Reduced-order Electrochemical Model Research Articles

Optimization of electrode thickness of lithium-ion batteries for maximizing energy density

AbstractThe demand for high capacity and high energy density lithium-ion batteries (LIBs) has drastically increased nowadays. One way of meeting that rising demand is to design LIBs with thicker electrodes. Increasing electrode thickness can enhance the energy density of LIBs at the cell level by reducing the ratio of inactive materials in the cell. However, after a certain value of electrode thickness, the rate of energy density increase becomes slower. On the other hand, the impact of associated limitations becomes stronger, reducing the practical applicability of LIBs with thicker electrodes. Hence, an optimum value of thickness is of utmost importance for the practicability of thicker electrode design. In this paper, both the cathode thickness and the anode thickness of an NCM LIB cell were optimized by applying response surface methodology (RSM) with a Box-Behnken design (BBD) to maximize the energy density. Moreover, the influence of electrode porosity, together with the interaction of porosity with cathode and anode thickness, was incorporated into the optimization. A full factorial design of 3-level, 3-factor was used to generate 15 simulation conditions in accordance with the design of experiment (DoE) achieved through BBD. Then, those conditions were used to achieve 15 responses by simulating a reduced-order electrochemical model. Finally, the statistical technique analysis of variance (ANOVA) was used to analyze and validate the results of RSM. The results show that the RSM-BBD optimization method, coupled with ANOVA, has successfully optimized the thicknesses of both positive and negative electrodes for maximum energy density, despite the nonlinearity of the electrochemical system. The findings suggest an optimized cathode thickness of 401.56 µm and anode thickness of 186.36 µm for a maximum energy density of 292.22 of an NCM LIB cell, while electrode porosity is preferred to be 0.2.

Towards Rational Design of Battery Formation Protocols: From Electrochemical Modeling to Factory Deployment

In battery manufacturing, the formation process is both paramount and problematic. It is paramount since all batteries undergo formation charging and aging steps to build a resilient solid electrolyte interphase (SEI) and to screen for defects. It is problematic because the formation process is expensive to operate, remains a major source of factory energy demand, requires larger factory footprints, and takes an order of magnitude longer than nearly every other manufacturing step. Despite the centrality of the formation process in battery manufacturing, steps taken to optimizing formation protocols remain ad-hoc in the absence of design principles and physical models.We highlight recent progress in developing a principled understanding of the SEI formation process applied towards industrial battery formation process (i.e. in the context of a full cell). We review a reduced-order electrochemical model of the formation process that captures first cycle charge dynamics and subsequent cycling and aging degradation behavior [1]. The model predicts measurable quantities such as first cycle efficiency (FCE) and irreversible thickness growth under multiple formation protocols. We discuss opportunities to integrate the electrochemical formation model with differential voltage analysis (DVA) to visualize the connection between first cycle charge dynamics from the perspective of shifting electrode stoichiometries [2] and comment on applications towards physics-informed battery lifetime prediction [3]. We finally share recent data elucidating the effect of formation temperature, pressure, and protocol on cycle life.

A reduced-order electrochemical battery model for wide temperature range based on Pareto multi-objective parameter identification method

Lithium-ion batteries (LIBs) are critical components of electric vehicles and energy storage systems. However, low ambient temperatures can significantly slow down the electrochemical reaction rate and increase polarization within the battery, resulting in a reduction in capacity and power. In this paper, an accurate reduced-order electrochemical model is developed targeting for a wide temperature range (−20 to 40 °C). The model considers the excess driving force of Li + (de)intercalation in the charge transfer reaction for ion-intercalation materials by adopting adjustment in the Butler-Volmer (BV) equation. Moreover, concentration-dependent solid-phase diffusion coefficients are utilized to improve the accuracy of the model in the voltage recovery session under different charge/discharge rate conditions. To address the multi-objective optimization challenge in parameter identification across a wide range of operating conditions, the Pareto optimization method is employed. The parameters of the proposed model are identified using experimental data under different discharge conditions, including 0.2C, 0.33C, 0.5C, 1C CC discharge, and the UDDS driving cycle. To further validate the model, three dynamic conditions for testing are selected, and the model agrees well with real-world data with an average RMSE of 20 mV at different temperatures and test cycles, exhibiting its capability and robustness in predicting the battery performance under various conditions and temperatures.

