Optimal experimental design for large-scale inverse problems for battery models via PDE-Constrained optimization

Lithium ion (li-ion) batteries are gradually increasing their volumetric power and energy densities due to requirements imposed by electric vehicles and portable electronics applications. In consequence, modern lithium ion battery cells and battery packs are having much more compact design where a high amount of energy is encapsulated in a small volume. This imposes a need for an accurate thermal modeling of li-ion batteries in order to avoid battery cells overheating, which leads to safety concerns (e.g. thermal runaways) and accelerates battery cell's performance degradation. Conventional methods for two- and three-dimensional thermal modeling of lithium ion batteries, usually require a coupled electrochemical battery cell model and a lot of knowledge about battery cell internal composition which are not provided and often protected by the battery cell’s manufacturers. This paper proposes an alternative method for battery cell 2D modeling based on the extended concept of electrothermal impedance spectroscopy. Moreover, a verification of the accuracy of the obtained results will be performed. Electrothermal impedance spectroscopy and proposed approach Barsoukov et al. were the first ones presenting the concept of electrothermal impedance spectroscopy (ETIS) in [1]. Later the concept was further extended and improved by Schmidt et al., by introducing internal heat excitation and a more accurate frequency-based measurement method [2]. ETIS is a non-destructive, relatively easy to implement, ‘entropy-free’ and not requiring any a priori knowledge about cell internal composition method, which is used for determining battery cell’s heat capacity and heat conductivity. The ETIS method is based on applying a specific heat flow to the battery and measuring the amplitude and phase delay of resulting battery temperature response [2]. For the frequency-based method, this procedure is repeated for several heat excitation frequencies and thus the thermal impedance function is defined. So far, the ETIS method has been used for one point measurements. This work is extendig the ETIS concept to multi-point measurements and study the accuracy of the two-dimensional thermal model parameterized by the means of the multi-point ETIS measurement. Laboratory setup and description of experiments A laboratory setup was built based on a high-bandwidth Kepco galvanostat and several high-precision temperature sensors located at different points of the high-power LiMO2/Li4Ti5O12 battery cell (Fig.1). Frequency-based method with internal heat generation was applied and results for two spots, on the battery cell surface, are presented in Fig.2. The obtained results are demonstrating different ‘local’ thermal impedances of the battery cell and in consequence uneven battery cell heating (Fig.3 ad Fig.4). These spot dependent thermal impedance functions can be used later for two-dimensional thermal modeling of battery cell by means e.g. Cauer model [3]. Final version of the paper In the final version of the paper, the detailed results from multi-point ETIS will be presented and discussed. A two-dimensional equivalent thermal circuit based battery model will be developed and parametrized by means of multi-point ETIS. Finally, the accuracy of this two-dimensional battery thermal model [3], developed based on the multi-point ETIS, will be analyzed.

Mathematical modeling is an essential tool for the design, construction, and operation of battery cells and systems. There is a large number of battery models with various complexities and length scales of consideration. At the highest level there are system models which usually do not have a spatial dimension. The smallest component is typically a battery cell. Often they are real-time capable and can be used in HIL (hardware-in-the-loop) simulations. At the next level (length scales ~cm to ~mm) electrothermal models are used. They are capable of predicting the temperature distribution in battery cells, modules and packs and are typically based on empirical current/voltage relationships. For the length scales ~mm to ~mm electrochemical transport models are applied. In these models reaction layers are spatially resolved and basic physical transport mechanisms, e.g. the diffusion of lithium ions, are calculated. Finally, for the smallest scales (~mm to ~nm) molecular models exist which are used in material science.This work focuses on the comparison of the electrothermal and electrochemical models implemented in the multiphysics software package FIRE ®developed by AVL List GmbH [1]. Whereas electrothermal models often are three-dimensional, electrochemical models usually contain only one spatial direction (e.g. normal to the separator). In the present work, both models are used in three dimensions allowing for a detailed comparison of their results in temporal evolutions as well as spatial distributions. The most important characteristics of both models are listed in Table 1. The pros and cons are obvious: The electrochemical model provides a higher accuracy but also requires a higher calculation time. The electrothermal model needs a larger number of fitting parameters whereas the electrochemical model needs a larger number of material parameters but, hence, also allows for the investigation of different material properties of the reaction layers.This work is divided into the following three main parts:Theoretical background of both modelsModel fitting to experimental dataModel application to a realistic case with experimental validation For the second and third part a high energy lithium-ion battery of the pouch type, typically used in electric vehicles, is considered. In the second part, both the electrothermal and the electrochemical model are adjusted independently from each other to experimental discharge curves for different constant current loads and temperatures – see Figure 1 for the electrothermal model. After that, the calculation results obtained with both models are compared to each other: The temporal evolution of averaged quantities (e.g. cell voltage) as well as the spatial distribution of quantities (e.g. temperature or state of charge – see Figure 2) at fixed time steps are shown. Moreover, a deep insight into the battery is given with the electrochemical model via a visualization of results in the electrodes (e.g. lithium concentration). In the third part, finally, the models are tested for a case where the battery is loaded with a realistic current profile – see Figure 3. Voltage response and temperature evolution are being compared to experimental data. Moreover, the results of both models are being compared to each other once more.With the results shown in this work the strengths and weaknesses of electrothermal and electrochemical models are pointed out clearly. Conclusions to adequate application fields for both models can be drawn from the results. Future work includes the application of the electrothermal and electrochemical model to battery modules and stacks as well as an investigation of degradation phenomena in batteries with the electrochemical model.[1] FIRE ® v2013, Electrification / Hybridization Manual, AVL List GmbH, 2013.[2] M. Doyle and J. Newman, J. Electrochem. Soc. 143, 1890-1903, 1996.

