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A two-dimensional single-phase model is developed for the steady-state and transient analysis of polymer electrolyte membrane fuel cells (PEMFC). Based on diluted and concentrated solution theories, viscous flow is introduced into a phenomenological multi-component modeling framework in the membrane. Characteristic variables related to the water uptake are discussed. A Butler–Volmer formulation of the current-overpotential relationship is developed based on an elementary mechanism of electrochemical oxygen reduction. Validated by using published V–I experiments, the model is then used to analyze the effects of operating conditions on current output and water management, especially net water transport coefficient along the channel. For a power PEMFC, the long-channel configuration is helpful for internal humidification and anode water removal, operating in counterflow mode with proper gas flow rate and humidity. In time domain, a typical transient process with closed anode is also investigated.
The state-of-the-art electrochemical impedance spectroscopy (EIS) calculations have not yet started from fully multi-dimensional modeling. For a polymer electrolyte membrane fuel cell (PEMFC) with long flow channel, the impedance plot shows a multi-arc characteristic and some impedance arcs could merge. By using a step excitation/Fourier transform algorithm, an EIS simulation is implemented for the first time based on the full 2D PEMFC model presented in the first part of this work. All the dominant transient behaviors are able to be captured. A novel methodology called ‘configuration of system dynamics’, which is suitable for any electrochemical system, is then developed to resolve the physical meaning of the impedance spectra. In addition to the high-frequency arc due to charge transfer, the Nyquist plots contain additional medium/low-frequency arcs due to mass transfer in the diffusion layers and along the channel, as well as a low-frequency arc resulting from water transport in the membrane. In some case, the impedance spectra appear partly inductive due to water transport, which demonstrates the complexity of the water management of PEMFCs and the necessity of physics-based calculations.
A balcony photovoltaic (PV) system, also known as a micro-PV system, is a small PV system consisting of one or two solar modules with an output of 100–600 Wp and a corresponding inverter that uses standard plugs to feed the renewable energy into the house grid. In the present study we demonstrate the integration of a commercial lithium-ion battery into a commercial micro-PV system. We firstly show simulations over one year with one second time resolution which we use to assess the influence of battery and PV size on self-consumption, self-sufficiency and the annual cost savings. We then develop and operate experimental setups using two different architectures for integrating the battery into the micro-PV system. In the passive hybrid architecture, the battery is in parallel electrical connection to the PV module. In the active hybrid architecture, an additional DC-DC converter is used. Both architectures include measures to avoid maximum power point tracking of the battery by the module inverter. Resulting PV/battery/inverter systems with 300 Wp PV and 555 Wh battery were tested in continuous operation over three days under real solar irradiance conditions. Both architectures were able to maintain stable operation and demonstrate the shift of PV energy from the day into the night. System efficiencies were observed comparable to a reference system without battery. This study therefore demonstrates the feasibility of both active and passive coupling architectures.
Fast charging of lithium-ion batteries remains one of the most delicate challenges for the automotive industry, being seriously affected by the formation of lithium metal in the negative electrode. Here we present a physicochemical pseudo-3D model that explicitly includes the plating reaction as side reaction running in parallel to the main intercalation reaction. The thermodynamics of the plating reaction are modeled depending on temperature and ion concentration, which differs from the often-used assumption of a constant plating condition of 0 V anode potential. The reaction kinetics are described with an Arrhenius-type rate law parameterized from an extensive literature research. Re-intercalation of plated lithium was modeled to take place either via reverse plating (solution-mediated) or via an explicit interfacial reaction (surface-mediated). At low temperatures not only the main processes (intercalation and solid-state diffusion) become slow, but also the plating reaction itself becomes slower. Using this model, we are able to predict typical macroscopic experimental observables that are indicative of plating, that is, a voltage plateau during discharge and a voltage drop upon temperature increase. A spatiotemporal analysis of the internal cell states allows a quantitative insight into the competition between intercalation and plating. Finally, we calculate operation maps over a wide range of C-rates and temperatures that allow to assess plating propensity as function of operating condition.
Passive hybridization refers to a parallel connection of photovoltaic and battery cells on the direct current level without any active controllers or inverters. We present the first study of a lithium-ion battery cell connected in parallel to a string of four or five serially-connected photovoltaic cells. Experimental investigations were performed using a modified commercial photovoltaic module and a lithium titanate battery pouch cell, representing an overall 41.7 W-peak (photovoltaic)/36.8 W-hour (battery) passive hybrid system. Systematic and detailed monitoring of this system over periods of several days with different load scenarios was carried out. A scaled dynamic synthetic load representing a typical profile of a single-family house was successfully supplied with 100 % self-sufficiency over a period of two days. The system shows dynamic, fully passive self-regulation without maximum power point tracking and without battery management system. The feasibility of a photovoltaic/lithium-ion battery passive hybrid system could therefore be demonstrated.
