He plans to write a book on this topic. The title is inspired by the movie “Where Eagles Dare” (without any political affiliation). The importance/relevance/pitfalls of disguise and deception carry over to simulation/optimization and control of electrochemical systems and batteries. M. Pathak, D. Sonawane, S. Santhanagopalan, R. D. Braatz, and V. R. Subramanian, “Analyzing and Minimizing Capacity Fade through Optimal Model-based Control – Theory and Experimental Validation”, ECS Trans., 75 (23), 51-75 (2017). T. K. Telmasre, A.C. Concepción, S. Kolluri, L. Mishra, R.S. Thiagarajan, A.N. Matam, A. Subramaniam, T.R. Garrick, and V. R. Subramanian, “Perspective—Moving Next-Generation Phase-Field Models to BMS Applications: A Case Study that Confirms Professor Uzi Landau’s Foresight”, J. Electrochem. Soc., 171, 063507 (2024). L. Petzold, “Differential/Algebraic Equations are not ODE’s”, SIAM Journal on Scientific and Statistical Computing 3 (3), 367 (1982). T. Jang, L. Mishra, S. A. Roberts, B. Planden, A. Subramaniam, M. Uppaluri, D. Linder, M. P. Gururajan, J. Zhang, and V. R. Subramanian, “BattPhase—A Convergent, Non-Oscillatory, Efficient Algorithm and Code for Predicting Shape Changes in Lithium Metal Batteries Using Phase-Field Models: Part I. Secondary Current Distribution”, J. Electrochem. Soc., 169, 080516 (2022). https://www.ibm.com/think/topics/what-is-a-digital-twin https://battgenie.life/functional-mock-up-units-fmus-part-1/ Figure 1

Physics-based electrochemical models are essential tools that facilitate the simulation of electrochemical and transport phenomena inside the battery, enable battery parameter estimation, and inform control-based cycling protocols for lithium-ion (Li-ion) batteries. BMS implemented with physics-based models can also estimate the state-of-charge (SOC) and state-of-health (SOH) of the battery. Physics-based models consist of coupled differential-algebraic equations (DAEs) that can vary in complexity and simulation time based on the granularity of the underlying physics. [1] Modeling solid phase diffusion within electrodes during charging/discharging is an integral part of model development for simulating batteries. Fick’s laws have been traditionally used for this purpose, which provides reasonable predictions in the case of non-phase transforming materials. [2] However, studies have revealed that cathode materials such as LFP and anode materials such as graphite can and do undergo phase transformation when subjected to intercalation/deintercalation.[3] In such models, fickian diffusion does not provide a sufficient mathematical description of the underlying physical phenomenon.[4] In our previous work, we sought to replace the fickian model for diffusion with phase field models.[5] Being thermodynamically more accurate, phase field models are powerful with fewer parameters to fit and provide an excellent description of phase change behavior. It was shown that by tweaking the ‘Enthalpy of Mixing per site’ parameter i.e. ω, it is possible to obtain a single-phase behavior (low ω) as well as multi-phase behavior (high ω). Further, optimization studies were performed on the phase field models to reduce the concentration gradient within the particle to achieve faster rate of charging without sacrificing the particle integrity via Cracking.[5] Phase field models predicted a pulsed profile which is a proven fast charging protocol utilized by many in the experimental setting.[6] In this work, we evaluate the validity of phase field models with the addition of more physics and domains in the system. We evaluate a full cell model where cathode is represented by LFP type phase change material (Phase field) and anode is non phase change material (Fick’s law). We further seek to implement optimization studies and try to find an answer to the question: Are pulsed profiles really the truly optimized profiles when dealing with a phase change material electrode? References: [1] V. Ramadesigan, P. W. C. Northrop, S. De, S. Santhanagopalan, R. D. Braatz, and V. R. Subramanian, “Modeling and simulation of lithium-ion batteries from a systems engineering perspective,” J Electrochem Soc , vol. 159, no. 3, pp. R31–R45, 2012, doi: 10.1149/2.018203jes. [2] M. Doyle, T. F. Fuller, and J. Newman, “Modeling of galvanostatic charge and discharge of the lithium/polymer/insertion cell,” J Electrochem Soc , vol. 140, no. 6, pp. 1526–1533, 1993, doi: 10.1149/1.2221597. [3] K. E. Thomas-Alyea, C. Jung, R. B. Smith, and M. Z. Bazant, “In Situ Observation and Mathematical Modeling of Lithium Distribution within Graphite,” J Electrochem Soc , vol. 164, no. 11, pp. E3063–E3072, 2017, doi: 10.1149/2.0061711jes. [4] Y. Zeng and M. Z. Bazant, “Phase separation dynamics in isotropic ion-intercalation particles,” SIAM J Appl Math , vol. 74, no. 4, pp. 980–1004, 2014, doi: 10.1137/130937548. [5] T. K. Telmasre et al. , “Moving next-generation phase-field models to BMS applications - A case study that confirms Professor Uzi Landau’s foresight,” Journal of Electrochemical Society , vol. Under Review. [6] B. K. Purushothaman and U. Landau, “Rapid Charging of Lithium-Ion Batteries Using Pulsed Currents,” J Electrochem Soc , vol. 153, no. 3, p. A533, 2006, doi: 10.1149/1.2161580.

The lifetime of lithium-ion batteries in electric vehicles is partly influenced by dynamic operational conditions, such as driver behavior 1 . Although numerous optimized charging protocols have been proposed to reduce charging time and degradation, most are fixed full-charge strategies that do not account for battery aging or real-world usage patterns 2 . This study investigates how adapting charging strategies based on different degradation levels affects both battery lifetime and charging time. We first designed four charging strategies to simulate typical daily commuting scenarios. Two sets of aging tests were then conducted: in the first, four cells were aged using the fixed charging scenarios; in the second, four cells were aged using dynamically adjusted charging scenarios triggered by a 5% loss in state of health (SOH). Results showed that switching from fast to slower charging can extend battery lifetime while reducing overall charging time. The SOH gains for three pairs of compared cells were 1.3%, 0.39%, and 0.65% under the same full-equivalent cycles, accompanied by reductions in total charging time of 11.86%, 9.31%, and 4.92%, respectively. Incremental capacity analysis and differential voltage analysis revealed that the rates of lithium inventory loss and active material degradation varied across charging scenarios. To explore additional incomplete tests and alternative optimal charging strategies, we developed a tanks-in-series coupled with thermal and aging models 3 . The model demonstrated high accuracy, with maximum RMSEs of 88.5 mV (voltage), 0.9 K (temperature), and 1.36% (SOH), while requiring a maximum simulation time of 76.88 hours for two-year aging tests. Simulations indicated that switching charging strategies could extend the battery by up to 253 full-equivalent cycles at the same degradation level. X. Li, D. Yu, S. B. Vilsen, and D. I. Stroe, Journal of Energy Chemistry , 92 , 591–604 (2024). X. Li, D. Yu, S. B. Vilsen, V. R. Subramanian, and D.-I. Stroe, IEEE Transactions on Transportation Electrification , 11 , 6279–6290 (2025). A. Subramaniam, S. Kolluri, S. Santhanagopalan, and V. R. Subramanian, J Electrochem Soc , 167 , 113506 (2020).

Batteries are vital for enabling renewable energy use and reducing carbon emissions, with lithium-ion (Li-ion) batteries favored for their high energy density and fast charging. High-voltage battery packs use thousands of Li-ion cells in complex configurations, where extrapolation from single-cell models fails to predict overall behavior due to cell-to-cell variation, thermal gradients, and manufacturing inconsistencies. Over-discharge, an electrochemical abuse condition, can lead to irreversible damage, capacity degradation, and safety hazards. Critically, overdischarge events cannot be reliably detected at the individual cell level within a module, complicating early intervention strategies. To address this, a simulation framework using the thermal Tanks-in-Series (TiS) model for individual cells and packs has been developed and compared with the pseudo-two-dimensional (p2D) model, enabling accurate and fast prediction of pack-level performance. The model is utilized to investigate the impact of non-uniformities among the cells in the pack both at initial cycle and after long-term cycling. The impact of their spatial location within the pack is also studied. The developed framework accurately predicts the pack-level performance and is computationally efficient, facilitating improved battery design.