Besides their widespread use in consumer electronics, lithium-ion batteries are becoming a technology of choice in automotive as well as grid and off-grid energy storage applications. Such large-scale battery systems require advanced and more accurate diagnostics and prognostics tools to maximize the battery performance over its lifetime. Increasingly, researchers are investigating the use of electrochemical lithium-ion battery models to enhance the state estimation and prediction capabilities of battery management systems (BMSs), for example by allowing fast charging while minimizing degradation [1-3]. In recent work [4-5], we have shown that electrochemical models, including the Newman pseudo two-dimensional (P2D) model [6], can indeed be used for lithium-ion battery state estimation. However, an important challenge remains as whether the parameters of such electrochemical models, including geometrical and physical properties, can be estimated for commercial cells from available measurements of voltage, current and temperature. Previous studies attempting the parameter estimation of electrochemical lithium-ion battery models, such as [7-8], have revealed the challenging nature of this identification problem and shown that not all model parameters can be estimated uniquely and with satisfactory confidence. We present a structural and practical identifiability analysis of an electrochemical model for lithium-ion batteries, namely the single particle model (SPM) [9]. The SPM is a simplification of the Newman P2D model, valid at low currents, which neglects electrolyte dynamics and assumes uniform reaction rate across each electrode. The structural identifiability approach allows determining which parameters of the SPM can be identified in principle. The approach first involves grouping the parameters and partially non-dimensionalizing the SPM to reveal that there are only a small number of unique grouped parameters, excluding open-circuit potential functions, required to fully parameterize the SPM. Then, by asking whether the transfer function of the linearized SPM is unique, we show that these parameters can be identified from experimental data provided that the electrode open-circuit potential functions have a known and non-zero gradient, the electrode parameters are ordered and the parameters describing kinetics in both electrodes are combined into a single charge-transfer resistance term. Considering results from this structural identifiability analysis, we then investigate the practical identifiability of the SPM by performing parameter estimation against experimental frequency-domain electrochemical impedance spectroscopy (EIS) data at various depth-of-discharge (DoD). This practical analysis confirms the crucial role played by the electrode OCP dependency on DoD for the parameter estimation of lithium-ion battery electrochemical models. If the gradient of an electrode open-circuit potential is zero, the parameters associated with this electrode cannot be identified; this is shown on figure 1a where the minimum of the cost function is extended along the anode parameter (i.e. the anode parameter is unidentifiable) since the anode OCP is almost flat at this DoD. At 10% DoD however, figure 1b, both the anode and cathode OCPs have a significant gradient resulting in a smaller minimum region, i.e.more accurate parameter estimate. Finally, assuming that the Fickian diffusivity in each electrode material is constant, we show that EIS data at several DoDs must be combined to obtain confident parameter estimates for both electrodes, figure 1c. Especially, EIS data at complementary DoDs where in turn the anode and cathode OCP slopes are significant must be combined to yield good parameter estimates. Acknowledgements Financial support is gratefully acknowledged from EPSRC UK (EP/K002252/1) and from Samsung Electronics Co. Ltd. through a collaborative research project between the Samsung Advanced Institute of Technology and the University of Oxford.