Batteries typically consist of multiple individual cells connected in series. Here we demonstrate single-cell state of charge (SOC) and state of health (SOH) diagnosis in a 24 V class lithium-ion battery. To this goal, we introduce and apply a novel, highly efficient algorithm based on a voltage-controlled model (VCM). The battery, consisting of eight single cells, is cycled over a duration of five months under a simple cycling protocol between 20 % and 100 % SOC. The cell-to-cell standard deviations obtained with the novel algorithm were 1.25 SOC-% and 1.07 SOH-% at beginning of cycling. A cell-averaged capacity loss of 9.9 % after five months cycling was observed. While the accuracy of single-cell SOC estimation was limited (probably owed to the flat voltage characteristics of the lithium iron phosphate, LFP, chemistry investigated here), single-cell SOH estimation showed a high accuracy (2.09 SOH-% mean absolute error compared to laboratory reference tests). Because the algorithm does not require observers, filters, or neural networks, it is computationally very efficient (three seconds analysis time for the complete data set consisting of eight cells with approx. 780.000 measurement points per cell).
The accurate diagnosis of state of charge (SOC) and state of health (SOH) is of utmost importance for battery users and for battery manufacturers. State diagnosis is commonly based on measuring battery current and using it in Coulomb counters or as input for a current-controlled model. Here we introduce a new algorithm based on measuring battery voltage and using it as input for a voltage-controlled model. We demonstrate the algorithm using fresh and pre-aged lithium-ion battery single cells operated under well-defined laboratory conditions on full cycles, shallow cycles, and a dynamic battery electric vehicle load profile. We show that both SOC and SOH are accurately estimated using a simple equivalent circuit model. The new algorithm is self-calibrating, is robust with respect to cell aging, allows to estimate SOH from arbitrary load profiles, and is numerically simpler than state-of-the-art model-based methods.
Lithium-ion batteries exhibit a dynamic voltage behaviour depending nonlinearly on current and state of charge. The modelling of lithium-ion batteries is therefore complicated and model parametrisation is often time demanding. Grey-box models combine physical and data-driven modelling to benefit from their respective advantages. Neural ordinary differential equations (NODEs) offer new possibilities for grey-box modelling. Differential equations given by physical laws and NODEs can be combined in a single modelling framework. Here we demonstrate the use of NODEs for grey-box modelling of lithium-ion batteries. A simple equivalent circuit model serves as a basis and represents the physical part of the model. The voltage drop over the resistor–capacitor circuit, including its dependency on current and state of charge, is implemented as a NODE. After training, the grey-box model shows good agreement with experimental full-cycle data and pulse tests on a lithium iron phosphate cell. We test the model against two dynamic load profiles: one consisting of half cycles and one dynamic load profile representing a home-storage system. The dynamic response of the battery is well captured by the model.
Lithium-ion batteries exhibit slow voltage dynamics on the minute time scale that are usually associated with transport processes. We present a novel modelling approach toward these dynamics by combining physical and data-driven models into a Grey-box model. We use neural networks, in particular neural ordinary differential equations. The physical structure of the Grey-box model is borrowed from the Fickian diffusion law, where the transport domain is discretized using finite volumes. Within this physical structure, unknown parameters (diffusion coefficient, diffusion length, discretization) and dependencies (state of charge, lithium concentration) are replaced by neural networks and learnable parameters. We perform model-to-model comparisons, using as training data (a) a Fickian diffusion process, (b) a Warburg element, and (c) a resistor-capacitor circuit. Voltage dynamics during constant-current operation and pulse tests as well as electrochemical impedance spectra are simulated. The slow dynamics of all three physical models in the order of ten to 30 min are well captured by the Grey-box model, demonstrating the flexibility of the present approach.
The lifetime of a battery is affected by various aging processes happening at the electrode scale and causing capacity and power fade over time. Two of the most critical mechanisms are the deposition of metallic lithium (plating) and the loss of lithium inventory to the solid electrolyte interphase (SEI). These side reactions compete with reversible lithium intercalation at the graphite anode. Here we present a comprehensive physicochemical pseudo-3D aging model for a lithium-ion battery cell, which includes electrochemical reactions for SEI formation on graphite anode, lithium plating, and SEI formation on plated lithium. The thermodynamics of the aging reactions are modeled depending on temperature and ion concentration, and the reactions kinetics are described with an Arrhenius-type rate law. The model includes also the positive feedback of plating on SEI growth, with the presence of plated lithium leading to a higher SEI formation rate compared to the values obtained in its absence at the same operating conditions. The model is thus able to describe cell aging over a wide range of temperatures and C-rates. In particular, it allows to quantify capacity loss due to cycling (here in % per year) as function of operating conditions. This allows the visualization of aging colormaps as function of both temperature and C-rate and the identification of critical operation conditions, a fundamental step for a comprehensive understanding of batteries performance and behavior. For example, the model predicts that at the harshest conditions (< –5 °C, > 3 C), aging is reduced compared to most critical conditions (around 0–5 °C) because the cell cannot be fully charged.