The Li-ion battery is an electrochemical system having only voltage and current as measurable outputs; therefore, it requires special characterization techniques and mathematical models to non-destructively diagnose the state of battery. [1] Electrochemical impedance spectroscopy (EIS) being a non-invasive technique[2] (due to the small amplitude input modulation) is a powerful diagnostic and prognostic tool for batteries. EIS deconvolutes electrochemical processes occurring simultaneously based on their characteristics phenomenological time scales[3]. In batteries, EIS has been used to measure electrode materials' kinetics and transport properties[4], [5] and estimate the state of health and remaining useful life. Physics based EIS models have been reported in literature [6], [7], [8] and improve upon shortcomings of traditional equivalent circuit-based models such as their empirical nature and degeneracy. [9], [10] Physics based EIS models can thus estimate experimentally measurable thermodynamic and kinetic parameters with more accuracy and fidelity to the physical processes occurring within the batteries. This work extends EIS models based on SPM and Tanks-in-series models [11] with the addition of degradation mechanisms such as SEI layer, Lithium plating, Particle cracking and more. It presents an alternative framework for estimating thermodynamic, kinetic and degradation parameters with the help of impedance models (frequency domain) providing support to the existing voltage time (time domain) estimation models. We identify the relative strengths of the frequency domain and time domain models in estimating various battery parameters. Finally, we provide meaningful predictions about the State of Health (SOH) and Remaining Useful Life (RUL) of the battery using model predictions and compare it with experimental data collected at different stages of life of a commercial cell. References: [1] V. Ramadesigan, P. W. C. Northrop, S. De, S. Santhanagopalan, R. D. Braatz, and V. R. Subramanian, “Modeling and simulation of lithium-ion batteries from a systems engineering perspective,” J Electrochem Soc , vol. 159, no. 3, pp. R31–R45, 2012, doi: 10.1149/2.018203jes. [2] A. Barai et al. , “A comparison of methodologies for the non-invasive characterisation of commercial Li-ion cells,” Prog Energy Combust Sci , vol. 72, pp. 1–31, 2019, doi: 10.1016/j.pecs.2019.01.001. [3] D. D. Macdonald, “Why electrochemical impedance spectroscopy is the ultimate tool in mechanistic analysis!,” ECS Trans , vol. 19, no. 20, pp. 55–79, 2009, doi: 10.1149/1.3247566. [4] S. Gao, X. Zhan, and Y. T. Cheng, “Structural, electrochemical and Li-ion transport properties of Zr-modified LiNi0.8Co0.1Mn0.1O2 positive electrode materials for Li-ion batteries,” J Power Sources , vol. 410–411, no. September 2018, pp. 45–52, 2019, doi: 10.1016/j.jpowsour.2018.10.094. [5] R. Shanmugam and W. Lai, “Study of transport properties and interfacial kinetics of Na 2/3 [Ni 1/3 Mn x Ti 2/3-x ]O 2 (x = 0,1/3) as electrodes for Na-Ion batteries,” J Electrochem Soc , vol. 162, no. 1, pp. A8–A14, 2015, doi: 10.1149/2.0201501jes. [6] M. Doyle, J. P. Meyers, and J. Newman, “Computer Simulations of the Impedance Response of Lithium Rechargeable Batteries,” J Electrochem Soc , vol. 147, no. 1, p. 99, 2000, doi: 10.1149/1.1393162. [7] M. Pathak et al. , “Fast impedance simulation of lithium-ion batteries with pseudo-two dimensional electrochemical models,” J Electrochem Soc , vol. 165, no. 7, pp. A1324–A1337, 2018, doi: 10.1149/2.0831805jes. [8] L. Teo, V. R. Subramanian, and D. T. Schwartz, “Dynamic electrochemical impedance spectroscopy of lithium-ion batteries: revealing underlying physics through efficient joint time-frequency modeling,” J Electrochem Soc , vol. 168, no. 1, p. 010526, 2021, doi: 10.1149/1945-7111/abda04. [9] N. Meddings et al. , “Application of electrochemical impedance spectroscopy to commercial Li-ion cells: A review,” J Power Sources , vol. 480, no. July, 2020, doi: 10.1016/j.jpowsour.2020.228742. [10] T. K. Telmasre, T. Jang, N. Goswami, A. Concepcion, and V. R. Subramanian, “Simulation Strategies for Measuring Impedance Response of Lithium-Ion Batteries,” ECS Meeting Abstracts , vol. MA2022-02, no. 2, pp. 115–115, Oct. 2022, doi: 10.1149/MA2022-022115mtgabs. [11] A. Subramaniam, S. Kolluri, C. D. Parke, M. Pathak, S. Santhanagopalan, and V. R. Subramanian, “Properly Lumped Lithium-ion Battery Models: A Tanks-in-Series Approach,” J Electrochem Soc , vol. 167, no. 1, p. 013534, 2020, doi: 10.1149/2.0342001jes.

Batteries as an energy storage solution will play a key role in the transition to renewable sources of energy and reducing the impact of carbon emissions on the climate. Lithium-ion (Li-ion) batteries are a prime candidate for such applications due to their high energy density, low self-discharge rate and fast charging capabilities for applications like electric vehicles and grid storage. 1,2 High voltage battery systems used in the applications stated above make use of several thousand single Li-ion cells, connected in series and parallel combinations, to achieve the desired pack level characteristics. Rich literature is available on the physics-based modeling of single Li-ion cells, however simple extrapolations of these models from the cell level to the pack level cannot accurately predict pack behavior, especially during short times 3 . Cell-to-cell variations, quality of manufacturing, series-parallel connection combinations, and thermal gradients across a pack affect the behavior of each component cell in a single cycle and the overall pack behavior with cycling. 4 Li-ion batteries, where several cells are connected in series, can also experience inadvertent over-discharge due to differences between cells 5,6 . Over-discharge is a condition when the manufacturer recommended lower voltage limit is crossed while discharging, which results in various side effects such as electrolyte decomposition, capacity degradation due to copper dissolution and deposition, SEI layer breakdown, possible internal short circuit and an increase in cell temperature. These effects on a single cell level can lead to the under-utilization of cells connected in series and over-utilization of the cells connected in parallel, due to nature of current distribution in different regions of the pack dictated by the net resistance from each cell. Temperature plays a key role in the electrochemical performance of individual cells, affecting the diffusivities of ions, kinetics of reactions and rate of degradation of the battery. A physics-based modeling approach coupled with a thermal model, with appropriate voltage and current relations, can be employed to scale simulations from a cell to a pack level. This model can then enhance understanding on the effect of over-discharge on the electrochemical behavior of Li-ion batteries and provides insights into the interplay of various degradation mechanisms that cause capacity fade in battery-packs. The Thermal Tank-In-Series 7,8 (TTiS) model is a systematically volume-averaged form of the standard pseudo-2-dimensional (p2D) 9 model with energy balance equations for the current collectors, cathode, separator and anode. This reduces the number of equations in the model and improves computational speed while maintaining accuracy. Including variable-diffusivity in the solid phase equations 10 helps predict the overpotential changes to the cell electrodes and thermodynamics, as it is sensitive to changes in the battery’s state of charge (SOC) and cell temperature. We have developed a framework to simulate each cell in a pack using individual TTiS models, appropriately connected in parallel and series, to quantify the impact of cell-to-cell variation on pack level outputs as a function of cell capacity, internal parameter variations and cell location in the pack. A comparison of computational times will highlight the benefits of using such an approach to scale cell level models reliably, with robust initialization of the model and ensuring accuracy of predictions. References S. P. S. Badwal, S. S. Giddey, C. Munnings, A. I. Bhatt, and A. F. Hollenkamp, Frontiers in Chemistry , 2 , 28 (2014). G. Pistoia. Lithium-Ion Batteries: Advances and Applications . First ed. Elsevier, Amsterdam (2014). T.K. Telmasre, L. Mishra, R.S. Thiagarajan, A. Subramaniam, V. Ramadesigan, T.R. Garrick, V. R. Subramanian, J. Electrochem. Soc. 170 103512 (2023) T. Weaver, A. Allam and S. Onori, 2020 American Control Conference (ACC) pp. 365-372 R. Guo, L. Lu., M. Ouyang, X. Feng. Sci Rep 6 , 30248 (2016). G. Zhang, X. Wei, S. Chen, J. Zhu, G. Han, and H. Dai. J. Power Sources , 521 , 230990. (2022). Subramaniam, S. Kolluri, S. Santhanagopalan, and V. R. Subramanian, J. Electrochem. Soc. , 167 , 113506 (2020). 8. A. Subramaniam, S. Kolluri, C. D. Parke, M. Pathak, S. Santhanagopalan, and V. R. Subramanian, J. Electrochem. Soc. , 167 , 013534 (2020). Doyle, T. F. Fuller, and J. Newman, J. Electrochem. Soc ., 140 , 1526 (1993) S. Thiagarajan, A. Subramaniam, S. Kolluri, T. R. Garrick, Y. Preger, V. D. Angelis, J. Lim, and V. R. Subramanian, J. Electrochem. Soc. , 170 , 010528 (2023)