A computational analysis of effects of electrode thickness on the energy density of lithium-ion batteries

Elevated energy density is a prime requirement for many lithium-ion battery (LIB) applications, including electric vehicles (EVs). At the cell level, the enhanced energy density of LIBs is achievable by designing thicker electrodes, which decreases the weight of the inactive materials. This work proposed a reduced order electrochemical model (ROEM), which is an extended single particle model (extended SPM), by incorporating the existing order reduction techniques of Pseudo-Two-Dimensional (P2D) model. The proposed model was used to predict the energy density values of a nickel cobalt manganese (NCM) cell with respect to the cathode thickness. The validation of the model was performed by using MATLAB at 0.05C, 1C, 2C, 3C and 5C rates by comparing the simulation results with the experimental data obtained from the Stanford Energy Control Laboratory. The verified model was then used to simulate the effect of varying thicknesses of the cathode on energy density of the NCM cell at C/5, C/2, 1C, and 2C rates of discharge. The simulation results illustrate that increasing the cathode thickness has a profound positive impact on the energy density of the cell. The prediction of the model demonstrates a total of around 20 % increase in energy density for the increment of cathode thickness from 75 μm to 600 μm. The proposed model can replicate energy density for thicker electrodes up to 2C rate reasonably correctly. However, the incorporation of thermal characteristics and other influential parameters of LIB cells into the proposed model can yield a better prediction of energy density of thicker electrodes at higher C-rates, which is recommended for further study.

Electrochemical-Thermal Modeling of Large-Format, Thin-Film, Lithium-Ion Batteries with Cocurrent and Countercurrent Tab Connections Using a Reduced-Order Model

We derive and implement a new reduced-order model for the simulation of large-format, thin-film batteries with cocurrent and countercurrent tab connections. We employ the multi-site, multi-reaction (MSMR) framework to describe the solid phase thermodynamics as well as irreversible phenomena associated with diffusion and electrochemical reactions for a graphite negative and a spinel manganese oxide positive. The calculations are streamlined by using the reduced-order electrochemical model for a porous electrode derived by means of a perturbation analysis, which we term ROM1. For discharge rates less than 1 C, where the 1 C rate corresponds to the current needed to fully discharge the cell in 1 h, ROM1 yields accurate results for traction-battery electrodes. We employ ROM1 in the cell energy balance, with the overall results allowing one to clarify the current and temperature distributions within the cell during discharge and isolate and identify the different heat sources. The governing partial differential equations are coupled and nonlinear in part due to the temperature dependence of the physicochemical properties. We show how cocurrent tab locations yield higher cell energy densities, while countercurrent tab locations yield more uniform current and temperature distributions. Sensitivity analyses underscore the flexibility of the approach. Overall, the equation system and open-source (Python) software enables an efficient and rational tool for cell design and integration.

Reduced-order electrochemical models with shape functions for fast, accurate prediction of lithium-ion batteries under high C-rates

This paper proposes physical-based, reduced-order electrochemical models that are much faster than the pseudo-2D (P2D) model, while providing high accuracy even under the challenging conditions of high C-rate and strong polarization of lithium ion concentration and potential. In particular, a weak form of equations are developed by using shape functions, which reduces the fully coupled electrochemical and transport equations to ordinary differential equations, and provides self-consistent solutions for the evolution of polynomial coefficients. Results show that the models, named as revised single-particle model (RSPM) and fast-calculating P2D model (FCP2D), give reliable prediction of battery operations, including under dynamic driving profiles. They can calculate battery parameters, such as terminal voltage, over-potential, interfacial current density, lithium-ion concentration distribution, and electrolyte potential distribution with a relative error less than 2%. Applicable for moderately high C-rates (below 2.5C), the RSPM is up to more than 33 times faster than the P2D model. The FCP2D is applicable for high C-rates (above 2.5C) and is about 8 times faster than the P2D model. With their high speed and accuracy, these physics-based models can significantly improve the capability and performance of the battery management system and accelerate battery design optimization.