This article presents the development, parameterization, and experimental validation of a pseudo-three-dimensional (P3D) multiphysics aging model of a 500 mAh high-energy lithium-ion pouch cell with graphite negative electrode and lithium nickel manganese cobalt oxide (NMC) positive electrode. This model includes electrochemical reactions for solid electrolyte interphase (SEI) formation at the graphite negative electrode, lithium plating, and SEI formation on plated lithium. The thermodynamics of the aging reactions are modeled depending on temperature and ion concentration and the reactions kinetics are described with an Arrhenius-type rate law. Good agreement of model predictions with galvanostatic charge/discharge measurements and electrochemical impedance spectroscopy is observed over a wide range of operating conditions. The model allows to quantify capacity loss due to cycling near beginning-of-life as function of operating conditions and the visualization of aging colormaps as function of both temperature and C-rate (0.05 to 2 C charge and discharge, −20 °C to 60 °C). The model predictions are also qualitatively verified through voltage relaxation, cell expansion and cell cycling measurements. Based on this full model, six different aging indicators for determination of the limits of fast charging are derived from post-processing simulations of a reduced, pseudo-two-dimensional isothermal model without aging mechanisms. The most successful aging indicator, compared to results from the full model, is based on combined lithium plating and SEI kinetics calculated from battery states available in the reduced model. This methodology is applicable to standard pseudo-two-dimensional models available today both commercially and as open source.
This article presents the development, parameterization, and experimental validation of a pseudo-three-dimensional (P3D) multiphysics model of a 350 mAh high-power lithium-ion pouch cell with graphite anode and lithium cobalt oxide/lithium nickel cobalt aluminum oxide (LCO/NCA) blend cathode. The model describes transport processes on three different scales: Heat transport on the macroscopic scale (cell), mass and charge transport on the mesoscopic scale (electrode pair), and mass transport on the microscopic scale (active material particles). A generalized description of electrochemistry in blend electrodes is developed, using the open-source software Cantera for calculating species source terms. Very good agreement of model predictions with galvanostatic charge/discharge measurements, electrochemical impedance spectroscopy, and surface temperature measurements is observed over a wide range of operating conditions (0.05C to 10C charge and discharge, 5°C to 35°C). The behavior of internal states (concentrations, potentials, temperatures) is discussed. The blend materials show a complex behavior with both intra-particle and inter-particle non-equilibria during cycling.
Aqueous lithium–oxygen batteries are promising candidates for electric energy storage. In this paper we present and discuss a multiphase continuum model of an aqueous lithium–oxygen single cell including reactions and transport in a porous gas diffusion electrode (GDE). The model is parameterized using in-house half-cell experiments and available literature data on aqueous electrolytes. We validate our transport model with cyclic voltammetry and electrochemical impedance spectroscopy measurements over a wide range of temperatures (25, 40, 55 °C) and electrolyte concentrations (0.1–2 M). We observe very good agreement between simulations and measurements during oxygen reduction conditions. A sensitivity analysis of the validated model demonstrates the influence of the porous structure on GDE performance and gives directions for the future development of electrodes.
On the Fundamental and Practical Aspects of Modeling Complex Electrochemical Kinetics and Transport
(2018)
Numerous technologies, such as batteries and fuel cells, depend on electrochemical kinetics. In some cases, the responsible electrochemistry and charged-species transport is complex. However, to date, there are essentially no general-purpose modeling capabilities that facilitate the incorporation of thermodynamic, kinetic, and transport complexities into the simulation of electrochemical processes. A vast majority of the modeling literature uses only a few (often only one) global charge-transfer reactions, with the rates expressed using Butler–Volmer approximations. The objective of the present paper is to identify common aspects of electrochemistry, seeking a foundational basis for designing and implementing software with general applicability across a wide range of materials sets and applications. The development of new technologies should be accelerated and improved by enabling the incorporation of electrochemical complexity (e.g., multi-step, elementary charge-transfer reactions and as well as supporting ionic and electronic transport) into the analysis and interpretation of scientific results. The spirit of the approach is analogous to the role that Chemkin has played in homogeneous chemistry modeling, especially combustion. The Cantera software, which already has some electrochemistry capabilities, forms the foundation for future capabilities expansion.