Digital twins are virtual replicas of physical systems updated real time and are increasingly vital for complex systems like batteries. Being electrochemical black boxes that degrade with use, batteries benefit from digital twins for monitoring, predictive maintenance, and optimization. Data-driven models are popular for digital twins due to their adaptability, but are limited by dependence on large datasets, weak interpretability, and poor generalization. In contrast, physics-based approaches such as Pseudo-2D (P2D) models offer higher accuracy, interpretability, and require less operational data. These models can be packaged and deployed across platforms using the Functional Mock-up Interface (FMI) and Functional Mock-up Units (FMUs). Through FMU deployment, this paper illustrates the challenges and opportunities in balancing model depth and scale across battery chemistries, as well as the robustness and efficiency of numerical methods across spatial and temporal scales. In addition, the paper highlights how classical models already integrate data-driven elements, such as empirical fits for open-circuit voltage and electrolyte properties. The most effective digital twin strategies for batteries are therefore hybrid models that combine the rigor of physics-based methods with the flexibility of data-driven tools. Such interpretable, scalable, and fast models are essential for advancing energy storage and enabling real-time control in diverse applications.

The evolution of metal anode–electrolyte interfaces critically impacts the safety and performance of metal anode batteries. We present a two-dimensional moving boundary model that simulates zinc anode morphology under high discharge–low charge (HD–LC) and low discharge–low charge (LD–LC) protocols. The model predicts interface flattening with HD–LC cycling, validated by symmetric cell experiments. LD–LC, however, does not show any change in interface morphology. Our findings highlight that cathode current modulation can effectively control interfacial morphology, offering a simple, yet powerful strategy to suppress dendrite growth and enhance metal anode battery safety through current control.

Digital twins are virtual replicas of physical systems updated in real-time and are becoming vital in modern engineering, especially for complex systems like batteries. Batteries, being electrochemical black boxes that degrade over time, benefit greatly from digital twins for real-time monitoring, predictive maintenance, and performance optimization. This paper explores various modeling approaches that underpin battery digital twins, including Equivalent Circuit Models (ECMs), Physics-based electrochemical models, and Data-driven or hybrid methods. While data-driven models are gaining popularity and offer wide ranging adaptability, they face limitations due to dependency on data, poor interpretability, and generalization. In contrast, physics-based models such as the Single Particle Model (SPM) and Pseudo-2D (P2D) models provide high accuracy, physical interpretability, and require less operational data. The paper also discusses the use of Functional Mock-up Interface (FMI) and Functional Mock-up Units (FMUs) to package and deploy these models across platforms. Comparative results from different numerical schemes like finite difference and galerkin weak form formulations are presented, showing the trade-off in accuracy and computational efficiency. Advanced modeling approaches for phase-separating chemistries like LFP are explored using phase-field methods as well as conversion chemistries like LiS using cell centered finite different techniques. Importantly, the paper highlights how classical models already integrate data-driven elements, such as empirical fits for open-circuit voltage and electrolyte properties. In conclusion, the most effective digital twin strategies for batteries lie in hybrid models that combine the rigor of physics-based modeling with the flexibility of data-driven methods. These fast, interpretable, and scalable models are essential for advancing energy storage technologies and enabling real-time control and optimization in diverse applications.

Digital twins are virtual replicas of physical systems updated in real-time and becoming vital in modern engineering, especially for complex systems like batteries. Batteries, being electrochemical black boxes that degrade over time, benefit greatly from digital twins for real-time monitoring, predictive maintenance, and performance optimization. This paper explores various modeling approaches that underpin battery digital twins, including Equivalent Circuit Models (ECMs), physics-based electrochemical models, and data-driven or hybrid methods. While data-driven models offer adaptability, they face limitations due to data dependency, poor interpretability, and generalization issues. In contrast, physics-based models such as the Single Particle Model (SPM) and Pseudo-2D (P2D) models provide high accuracy, physical interpretability, and require less operational data.The paper also discusses the use of Functional Mock-up Interface (FMI) and Functional Mock-up Units (FMUs) to package and deploy these models across platforms. Comparative results from different numerical schemes like finite difference and Galerkin methods are presented, showing trade-offs in accuracy and computational efficiency. Advanced modeling approaches for phase-separating chemistries like LFP are explored using phase-field methods. Importantly, the paper highlights how classical models already integrate data-driven elements, such as empirical fits for open-circuit voltage and electrolyte properties.In conclusion, the most effective digital twin strategies for batteries lie in hybrid models that combine the rigor of physics-based modeling with the flexibility of data-driven methods. These fast, interpretable, and scalable models are essential for advancing energy storage technologies and enabling real-time control and optimization in diverse applications.

Abstract: Developing resilient aqueous energy storage systems, such as Zn batteries, is essential for advancing energy sustainability powered by renewable sources. A key challenge in these systems is the dendritic growth of metals, which causes internal short circuits and poor rechargeability in battery electrodes. Classical diffusion-limited dendritic growth is predicted to occur when the charging rate exceeds the intrinsic limiting current density Jlim determined by electrolyte transport properties. Understanding the electrochemical behavior of dendrites is critical for designing fast-charging metal electrodes, where dendritic growth is not only likely but sometimes unavoidable due to uneven current distribution. Conventional wisdom suggests that these high-aspect-ratio fractal dendrites are susceptible to bottom-initiated dissolution, leading to mechanical break-off from the current collector and the formation of “dead” / orphaned metal. Surprisingly, our results show that near-unity charge-discharge reversibility can be achieved even with highly-ramified classical diffusion-limited dendritic Zn metal structures. In particular, the reversibility improves with an increasing discharge rate, demonstrating a strong positive correlation. A two-orders-of-magnitude (~200x) enhancement in cycle life is observed when Zn electrodes charged identically fast, but discharged at higher versus lower rates! Operando visualization reveals that dendrite fragmentation is significantly suppressed at higher discharge rates. Complementary post-mortem microstructural analysis shows that, consistent with predictions based on the Wagner number (Wa), high-rate discharges promote preferential tip-initiated stable retraction, whereas low-rate discharges induce “pitting” corrosion that mechanically weakens the dendrite backbone and promotes fragmentation. These findings challenge the prevailing assumption that dendritic growth necessarily limits the electrode reversibility and offer new design principles for metal battery electrodes operating at unprecedented high rates approaching the diffusion limit.

Abstract: Developing resilient aqueous energy storage systems, such as Zn batteries, is essential for advancing energy sustainability powered by renewable sources. A key challenge in these systems is the dendritic growth of metals, which causes internal short circuits and poor rechargeability in battery electrodes. Classical diffusion-limited dendritic growth is predicted to occur when the charging rate exceeds the intrinsic limiting current density Jlim determined by electrolyte transport properties. Understanding the electrochemical behavior of dendrites is critical for designing fast-charging metal electrodes, where dendritic growth is not only likely but sometimes unavoidable due to uneven current distribution. Conventional wisdom suggests that these high-aspect-ratio fractal dendrites are susceptible to bottom-initiated dissolution, leading to mechanical break-off from the current collector and the formation of “dead” / orphaned metal. Surprisingly, our results show that near-unity charge-discharge reversibility can be achieved even with highly-ramified classical diffusion-limited dendritic Zn metal structures. In particular, the reversibility improves with an increasing discharge rate, demonstrating a strong positive correlation. A two-orders-of-magnitude (~200x) enhancement in cycle life is observed when Zn electrodes charged identically fast, but discharged at higher versus lower rates! Operando visualization reveals that dendrite fragmentation is significantly suppressed at higher discharge rates. Complementary post-mortem microstructural analysis shows that, consistent with predictions based on the Wagner number (Wa), high-rate discharges promote preferential tip-initiated stable retraction, whereas low-rate discharges induce “pitting” corrosion that mechanically weakens the dendrite backbone and promotes fragmentation. These findings challenge the prevailing assumption that dendritic growth necessarily limits the electrode reversibility and offer new design principles for metal battery electrodes operating at unprecedented high rates approaching the diffusion limit.