System identification and state estimation of a reduced-order electrochemical model for lithium-ion batteries

Lithium-ion batteries commonly used in electric vehicles are an indispensable part of the development process of decarbonization, electrification, and intelligence in transportation. From intelligent designing, manufacturing to controlling, an intelligent battery management system plays a crucial role in the long life, high efficiency, and safe operation of lithium-ion batteries. As a first-principle model, the electrochemical parameters of the electrochemical model have physical meanings and reflect the internal state of the lithium-ion batteries. The application of electrochemical models in an advanced intelligent battery management system is a future trend that promises to mitigate battery life degradation and prevent safety incidents. The reduced-order electrochemical model is expected to alleviate the requirements of advanced battery management systems for high accuracy and fast computing of lithium-ion battery models. However, the existing model order reduction methods have the drawbacks of high computational complexity and small application scope, so that inconvenient to apply onboard. In order to solve the existing obstacles, this paper applies the pseudo-spectral method to solve the solid-phase diffusion equation, while the liquid-phase concentration equation is simplified by the Galerkin method. Subsequently, a particle swarm optimization algorithm is used to identify 11 parameters of the electrochemical model. To further improve the accuracy of the electrochemical model, the above system identification method is applied to segment identification, especially for high or low state-of-charge (SoC) conditions in this study. Finally, based upon the derived model, estimation of SoC is performed using a particle filter. The results show that the proposed reduced-order electrochemical model achieves a low Mean Absolute Error (MAE) of 8.4 mV and a MAE of 0.54 % on estimation of SoC based on the envisaged particle filter. This work is expected to provide the basis for the subsequent development of lithium-ion battery electrochemical models with smaller identification parameters and faster identification processes.

(Invited) Novel Reduced-Order Electrochemical Models for Battery Simulation

It is important to accelerate the calculation efficiency while guaranteeing the accuracy for battery simulations. We introduced novel reduced-order physical-based models for simplifying the P2D electrochemical model, where the lithium-ion concentration and electrolyte potential distribution are represented by polynomial equations. Two types of reduced-order models, including the revised single-particle model (RSPM) and the fast-calculating P2D model (FCP2D) are developed based on the simplification method of lithium diffusion modeling within particles. Results show that the proposed RSPM and FCP2D reduced-order model can represent the battery parameters, including terminal voltage, overpotential, interfacial current density, lithium-ion concentration distribution, and electrolyte potential distribution, accurately and efficiently. Specifically, the RSPM owns a better performance (embodied in a satisfactory accuracy and a higher efficiency) under lower C rates operation, while the FCP2D can guarantee a higher accuracy than the RSPM under higher C rates operation. This work provides a novel approach for promoting battery simulation efficiency through physical-based models, which contributes to the acceleration of the responding speed of battery management systems and the battery development period.

A reduced-order electrochemical model for analyzing temperature distributions in a tubular solid oxide fuel cell stack