Lithium-oxygen cells with organic electrolyte suffer high overpotentials during charge, indicating asymmetric charge/discharge reaction mechanisms. We present a multi-physics dynamic modeling and simulation study of the Li/O2 cell cycling behavior. We present three different multi-step mechanisms of the 2 Li + O2 ⇄ Li2O2 cell reaction, (A) a reversible 5-step mechanism, (B) a partially irreversible 6-step mechanism, and (C) a partially irreversible 8-step mechanism that includes reactions of a redox mediator. Model predictions are compared to experimental galvanostatic cycling data of Swagelok cells without and with 2,2,6,6-tetramethylpiperidinyloxyl (TEMPO) as redox mediator. All mechanisms are able to predict the discharge behavior in good agreement to the experimental results. The experimentally observed high charge overpotentials as well as their reduction by using a redox mediator can be qualitatively reproduced with the irreversible reaction mechanisms. However, the particular shape of the experimental charge curve with continuously increasing charge overpotential cannot be reproduced with the present mechanisms.
Lithium–oxygen cells with nonaqueous electrolyte show high overpotentials during charge, indicating asymmetric charge/discharge reaction mechanisms. We present a kinetic modeling and simulation study of the lithium–oxygen cell cycling behavior. The model includes a multistep reaction mechanism of the cell reaction (2Li + O2 ⇄ Li2O2) forming lithium peroxide by precipitation, coupled to a 1D porous-electrode transport model. We apply the model to study the asymmetric discharge/charge characteristics and analyze the influence of a redox mediator dissolved homogeneously in the liquid electrolyte. Model predictions are compared to experimental galvanostatic cycling data of cells without and with 2,2,6,6-tetramethylpiperidinyloxyl (TEMPO) as redox mediator. The predicted discharge behavior shows good agreement with the experimental results. A spatiotemporal analysis of species concentrations reveals inhomogeneous distributions of dissolved oxygen and reaction products within the cathode during discharge. The experimentally observed charge overpotentials as well as their reduction by using a redox mediator can be qualitatively reproduced with a partially irreversible reaction mechanism. However, the proposed models fail to reproduce the particular shape of the experimental charge curve with continuously increasing charge overpotential, which implies that part of the reaction mechanism is still open for investigation in future work.
Seven cell design concepts for aqueous (alkaline) lithium–oxygen batteries are investigated using a multi-physics continuum model for predicting cell behavior and performance in terms of the specific energy and specific power. Two different silver-based cathode designs (a gas diffusion electrode and a flooded cathode) and three different separator designs (a porous separator, a stirred separator chamber, and a redox-flow separator) are compared. Cathode and separator thicknesses are varied over a wide range (50 μm–20 mm) in order to identify optimum configurations. All designs show a considerable capacity-rate effect due to spatiotemporally inhomogeneous precipitation of solid discharge product LiOH·H2O. In addition, a cell design with flooded cathode and redox-flow separator including oxygen uptake within the external tank is suggested. For this design, the model predicts specific power up to 33 W/kg and specific energy up to 570 Wh/kg (gravimetric values of discharged cell including all cell components and catholyte except housing and piping).
Electrochemical impedance spectroscopy (EIS) is a widely-used diagnostic technique to characterize electrochemical processes. It is based on the dynamic analysis of two electrical observables, that is, current and voltage. Electrochemical cells with gaseous reactants or products (e.g., fuel cells, metal/air cells, electrolyzers) offer an additional observable, that is, the gas pressure. The dynamic coupling of current and/or voltage with gas pressure gives rise to a number of additional impedance definitions, for which we use the term electrochemical pressure impedance. It also gives rise to different experimental probing approaches. In this article we present a model-based study of electrochemical pressure impedance spectroscopy (EPIS). Possible quantifications and realizations of EPIS are discussed. The study of generic cell geometries consisting of gas reservoir, diffusion layer(s) and electrochemically active layer(s) reveals distinct spiral-shaped features in the Nyquist plot. Using the example of a sodium/oxygen (Na/O2) cell, the dynamic spatiotemporal behavior of the state variables is quantified and interpreted. Results are compared to first experimental EPIS measurements by Hartmann et al. [J. Phys. Chem. C118, 1461, 2014]. A sensitivity analysis highlights the properties of EPIS with respect to geometric, transport, and kinetic parameters. We demonstrate that EPIS is sensitive to transport parameters that are not well-accessible with standard EIS.