Off-nominal circumstances inducing cell over-discharge in a battery are of concern due to electrolyte decomposition and prolonged degradation effects. Copper dissolution and its subsequent deposition could lead to high cell temperatures and, in some cases, catastrophic failures during the battery’s operational lifetime. Extending physics-based lithium-ion battery models for the purpose of simulating over-discharge conditions require key considerations in model parameters, constitutive equations, and the state of charge window of operation. This paper reports a reduced-order model for over-discharge and simulating its effect under various scenarios using a thermal tanks-in-series (TTiS) approach. The model was used to compare voltage-time behavior and capacity fade during cycling for different over-discharge cycling protocols. The efficacy of cycling simulations was validated with experimental data, and the TTiS model demonstrates reasonable agreement with the voltage, temperature, and capacity fade trends under the given experimental cycling regimes.

Electrochemical impedance spectroscopy (EIS) can be useful for the mechanism analysis and diagnosis of proton-exchange membrane fuel cell (PEMFC) performance degradation. This review summarizes the potential of using EIS for real-time fault detection and isolation of the PEMFC by data-driven methods from the following aspects. First, the data-driven diagnosis strategy of PEMFC based on EIS is overviewed; the typical faults and EIS measurement for data collection are briefly introduced. Then, the application of EIS in the online data-driven diagnosis of PEMFC is analyzed and discussed, focusing on feature extraction from EIS, diagnosis models employing various machine learning methods, and the corresponding EIS features for each machine learning method. Finally, the feasibility of using EIS for online data-driven fault diagnosis of PEMFC is briefly summarized, and the research challenges and prospects are proposed. This review aims to provide inspiration and new insights for future research on online PEMFC diagnosis, prognostics, and health management.

Lithium-ion batteries have emerged as the primary power source for electric mobilities. Accurate remaining useful lifetime (RUL) prediction is required to ensure the safe operation of the batteries throughout their lifespan. This article proposes combination strategies that integrate two different Gaussian process regression (GPR) methods and model-based methods to enhance the robustness of the prediction. The first GPR is based on the forward extrapolation of the measured capacity sequence. The second GPR is based on the extrapolation of the measured feature and then inputs the predicted feature into a capacity estimation model. The first ensemble strategy is the weighted ensemble method, which uses the least squares method to determine the weighted coefficients. The second strategy is a more conservative method, which chooses the fastest degradation path between two basic methods at each prediction step. The third strategy is particle filter (PF), which combines the predicted data from different methods. The batteries aged by a real forklift aging profile and open access dataset are used to verify the proposed methods. The results of all methods based on different percentages of data are analyzed. The results show that individual methods may obtain different prediction results, while ensemble strategies have accurate and robust predictions. The PF for capacity-based and feature-based methods has the best performance with the absolute error of RUL less than 23 full equivalent cycles (FECs), error of prediction steps less than 1, and negligible simulation time for the forklift dataset.

Advancing electric transportation technologies is critically dependent on maximizing the performance and lifetime of Electric Vehicle (EV) batteries, which is directly related to the economic case for EVs. Large-format cells are achieving increasing popularity for EVs to maximize volumetric and gravimetric energy densities. However, due to the large format size cells that are currently used for applications within the electrification portfolio, challenges have arisen that are tied to thermal management. Spatial variations in current density and temperature negatively affect overall cell utilization and lifetime. Identifying these temperature-dependent degradation mechanisms often relies on post-mortem analysis of cells, while model systems may be required to characterize spatial non-uniformities in degradation adequately. These challenges, along with the need for a comprehensive understanding of the degradation mechanisms, have led to the development of rigorous electrochemical models for battery degradation. These physics-based models serve as a powerful complementary tool in obtaining a better understanding of such phenomena. Simulating performance scenarios can lead to significant improvements in cell design and experimentation while also reducing time and cost commitments for the same. The primary goal of this work is the development and validation of a physicochemical degradation model for automotive lithium-ion batteries to predict the degradation in the large-format cell in the presence of temperature gradients using the given set of cell parameters with temperature dependence. It was realized that the degradation trajectories are a result of a combination of both LLI (loss of lithium inventory) and LAM (loss of active material) fade mechanisms at both anode and cathode. The state-of-the-art electrochemical models for lithium-ion cells (p2D models) were implemented in both in-house codes and equation-based formulation in COMSOL Multiphysics with weak form implementation. The codes were validated against experimental cycling data at different operating temperatures at each stage of the model development. The validated codes possess the ability to estimate local degradation rates and consequent impact on cycling performance for EV batteries at a range of operating conditions. Thus, the model is a means of identifying dominant degradation mechanisms in practically relevant operating conditions, helping improve overall lifetime, and enhancing value by helping identify improvements in battery operations. The errors between the model and data can provide information on the suitability of a given model and the relative prevalence of a given degradation mechanism, with useful, practical benefits.

In the rapidly evolving landscape of battery research and development, the need for efficient, standardized, and automated testing processes has become increasingly apparent. Battery testing generates an immense volume of valuable electrochemical data, including critical parameters such as cell impedance, coulombic efficiency, capacity, and constant current/constant voltage (CC/CV) charging times. However, the lack of standardization and the limitations of traditional data management approaches have hindered the ability to extract meaningful insights from this wealth of data. LF Energy's Battery Data Alliance (BDA) is a collaboration between industry leaders, academic institutions, and open-source communities to drive innovation and accelerate the development of advanced battery technologies. [1] BDA has developed BattData, a user-friendly battery data management framework, that enables researchers and engineers to efficiently store, process, analyze, and visualize vast amounts of battery testing data. BattData includes the module BattDB, a time series relational database designed to handle the unique requirements of battery data storage. BattDB acts as a centralized repository, allowing seamless access and querying of specific datasets. Its optimized architecture ensures high performance and scalability, making it capable of handling the ever-growing volume of battery testing data. To streamline the data ingestion process and ensure data integrity, we introduce BattETL (Extract, Transform, Load), an open-source Python module. BattETL automates the extraction, transformation, and loading of battery cycler data into BattDB, supporting popular cycler platforms such as Maccor and Arbin. By eliminating manual data preprocessing tasks, BattETL significantly reduces the burden on researchers and minimizes the risk of errors. In this talk, we demonstrate the benefits of the same and introduce other supporting visualization, modeling, and parameter estimation tools. References Battery Data Alliance – LF Energy (https://lfenergy.org/projects/battery-data-alliance/)

Energy storage will play a crucial role in the transition to clean and renewable sources of energy production to meet future consumption demands. Lithium-ion (Li-ion) batteries are widely adopted due to their high energy density, low self-discharge rate and fast charging capabilities for applications like electric vehicles and grid storage. Lithium-ion batteries are typically modeled using porous electrode theory coupled with various transport and reaction mechanisms, along with suitable discretization or approximations for the solid-phase diffusion equation.The solid-phase diffusion equation represents the main computational burden for typical pseudo-2-dimensional (p2D) models since these equations in the pseudo r -dimension must be solved at each point in the computational grid. This substantially increases the complexity of the model as well as the computational time. Traditional approaches towards simplifying solid-phase diffusion possess certain significant limitations, especially in modeling emerging electrode materials which involve phase changes and variable diffusivities. A computationally efficient representation for solid-phase diffusion was shown in a previous paper 1 based on symmetric polynomials using Orthogonal Collocation and Galerkin formulation (weak form). This approach increases the accuracy of the approximation (p form in finite element methods) to enable efficient simulation with a minimal number of semi-discretized equations, ensuring mass conservation even for non-linear diffusion problems involving variable diffusivities. These methods will be reviewed for the full p2D model, illustrating their advantages in simulating high C-rates and short-time dynamic operation and extended to newer systems containing phase field models to analyze the robustness of the approximations. References R.S. Thiagarajan, A. Subramaniam, S. Kolluri, T. R. Garrick, Y. Preger, 729 V. De Angelis, J.-h. Lim, V. R. Subramanian et al 2023 J. Electrochem. Soc. 170 010528