It is challenging to improve the computational efficiency of temperature predictions in tubular solid oxide fuel cell (SOFC) stacks, since the complex flow field usually leads to non-uniform heat generations and heat dissipations. In this study, we have developed a reduced-order electrochemical model for fast predicting the temperature distributions in a kilowatt-class tubular solid oxide fuel cell stack. This model can explicitly express the current density as a function of operating potential, concentration, and temperature, which can significantly reduce the computational cost of stack level simulations. The influence of flow characteristics on temperature distributions in the tubular SOFC stack is examined and analyzed. We show that the vortices formed by flow separations could affect temperature uniformity among all cells. At a low inlet air velocity, few vortices are formed, and the convective heat transfer is too weak to dissipate the heat generations from electrochemical reactions. As the air velocity is increased, the number and strength of vortices are both increased. These vortices enable the air stream to flow spirally and enhance the local convective heat transfer. We also demonstrate that as the air inlet velocity is increased from 1 m/s to 4 m/s, the temperature uniformity among tubular solid oxide fuel cells first decreases and then increases, but the output power is reduced from 2.19 kW to 1.78 kW due to the decrease in operating temperatures. The inlet air velocity should be carefully selected to balance the electrical and thermal performance of the SOFC stack.

Efficient Electrochemical State of Health Model for Lithium Ion Batteries under Storage Conditions

Capacity degradation of batteries negatively impacts the lifetime of battery packs as well as the residual value of electric vehicles. Developing a degradation model for the prognosis of the state of health (SOH) under storage conditions is a critical aspect of developing algorithms to maximize the remaining useful lifetime of these systems. It is known that electrochemical degradation models have superior predictive ability compared to more empirical or data-driven models, but these still require improvement in terms of computational efficiency. In this work, we thus introduce a simple, reduced-order electrochemical degradation model for lithium-ion batteries. This model considers three key aging mechanisms with the ability to predict the SOH under various calendar aging conditions. Lumped model results are validated against a single particle-based degradation model and show close agreement, even as the simulation time is reduced by 2 orders of magnitude. This indicates significant potential in real-world applications to account and correct for the effects of storage on cell performance and lifetime.

Physics-Based Reduced Order Model for Sodium-Ion Batteries

Sodium-ion batteries (SIBs) are emerging as promising energy storage systems. SIBs share similar chemistry and performance metrics with lithium-ion batteries (LIBs), the workhorse energy storage technology. Abundant availability of sodium and its even distribution across the globe make SIBs a better alternative to LIBs. Electrochemical models enabling simulation of batteries help study performance and various parameters of batteries for a variety of operating conditions. Reduced order electrochemical models can be used to study batteries to obtain quick insights, especially, at lesser computational cost. In this work, we present a reduced order model for SIBs based on the single particle assumption. We validate the single particle model (SPM) by comparing the results with experimental data and predictions of detailed pseudo-two-dimensional (P2D) model. The SPM shows good accuracy in predicting voltage profiles over a range of discharge current densities. We further discuss the regime of operation where the SPM predictions deviate from that of the P2D model. Our study suggests that the SPM can be used as a viable alternative to the P2D model to design and test SIBs.

Lithium-ion battery design optimization based on a dimensionless reduced-order electrochemical model

Model-based optimal cell design is an efficient approach to maximize the energy density of lithium-ion batteries. This maximization problem is solved in this paper for a lithium iron phosphate (LFP) cell. We consider half-cells as opposed to full-cells typically considered, which are intermediate steps during battery manufacturing for electrode characterization and they are gaining popularity by themselves as lithium-metal batteries. First, a dimensionless reduced-order electrochemical model is used instead of high-order models. Second, sensitivity equations are analyzed to determine the ranking of the design parameters according to their effect on the energy density, which is often lacking in other contributions. Three parameters, namely electrode thickness, LFP particle radius and electrode cross sectional area, are shown to have the most influential effects. Third, a novel adaptive particle swarm optimization with a specific stopping criterion is used for LIB design optimization. The proposed optimization framework is tested in simulation on a LFP half-cell battery. The results show that the design optimization yields 250 Wh kg−1 for an LFP electrode of 310μm thickness, 10 nm particle radius and 2⋅10−4 m2 cross-sectional area, which is an increase of energy density of 61 Wh kg−1 with respect to an initial design proposed in the literature.