Physics-based electrochemical models play an important role in the model-based analysis, virtual engineering, estimation, control, and Battery Management Systems (BMS) of lithium-ion batteries. 1 The diffusion process within the cell electrodes has typically been modeled using Fick's laws of diffusion. 2 Recently, the intercalation of lithium ions in the electrode particle was posed as a phase change problem, where the area intercalated with lithium undergoes a phase change compared to the pristine material. Researchers have approached this phase change problem in terms of evolving core-shell problems 3,4 and, more recently, using phase field models 5,6 that may hold certain advantages over the traditional Fickian diffusion models. In this paper, we briefly discuss the relevance and rich physics of phase-field models and show how proper numerical analysis and schemes can help with implementing these models in BMS applications. Further, an investigation of phase-field model-based optimization framework predicts an impulse-like control profile and was designed to reduce capacity degradation. This work was partially inspired by the pulse-charging protocol proposed by Professor Landau in his 2006 work 7 . An open-source framework is developed and will be shared for predicting the (im)pulse protocol reported in this. References: Ramadesigan, V. et al. Modeling and simulation of lithium-ion batteries from a systems engineering perspective. J Electrochem Soc 159 , R31–R45 (2012). Doyle, M., Fuller, T. F. & Newman, J. Modeling of Galvanostatic Charge and Discharge of the Lithium/Polymer/Insertion Cell. J Electrochem Soc 140 , 1526–1533 (1993). Srinivasan, V. & Newman, J. Discharge Model for the Lithium Iron-Phosphate Electrode. J Electrochem Soc 151 , A1517 (2004). Dargaville, S. & Farrell, T. W. Predicting Active Material Utilization in LiFePO Electrodes Using a Multiscale Mathematical Model. J Electrochem Soc 157 , A830 (2010). Zeng, Y. & Bazant, M. Z. Phase Separation Dynamics in Isotropic Ion-Intercalation Particles. SIAM J Appl Math 74 , 980–1004 (2014). Singh, G. K., Ceder, G. & Bazant, M. Z. Intercalation dynamics in rechargeable battery materials: General theory and phase-transformation waves in LiFePO4. Electrochim Acta 53 , 7599–7613 (2008). Purushothaman, B. K. & Landau, U. Rapid Charging of Lithium-Ion Batteries Using Pulsed Currents. J Electrochem Soc 153 , A533 (2006). Figure 1

Electrochemical Impedance Spectroscopy (EIS) has emerged as a great alternative tool to characterize electrochemical energy storage systems such as batteries[1, 2]. A commercial Li-ion battery is essentially an electrochemical system having a multitude of physical and chemical processes occurring simultaneously while in operation. Each of these processes is governed by thermodynamic and kinetic principles and possesses a characteristic time scale. Thus, based on the frequency response, we can distinguish between individual mechanisms operating within the battery in a non-destructive manner via EIS [1]. Numerical modeling is a powerful tool to understand the battery system from a theoretical perspective[3]. Development of high-fidelity, multiscale, and multicomponent models can be a complex and computationally expensive endeavor. Therefore, simple models, with generalization about the physics of the problem, are sought initially upon which more complex models are built. Traditional EIS simulation approaches include Equivalent Circuit Models (ECMs), which aim to represent the battery as an electrical circuit. This approach suffers from its empirical nature and lack of physical interpretability[2]. Another approach is to work with physics-based models. Electrochemical Impedance Spectroscopy, being a frequency domain technique, needs a special treatment to be done on the time domain Partial Differential Equations (PDEs) to be solved in a steady state. This process involves linearizing the model equations and applying the Laplace transform. Physics-based models have a better fidelity but suffer from high computational costs.[4] Therefore, fast, reduced-order physics-based models have been developed by researchers that aim to maintain the physical consistency of the model at the same time reducing the computational cost.[5] Recent years have also seen a surge in new machine learning-based techniques to replicate the impedance response of the batteries.[6] The present work aims to summarize the various techniques of EIS modeling and discuss their relative advantages and shortcomings in a comprehensive manner. References: Gaberšček M (2021) Understanding Li-based battery materials via electrochemical impedance spectroscopy. Nat Commun 12:19–22. https://doi.org/10.1038/s41467-021-26894-5 Meddings N, Heinrich M, Overney F, et al (2020) Application of electrochemical impedance spectroscopy to commercial Li-ion cells: A review. J Power Sources 480:. https://doi.org/10.1016/j.jpowsour.2020.228742 Ramadesigan V, Northrop PWC, De S, et al (2012) Modeling and simulation of lithium-ion batteries from a systems engineering perspective. J Electrochem Soc 159:R31–R45. https://doi.org/10.1149/2.018203jes Meyers JP, Doyle M, Darling RM, Newman J (2000) The Impedance Response of a Porous Electrode Composed of Intercalation Particles. J Electrochem Soc 147:2930. https://doi.org/10.1149/1.1393627 Pathak M, Murbach MD, Pathak C, et al (2018) Fast impedance simulation of lithium-ion batteries with pseudo-two dimensional electrochemical models. J Electrochem Soc 165:A1324–A1337. https://doi.org/10.1149/2.0831805jes Telmasre T, Goswami N, Concepción A, et al (2022) Impedance response simulation strategies for lithium-ion battery models. Curr Opin Electrochem 36:101140. https://doi.org/10.1016/j.coelec.2022.101140

In this work, we present a water transport model to quantify the movement of water across Nafion® membranes in a proton exchange membrane electrolyzer as a function of varying operating conditions and membrane parameters. This physics-based model is based on the three main water transport mechanisms: diffusion, electro-osmotic drag, and pressure-driven flow. Three sets of equations are obtained to model the movement of water on the cathode side – I. Material balances for hydrogen and water in the flow channel ( z -direction), II. Water movement across the membrane in the x -direction, and III. Expressions for variable membrane properties to serve as model inputs. The condensation of water at the cathode is also modeled to understand the respective transport contributions from the vapor and liquid phases. The coupled equation sets are solved numerically with appropriate boundary conditions. An analytical solution is also obtained for the governing differential equation for the mole fraction of water in the vapor phase. This study is perhaps the first effort for a detailed physics-based transport model to predict the water transport in the electrolyzer in one dimension using the actual measured values for the physical parameters of the system. The model results are compared with the experimental data available for water transport, and a good agreement is observed over the wide range of current, temperature and pressure differentials. Further, with the help of this simple transport model, the numerical analysis is performed to delineate the effect of electrolyzer operating conditions on the net water transport across the membrane, water condensation at the cathode, individual contribution of the transport fluxes, and electrolyzer design. Finally, the model is exercised to simulate the dependence of water transport as a function of membrane thickness. This confirms the validity of the current approach of using thin reinforced membranes by electrolyzer fabricators. Figure 1

With the widespread use of Lithium-ion (Li-ion) batteries in multiple industries, the design space for the electrochemical cells has increased drastically. Li-ion batteries’ design parameters and material properties, such as porosity, electrode thickness, active material loading, and solid phase diffusivities, typically vary substantially due to different design goals and variation in the manufacturing process. Due to the difficulty in obtaining these and other battery cell parameters, many battery management systems used to control and monitor Li-ion batteries opt for the simplified resistor-capacitor circuit-based battery cell representation over using physics-based battery models. However, if these parameters become obtainable, scientists could implement advanced battery management systems using physics-based battery models to capture precisely how the battery performs and degrades in a wide variety of applications. Multiple methods, such as genetic algorithms, statistics, and machine learning, have been attempted for the parametrization and characterization of battery cells to enable accurate battery simulations using physics-based models [1, 2]. However, there are inherent limitations to the type and number of parameters that can be estimated using these methods if only the standard battery cell cycling data is utilized for this purpose. The present work elucidates and quantifies these limitations specifically in the case of machine learning based approach and provides deeper insights into these limitations that may help develop more robust cell characterization protocols. In the present work, model parameters are varied in a continuum-level physics-based model and simulations are performed for: (1) constant current-constant voltage (CC-CV) charging and constant current discharge at different C-rates, and (2) electrochemical impedance spectroscopy (EIS) in the relevant frequency range at multiple states of charge (SOCs). The Single Particle Model (SPM) is used for performing these simulations due to its relative simplicity and small number of parameters. At first, the charge and discharge data are input as training data in a long short-term memory neural network (LSTM NN), and EIS data is input as training data in a convolutional neural network (CNN). Once trained, the machine learning (ML) model uses charge, discharge, and EIS data withheld from the original training data set to predict SPM parameters. The estimated parameters are compared with the true parameters used for SPM charge, discharge, and EIS simulations, and the error in parameter estimation is quantified. A rigorous grid convergence analysis is performed to highlight the importance of ensuring that the spatial discretization error of the model does not influence the parameter estimation accuracy. For the first error quantification, all parameters are varied concurrently to obtain the parameter estimation error that arises from a large, undetermined parameter set. Parameters are then analyzed individually and in clusters related to diffusion, kinetics, and stoichiometric limitations. Further analysis is performed on parameters that could not be estimated with sufficient accuracy, including theoretical explanations for the incurred error and the effect of this error on the internal state predictions. References: Dawson-Elli, N., et al., On the creation of a chess-ai-inspired problem-specific optimizer for the pseudo two-dimensional battery model using neural networks. Journal of The Electrochemical Society, 2019. 166 (6): p. A886. Berliner, M.D., et al., Nonlinear identifiability analysis of the porous electrode theory model of lithium-ion batteries. Journal of The Electrochemical Society, 2021. 168 (9): p. 090546.

Li-metal batteries, often heralded as the next frontier in energy storage technology, boast remarkable characteristics such as high energy density. Unlike traditional lithium-ion batteries, Li-metal batteries incorporate a lithium-metal anode, which significantly enhances their energy density. This higher energy density translates to a substantial increase in the amount of energy that can be stored in a given volume or weight, paving the way for more powerful and longer-lasting energy solutions[1–3]. However, significant shortcomings do exist, such as uneven Li-Metal deposition, dendrite formation, dead lithium, etc.[4] These phenomena are different than what are usually observed in Li-ion batteries with intercalation chemistry and lead to issues such as poor cycle life, low coulumbic efficiency, and safety concerns.[3] Mathematical modeling is a promising approach to studying the physicochemical processes involved in the Li-metal batteries to understand the morphology changes during cycling. The plating and stripping of the lithium metal on the anode during cycling can be modeled as a moving boundary problem.[5] Since the process involves a shrinking and expanding electrolyte domain, moving boundary problems call for the fastidious implementation of the model equations and boundary conditions.[6] In this talk, we will show the general mathematical framework involved in modeling Li-metal batteries in 1D systems. It expands upon the importance of carefully implementing boundary conditions to tackle the mass conservation issue that arises while cycling the model. It will also touch upon the additional complexities associated with expanding the model to 2D systems with convection in the electrolyte phase. References: Wang J, Ge B, Li H, et al (2021) Challenges and progresses of lithium-metal batteries. Chem Eng J 420:129739. https://doi.org/10.1016/j.cej.2021.129739 Wang R, Cui W, Chu F, Wu F (2020) Lithium metal anodes: Present and future. J Energy Chem 48:145–159. https://doi.org/10.1016/j.jechem.2019.12.024 Liu J, Bao Z, Cui Y, et al (2019) Pathways for practical high-energy long-cycling lithium metal batteries. Nat Energy 4:180–186. https://doi.org/10.1038/s41560-019-0338-x Bandhauer TM, Garimella S, Fuller TF (2011) A Critical Review of Thermal Issues in Lithium-Ion Batteries. J Electrochem Soc 158:R1. https://doi.org/10.1149/1.3515880 Mishra L, Subramaniam A, Jang T, et al (2021) Perspective—Mass Conservation in Models for Electrodeposition/Stripping in Lithium Metal Batteries. J Electrochem Soc 168:092502. https://doi.org/10.1149/1945-7111/ac2091 Jang T, Mishra L, Subramaniam A, et al (2023) Addressing Mass Conservation in Two-dimensional Modeling of Lithium Metal Batteries with Electrochemically Plated/Stripped Interfaces. J Electrochem Soc 170:110516. https://doi.org/10.1149/1945-7111/ad0510

The classic ‘pseudo-2-dimensional’ (p2D) model of Newman and co-workers (also known as the DFN model), with its different efficient implementations, is the state-of-the-art physics-based electrochemical model for lithium-ion batteries. COMSOL Multiphysics (COMSOL) is a popular modeling software for multiphysics simulations, and its Battery Design Module for Li-ion batteries is widely used in the battery modeling community. This work presents a step-wise model development for reconstructing the detailed p2D model equations. This includes variable model parameters and weak-form implementation for the solid-phase diffusion, eliminating the need for non-local couplings and approximations for modeling the pseudo radial-dimension in the solid-phase. To model and understand the effect of capacity degradation on lithium-ion cells, the battery models in COMSOL need to incorporate additional degradation mechanisms. In this work, we have followed the approach of equation-based modeling in COMSOL, which enables us to formulate the p2D model using the mathematical PDE interfaces. This approach provides us with a lot of flexibility in terms of battery degradation model formulation and helps develop a clearer understanding of the p2D model equation in light of the assumptions involved. The steps in the model development involved writing the conservation equations in the generic mathematics interfaces coupled with the additional equations for the degradation mechanisms. The efforts were centered on developing cycling simulations based on Events for practically relevant operating conditions. This custom model was validated against the results from COMSOL’s Battery Interface, in-house codes, and baseline cycle data from the experiments and further developed as a physicochemical degradation model for automotive, large-format Lithium-ion batteries. This equation-based COMSOL model with cycling works without the requirement for Battery Design Module , LiveLink and Application Builder and can be adopted for a variety of electrochemical modeling applications by the electrochemical modeling community.

Electric aviation has been a major focus of research in recent years as the aviation industry has seen significant growth in electric aircraft development. Most of the current literature related to electric aircraft modeling focuses on aircraft design and optimization with less focus given to battery analysis. Many studies utilize battery simplifications such as linear voltage profiles 1 or the use of equivalent circuit models 2 . Similarly, in the emerging field of electric vertical takeoff and landing (eVTOL) aircraft, battery analysis is often reduced to considering just weight and empirical power parameters, failing to capture physical and electrochemical battery behavior and its effect on battery performance characteristics 3 . These simplifications can often lead to inaccurate battery state estimations and performance prediction. Moreover, the battery packs used for both electric conventional take-off and landing (eCTOL) aircraft and eVTOLs see unique operating requirements that are often significantly more demanding than those seen in electric vehicles (EVs) 4 . This suggests the need to analyze aircraft battery pack performance limitations in greater detail. However, only a few such studies have been reported in the literature 5-6 . In this work, we perform coupled simulation studies of longitudinal flight dynamics and physics-based battery dynamics. For modeling battery dynamics, we use the well-known single particle model 7 . This coupling enables more reliable estimation of battery states and can better inform battery pack design for electric aircraft. The simulations are performed to understand the interplay between battery cell dynamics governed by its design parameters and parameters associated with fixed-wing flight dynamics, such as cruise altitude, flight path angle, and velocity. Additionally, the simulations show how varying these parameters and operating temperature effects the range and endurance of electric aircraft which can be used to determine the optimal aircraft operating parameters for a desired mission profile. Results from this simulation study are compared against an equivalent circuit battery model to highlight the importance of detailed battery analysis in electric aviation. This work will be extended to develop similar coupled physics-based battery models with the state-of-the-art aircraft configurations emerging in the eVTOL field. References M. Kaptsov and L. Rodrigues, Journal of Guidance, Control, and Dynamics , 41 , 288 (2018). N. Biju and H. Fang, Applied Energy , 339 , 120905 (2023). L. Kiesewetter, K. H. Shakib, P. Singh, M. Rahman, B. Khandelwal, S. Kumar and K. Shah, Progress in Aerospace Sciences , 142 , 100949 (2023). X.-G. Yang, T. Liu, S. Ge, E. Rountree and C.-Y. Wang, Joule , 5 , 1644 (2021). A. Ayyaswamy, B. S. Vishnugopi and P. P. Mukherjee, Joule , 7 , 2016 (2023). M. Wang, S. Kolluri, K. Shah, V. R. Subramanian and M. Mesbahi, IEEE Transactions on Aerospace and Electronic Systems , 59 , 1084 (2022). S. Santhanagopalan, Q. Guo, P. Ramadass and R. E. White, Journal of power sources , 156 , 620 (2006).