State-of-health estimation of lithium-ion batteries for electrified vehicles using a reduced-order electrochemical model

The growing demand for Electric Vehicles (EVs) to operate reliably with ever-increasing driving ranges means that the lithium-ion batteries are more frequently working at their theoretical limits. Thus, it is essential that the Battery Management System (BMS) accurately monitors the internal states of the batteries with high precision and robustness. Although the Equivalent Circuit Model (ECM) is widely used in BMS, the Electrochemical Model (EM) outperforms the ECM in terms of inherent physics representation. In this paper, a novel coestimation scheme based on a fractional-order battery model is presented for simultaneous estimations of the internal resistance and capacity fade as State of Health (SOH) indicators. This approach avoids high computational cost due to a low number of calibration parameters, while maintaining high accuracy. Hence, it has great potential for BMS usage. First, the derivation of the fractional battery model from Partial Differential Equations (PDE) governing the Pseudo-Two-Dimensional model (P2D) is described. Next, the resistance estimation with an iterative model-based observer approach is developed to concurrently realize the adaptive estimation of the battery resistance and capacity. Finally, the effectiveness of the newly proposed approach is validated by experimental data considering different capacity aging levels, dynamic load profiles and initialization errors.

Reduced-order electrochemical modelling of Lithium-ion batteries

Complex systems require complex models to have high accuracy and a wide observation of internal dynamics. However, those models are not suitable for fast simulations or on-line applications. The modelling of Lithium ions batteries has been explored for years where model-based approaches and data-driven approaches have been developed. New techniques are being created to come up with simple enough models with high precision. A reduced order electrochemical model based on Dynamic Mode Decomposition with Control (DMDc) is developed here to achieve high prediction accuracy for state-of-charge (SOC) estimation without the need of a high-dimensional model. The performance is illustrated using numerical simulations and comparison with standard equivalent-circuit modelling with Coulomb counting.

Parameter identification of reduced-order electrochemical model simplified by spectral methods and state estimation based on square-root cubature Kalman filter

With the in-depth study of electrochemical models, the use of reduced-order electrochemical models on battery management systems becomes a possible future trend. In this paper, a two-step parameter identification method and the square-root cubature Kalman filter are employed. The pseudo-spectral method is used to solve the solid-phase diffusion equation, while the liquid-phase concentration equation is simplified by the Galerkin method. The reduced-order model maintains high accuracy while significantly reducing computational burden, the computation of a discharge voltage is within 1.5 s, which makes it possible to use the model for parameter identification and real-time state estimation. Next, the ant lion optimizer is applied to identify 21 parameters in the electrochemical model and 4 parameters in the thermal model respectively. Identification and validation results in constant current discharge and several dynamic tests demonstrate the high accuracy of the reduced-order model. Finally, the square-root cubature Kalman filter based on the identified parameters is developed to estimate the state of charge and temperature. The states are initialized with certain errors in two dynamic tests to verify the performance of the algorithm. The results show that the designed state observer has good convergence, robustness, and accuracy.

Measurement and Estimation of Heat Generation Rate of Lithium-Ion Battery Considering SEI Growth and Lithium Plating

Thermal management system for battery is one of the most crucial components required to guarantee high efficiency and reliable operation of the battery. The system requires a battery model for accurate characterization of heat generated from the battery during operation. In addition, the sustainability of the model accuracy over the lifetime of the battery is another important factor. Since the battery heat generation is increased due to the increase of the resistance during cycling, the model needs to predict the change accurately. In this presentation, a thermal model incorporated into a reduced-order electrochemical model is introduced. The model considers detailed irreversible heat source terms by considering internal processes of the battery during cycling. In addition, since the battery parameters are changed when the battery is aged, a degradation model considering growth of solid-electrolyte-interphase (SEI) and lithium plating is also incorporated, and therefore, the battery parameters can be updated after degradation. In order to experimentally validate the model, the battery heat generation rate is measured at the beginning, middle, and end of life. The validation results show good agreement of the increase of the heat generation rate with degradation estimated by the model with that measured by the experiment. In addition, the effect of the degradation on heat source term is analyzed, and the main cause for the increase in the heat generation rate is revealed.