Fast charging is researched heavily for the widespread implementation of lithium-ion batteries for electric vehicles. However, charging at high currents accelerates several parasitic reactions that lead to the degradation of the cell, affecting its lifetime. These reactions lead to loss of lithium inventory, loss of active material, and increased impedance in the cell. Examples of these side reactions include the growth of the solid-electrolyte interphase (SEI) layer, transition metal dissolution and deposition, lithium plating, solvent oxidation, etc. These mechanisms degrade the cell and reduce its cycle life. Physics-based multi-scale battery models solve for equations that govern the charge and mass balances within the cell. Using these detailed mathematical models, it is possible to study material degradation mechanisms and predict their impact on capacity loss under several operating conditions. These models can be used to design new batteries with appropriate materials and design parameters suited for any given purpose. More critically, these models can be integrated with battery management systems (BMS) to control the cell’s performance. These models can further be used to design novel charging protocols that enable safe and optimal cell performance and suppress cell material degradation. The BMS monitors and maintains the voltage, current, and temperature and estimates the internal states of the cell. Model-based BMS algorithms require fast codes that can predict and estimate battery parameters in real-time and control the battery’s performance under different loads. Currently, the BMS implements equivalent circuit models that inadequately predict the cell’s performance for various conditions and design parameters. This talk presents the current efforts to move the models for BMS for current and next-generation batteries for single-cell to pack-level simulations. Figure 1

Several previous studies have aimed to develop mathematical models based on the moving boundary approach for predicting changes in lithium morphology during the cycling of lithium metal batteries under different operating conditions. These modeling frameworks play a crucial role in enhancing our fundamental understanding of the transport and reaction mechanisms and guide experimentalists in developing safer and more efficient lithium metal anodes. In this work, using a simple two-dimensional model for the lithium metal battery, we aim to bring attention to the bulk convection in the liquid electrolytes induced by the movement of the lithium metal surface modeled as a moving boundary. The back-and-forth motion of the lithium metal surface during the plating and stripping of lithium introduces a weak fluid motion in the liquid electrolyte that should be incorporated in the model equations and corresponding boundary conditions (Figure 1). The results for the electrochemical signatures and morphology evolution thus obtained by solving a coupled fluid model are compared with the case where the velocity distribution in the liquid electrolyte is ignored. This work extends our previously reported perspective on the convective flux correction to rectify the mass conservation failures at moving boundaries in one-dimensional models to two dimensions. This careful implementation of the correct boundary conditions ensures the mass conservation of lithium in two-dimensional simulations for predicting the morphological evolution of lithium metal electrodes over cycles. These revised battery models with accurate mass and charge conservation are crucial for predicting the performance of the battery over cycles. Additionally, these relative fluxes at the moving and fixed boundaries are sometimes ignored by assuming a bulk concentration condition at the far end, especially at the cathode/separator interface. While it may not affect overpotential signatures at the anode, it leads to mass conservation issues with implications for the accuracy of cycling simulations. Thus, with the help of a two-dimensional electro-convection model, this study addresses the role of bulk convection in liquid electrolytes and the importance of ensuring mass conservation in lithium metal battery models. References: T. Jang et al. , J. Electrochem. Soc., 170 (11), 110516 (2023). T. Jang et al. , J. Electrochem. Soc., 169 (8), 080516 (2023). T. Jang et al ., ECS Trans , 104 (1), 131 (2021). L. Mishra et al ., J Electrochem Soc , 168 (9), 092502 (2021). Figure 1

Energy storage will play a crucial role in the transition to clean and renewable sources of energy production to meet future consumption demands. 1 Lithium-ion (Li-ion) batteries are widely adopted due to their high energy density, low self-discharge rate and fast charging capabilities for applications like electric vehicles and grid storage. 2 Li-ion batteries for high voltage operations, where several cells are connected in series, can experience inadvertent over-discharge due to differences between cells 3,4 . These differences can arise due to many reasons including changes in internal resistance and deviation in electrochemical degradation that can cause divergence of cell open circuit potentials. Over-discharge is a condition when the manufacturer recommended lower voltage limit is crossed while discharging 4 , which results in various side effects such as electrolyte decomposition, capacity degradation, cathode morphology change and possible internal short circuit 3,5 . Increase in anode potential during over-discharge leads to copper dissolution from the anode current collector and subsequent plating within cell components. The Solid Electrolyte Interface (SEI) can decompose during excessive deintercalation and reform during charging, reducing the lithium inventory. Excessive heat generation during cycle life testing after occurrences of over-discharges has been found in some cases to lead to temperatures >150 °C 6,7 . Temperature plays a key role in the electrochemical performance of individual cells, affecting the diffusivities of ions, kinetics of reactions and rate of degradation of the battery. A physics-based modeling approach coupled with a thermal model can be employed to understand the effect of over-discharge on the electrochemical behavior of Li-ion batteries and provides insights into the interplay of various degradation mechanisms that cause capacity fade. The Thermal Tank-In-Series 9,10 (TTiS) model is a systematically volume-averaged form of the standard pseudo-2-dimensional (p2D) 11 model with energy balance equations for the current collectors, cathode, separator and anode. This reduces the number of equations in the model and improves computational speed while maintaining accuracy, as it is sensitive to changes in the battery’s state of charge (SOC) and cell temperature, which helps predict the overpotential changes to the cell electrodes and thermodynamics. We use the TTiS model to identify the dominant fade mechanisms in 3 sets of experimental cycling and temperature data for the: i) normal operating window (2.5V – 4.2V), ii) 5 initial cycles of over-discharge (1.5V – 4.2V) followed by normal operation until 20% capacity fade and iii) 5 cycles of periodic over-discharge every 20 cycles, until 20% capacity fade. Analysis on the effect of C-rate, ambient temperature and voltage window of operation will facilitate understanding the temperature behavior and overpotentials in a cell sandwich. Key considerations required in extending a model to simulate over-discharge will also be examined. References S.P. S. Badwal, S. S. Giddey, C. Munnings, A. I. Bhatt, and A. F. Hollenkamp, Frontiers in Chemistry , 2 , 28 (2014). G. Pistoia. Lithium-Ion Batteries: Advances and Applications . First ed. Elsevier, Amsterdam (2014). R. Guo, L. Lu., M. Ouyang, X. Feng. Sci Rep 6 , 30248 (2016). G.Zhang, X. Wei, S. Chen, J. Zhu, G. Han, and H. Dai. J. Power Sources , 521 , 230990. (2022). C. Fear, D. Juarez-Robles, J. A. Jeevarajan, and P. P. Mukherjee, J. Electrochem. Soc ., 165 , A1639–A1647 (2018). D. Juarez-Robles, Purdue Univ. Ph.D. Thesis (2019). H. Maleki and J. N. Howard, J. Power Sources, 160 , 1395–1402 (2006). H. Liu, Z. Wei, W. He, and J. Zhao, Energy Convers. Manag. , 150 , 304–330 (2017). A. Subramaniam, S. Kolluri, S. Santhanagopalan, and V. R. Subramanian, J. Electrochem. Soc. , 167 , 113506 (2020). A. Subramaniam, S. Kolluri, C. D. Parke, M. Pathak, S. Santhanagopalan, and V. R. Subramanian, J. Electrochem. Soc. , 167 , 013534 (2020). M. Doyle, T. F. Fuller, and J. Newman, J. Electrochem. Soc ., 140 , 1526 (1993)

Battery models are useful tools that provide insight into electrochemical processes within an electrochemical device those can typically not be measured, either due to a lack of experimental methods or due to the transient nature of the phenomena in question. Irrespective of the complexity of the models involved, a general view is that a good model, once established for a single cell in terms of complexity of the physics, choice of simulation algorithm, value for the parameters, and mechanisms for fade, can be easily extended for pack simulations. 1–3 This type of extension is critical to the virtual engineering efforts currently underway at OEMs which rely on accurate and representative battery models. In this work, pitfalls during scaling up approximate and detailed physics-based models developed for single-cell to pack-level simulations are highlighted using representative examples. Interesting mathematical nuances were found depending on the numerical simulation approach used. A discrepancy at initial times (t = 0) during the pack-level simulations was highlighted and resolved using the Laplace transform approach to get an analytical solution for the simplest model. A few thoughts on numerical challenges and the index of differential-algebraic equations (DAEs) 4,5 while using the strong form and the weak form formulation of DAEs are also provided in the paper. 6 References: M. Dubarry, N. Vuillaume, and B. Y. Liaw, J. Power Sources , 186 , 500–507 (2009). T. Tranter et al., J. Open Source Softw. , 7 , 4051 (2022). J. M. Reniers and D. A. Howey, Appl. Energy , 336 , 120774 (2023) https://doi.org/10.1016/j.apenergy.2023.120774. E. Hairer and G. Wanner, Solving Ordinary Differential Equations II , 2nd ed., p. 15–37, Springer Berlin, Heidelberg, Berlin, (1996). L. Petzold, SIAM J. Sci. Stat. Comput. , 3 , 367–384 (1982). T. K. Telmasre et al., J. Electrochem. Soc. , 170 , 103512 (2023). Figure 1

Automotive manufacturers are working to improve individual cell, module, and overall pack design by increasing the performance, range, and durability, while reducing cost. One key piece to consider during the design process is the active material volume change, its linkage to the particle, electrode, and cell level volume changes, and the interplay with structural components in the rechargeable energy storage system. As the time from initial design to manufacture of electric vehicles decreases, design work needs to move to the virtual domain; therefore, a need for coupled electrochemical-mechanical models that take into account the active material volume change and the rate dependence of this volume change need to be considered. In this study, we illustrated the applicability of a coupled electrochemical-mechanical battery model considering multiple representative particles to capture experimentally measured rate dependent reversible volume change at the cell level through the use of an electrochemical-mechanical battery model that couples the particle, electrode, and cell level volume changes. By employing this coupled approach, the importance of considering multiple active material particle sizes representative of the distribution is demonstrated. The non-uniformity in utilization between two different size particles as well as the significant spatial non-uniformity in the radial direction of the larger particles is the primary driver of the rate dependent characteristics of the volume change at the electrode and cell level.

Corrigendum: Moving Next-Generation Phase-Field Models to BMS Applications -A Case Study that Confirms Professor Uzi Landau's Foresight [J. Electrochem. Soc., 171, 063507 (2024)], Tushar K. Telmasre, Anthony César Concepción, Suryanarayana Kolluri, Lubhani Mishra, Raghav S. Thiagarajan, Aditya Naveen Matam, Akshay Subramaniam, Taylor R. Garrick, Venkat R. Subramanian

Automotive manufacturers are working to improve individual cell, module, and overall pack design by increasing the performance, range, and durability, while reducing cost. One key piece to consider during the design process is the active material volume change, its linkage to the particle, electrode, and cell level volume changes, and the interplay with structural components in the rechargeable energy storage system. As the time from initial design to manufacture of electric vehicles decreases, design work needs to move to the virtual domain; therefore, a need for coupled electrochemical-mechanical models that take into account the active material volume change and the rate dependence of this volume change need to be considered. In this study, we illustrated the applicability of a coupled electrochemical-mechanical battery model considering multiple representative particles to capture experimentally measured rate dependent reversible volume change at the cell level through the use of an electrochemical-mechanical battery model that couples the particle, electrode, and cell level volume changes. By employing this coupled approach, the importance of considering multiple active material particle sizes representative of the distribution is demonstrated. The non-uniformity in utilization between two different size particles as well as the significant spatial non-uniformity in the radial direction of the larger particles is the primary driver of the rate dependent characteristics of the volume change at the electrode and cell level.

Physics-based electrochemical models play a prominent role in the model-based analysis, virtual engineering, and Battery Management Systems (BMS) of lithium-ion and next-generation batteries. In this paper, we demonstrate the rich physics of phase-field models and convey their potential in BMS applications. Our phase-field model-based optimization framework predicts an impulse-like control profile to reduce capacity degradation. This work was partially inspired by the pulse-charging protocol proposed by Professor Landau in his 2006 work [B. K. Purushothaman and U. Landau, J Electrochem Soc , 153 (3), A533 (2006)]. An open-source framework is shared for predicting the (im)pulse protocol reported in this paper.

Off-nominal circumstances inducing cell over-discharge in a battery, are of concern due to electrolyte decomposition and prolonged degradation effects. Copper dissolution and its subsequent deposition could lead to high cell temperatures and in some cases catastrophic failures during battery operational lifetime. Extending physics-based lithium-ion battery models for the purpose of simulating over-discharge conditions requires key considerations in model parameters, constitutive equations, and the SoC window of operation This paper reports a reduced-order model for over-discharge and simulating its effect under various scenarios using a Thermal Tanks-in-Series (TTiS) approach. The model was used to compare voltage-time behavior and capacity fade during cycling for different over-discharge cycling protocols. The efficacy of cycling simulations was validated with experimental data and the TTiS model demonstrates reasonable agreement with the voltage, temperature, and capacity fade trends under the given experimental cycling regimes.

Since 2009, the ENIGMA Consortium has brought together neuroimaging researchers from over 45 countries to perform some of the largest international studies of over 30 major brain disorders. The ENIGMA working groups tackle the growing challenge of data harmonization and standardization in analytic workflows, and address the need for well-powered, multi-center studies by providing a community-driven structure and platform for collaborations. The recently-formed ENIGMA Neuromodulation Working Group (ENIGMA-NeMo) includes subgroups representing individual neuromodulation modalities, supported by a machine learning/artificial intelligence core providing advanced analytic techniques within and across modulation modalities. The goals of this working group include: suggesting standards and standardizations for research in neuromodulation [e.g., neuromodulation extension of Brain Imaging Data Structure (BIDS)]; improving reproducibility of neuromodulation research findings; developing models for predicting and improving brain circuit engagement, safety, and clinical efficacy outcomes across modulation modalities; accelerating development of therapeutic parameters for clinical neuromodulation across disease populations; and evaluating existing neuromodulation methods and advancing these techniques to maximize individual treatment effects towards precision medicine. The ENIGMA-NeMo group applies standardized analytic pipelines for large-scale as well as single-patient analyses of multi-modal brain MRI, neuromodulation parameters and outcome data (e.g., neuropsychological, psychophysiological). Here, we discuss initial goals, challenges, and strategies for overcoming these challenges and gaps in the literature. We also outline both the current state of and opportunities to advance the field of multimodal neuromodulation and accelerate the translation of research findings to clinical practice.

In this work, we present a water transport model to quantify the movement of water across Nafion® membranes in a proton exchange membrane electrolyzer as a function of varying operating conditions and membrane parameters. This physics-based model is based on the three main water transport mechanisms: diffusion, electro-osmotic drag, and pressure-driven flow. Three sets of equations are obtained to model the movement of water on the cathode side – I. Material balances for hydrogen and water in the flow channel (z-direction), II. Water movement across the membrane in the x-direction, and III. Expressions for variable membrane properties to serve as model inputs. The condensation of water at the cathode is also modeled to understand the respective transport contributions from the vapor and liquid phases. The coupled equation sets are solved numerically with appropriate boundary conditions. An analytical solution is also obtained for the governing differential equation for the mole fraction of water in the vapor phase. This study is perhaps the first effort for a detailed physics-based transport model to predict the water transport in the electrolyzer in one dimension using the actual measured values for the physical parameters of the system. The model results are compared with the experimental data available for water transport, and a good agreement is observed over the wide range of current, temperature and pressure differentials. Further, with the help of this simple transport model, the numerical analysis is performed to delineate the effect of electrolyzer operating conditions on the net water transport across the membrane, water condensation at the cathode, individual contribution of the transport fluxes, and electrolyzer design. Finally, the model is exercised to simulate the dependence of water transport as a function of membrane thickness. This confirms the validity of the current approach of using thin reinforced membranes by electrolyzer fabricators. Figure 1