Lithium-ion battery cathode and anode potential observer based on reduced-order electrochemical single particle model

The fast charging of Lithium-ion batteries within electric vehicles can accelerate the side reaction of lithium plating due to an anode potential that occurs as state of charge increases. It is important to monitor the anode potential during battery charging, but it is not practical to measure the inside of the battery directly for a commercial cell. This paper proposes an observer for estimating the cathode and anode potentials based on the reduced-order electrochemical model, which only needs terminal voltage to track the cathode and anode potentials and their internal charge concentration. The observer design is based on the model order reduction and linearisation of a single particle model with electrolyte (SPMe) to achieve acceptable accuracy with a low calculation cost. The linearised model and the designed observer are validated by the experimental results of a three-electrode cell. The results show that the linearised model reduces the operation time by more than 99% compared with the full-order SPMe model using the same processor. The results also verify that the root mean square error of the cathode and anode potential estimated by the observer is less than 0.02 V for a charging current range from 0.3C to 1C. This shows that the developed cathode and anode potential observer based on the reduced-order electrochemical model can be used within real-time control applications to detect the anode potential in real time to avoid battery degradation caused by lithium plating.

A Computational Framework for Lithium Ion Cell-Level Model Predictive Control Using a Physics-Based Reduced-Order Model

Most state-of-the art battery-control strategies rely on voltage-based design limits to address performance and lifetime concerns. Such approaches are inherently conservative. However, by exploiting internal electrochemical quantities, it is possible to control battery performance right up to true physical bounds. This letter develops an extensible framework that combines model predictive control (MPC) with computationally efficient realization algorithm (xRA)-generated reduced-order electrochemical models for the advanced control of lithium-ion batteries. The approach is demonstrated on the fast-charge problem where hard constraints are imposed on problem variables to avoid lithium plating induced performance degradation. This letter establishes a general mathematical foundation for the incorporation of electrochemically rich reduced-order models directly into an MPC framework.

Dual Closed-Loops Capacity Evolution Prediction for Energy Storage Batteries Integrated with Coupled Electrochemical Model

The health assessment for energy storage batteries matters in the context of carbon neutrality. Dual closed-loops capacity framework integrated with a reduced-order electrochemical model including triple side reactions is put forward, realizing parameter correction for health evaluation. Simplified microgrid aging experiment is formulated to test the closed-loop matching between the aging mechanism and electrochemical model relying on incremental capacity analysis. In addition, taking into account the future degradation prediction for energy storage system, the reliable capacity output afterwards acts as references for closed-loop parameter updating in empirical model to predict degradation evolution. The framework proposed implements the closed-loop dynamic updating for aging parameters with ideal error within 2%, making up for the lack of aging mechanism interpretation of accustomed empirical or data-driven black box model in the field of energy storage batteries.

A Real Time Adaptive Charging Approach for Cycle Life Extension of Li Ion Cells

It is commonly understood that preservation of Lithium-ion battery life comes at the expense of charge time. In this manuscript a novel real time charging framework has been developed that charges the battery in a time as fast as the constant current-constant voltage (CC-CV) protocol while ensuring minimum loss of cycle life. The framework consists of a reduced order electrochemical model integrated with an optimization routine. The charge profile adapts to the battery state of charge and health in real time and is amenable for on-device implementation. Cycle life benefits of the novel adaptive charging profile are compared with traditional CC-CV charging of commercial Lithium-ion batteries cycled until end of life. In addition to the running capacity, battery health is also compared at a low probe C-rate(C/5) to assess the benefits in minimizing irreversible losses. A differential voltage analysis of Probe (C/5 rate) reveals that adaptive charging is able to eliminate the loss of active material (LAM) that is seen typically at advanced cycles. The framework is implemented on a mobile device and shows a substantial benefit in battery life without any penalty on the charge time.