Lithium-ion batteries (LiBs) are commonly used as energy storage for many applications due to LiBs’ high energy density, long cycle life, and reliable manufacturing processes. Hence, effective performance management of LiBs is important and is implemented via Battery Management Systems (BMS)[1]. A BMS also maximises the safety and lifetime of battery packs; and therefore, needs to estimate the internal battery states including State of Charge (SoC) and State of Safety (SoS). Accurate estimation of battery SoX can be realized with the use of an accurate battery model [2].From the different types of battery models, electrochemical models (EMs) are considered the most accurate type of cell-level model [2]. EMs are sets of coupled-partial differential equation systems, where each equation is derived from the physics occurring in the cell. Of the different EMs, the partial two-dimensional model (P2D) is considered the most accurate cell-level model. However, the P2D model is complex and has no analytical solution, thus requiring computationally expensive iterative solvers for implementation. This makes practical implementation of the P2D model on BMS platforms challenging. Hence, most approaches seen in the literature prefer the use of reduced order versions (ROM) of the P2D model, that require far less computational power for implementation [1].One such ROM is the extended Single Particle Model (eSPM) which differs from the P2D model in considering each electrode to be modelled by a single spherical particle and charge conservation in the electrolyte to follow a simplified linear relationship. These assumptions help the model be accurate up to a 2C operational current, and enable an analytical solution to be derived using direct solving of the eSPM equations [2]. Due to fast-forward computation and reasonable accuracy, we’ve chosen the eSPM model for parameter estimation (PE). For accurate measurements of the model parameters, expensive and invasive experimental techniques are typically required. Non-destructive PE methods have a growing interest in the literature as they are not only non-invasive but also enable real-time dynamic PE to be done on a BMS [3].PE of EMs can be categorized under two approaches: deterministic methods and stochastic methods. Deterministic methods require a form of gradient descent to be used, wherein partial derivatives of the parameters of the model need to be calculated. This approach tends to be computationally expensive and tends to converge to local minima instead of searching the parameter space thoroughly. Hence, stochastic approaches incorporate randomness, thus searching the entire parameter space for the global optima [3]. Multiple different stochastic approaches have been employed for PE of EMs [4]. In this work, we compare and contrast some of the commonly used approaches and propose a novel multi-stage approach based on a combination of different stochastic PE algorithms.The proposed algorithm (see Figure (1)) requires three sets of data as inputs: the cell chemistry, constant current discharge data and the parameter boundaries of the eSPM model. Using these definitions, the first stage of the algorithm estimates a parameter set that has a reasonable level of accuracy i.e. less than 4% root mean square error (RMSE). The second stage of the algorithm does region-by-region optimisation, where parameters sensitive to different regions of the discharge curve are varied until the overall error is reduced to less than 1% RMSE. The proposed approach significantly reduces the estimation processing time compared to the ‘brute-force’ approach and is flexible in selecting different algorithms for each stage.Thus, contributions of this paper are briefly explained as follows: A performance comparison of the proposed algorithm with different stochastic estimation algorithms including the genetic algorithm, the particle swarm optimisation algorithm and the simulated annealing algorithm.A fast and adaptable multi-stage PE algorithm for the LiB eSPM model, with average estimation speeds of less than fifteen minutes for a typical BMS processor, with accuracy of less than 1% RMSE error.Validation of the algorithm with dynamic drive-cycle test data. [1] L. Lu et al, “A review on the key issues for lithium-ion battery management in electric vehicles,” Journal of Power Sources, vol. 226. pp. 272–288, Mar. 15, 2013. doi: 10.1016/j.jpowsour.2012.10.060.[2] S. Abada et al, “Safety focused modeling of lithium-ion batteries: A review,” J Power Sources, vol. 306, pp. 178–192, Feb. 2016, doi: 10.1016/J.JPOWSOUR.2015.11.100.[3] E. Miguel et al, “Review of computational parameter estimation methods for electrochemical models,” Journal of Energy Storage, vol. 44. Elsevier Ltd, Dec. 15, 2021. doi: 10.1016/j.est.2021.103388.[4] A. Jokar et al, “An Inverse Method for Estimating the Electrochemical Parameters of Lithium-Ion Batteries,” J Electrochem Soc, vol. 163, no. 14, p. A2876, Oct. 2016, doi: 10.1149/2.0191614JES. Figure 1

This paper reports the development of a battery model and its parameter estimator that are readily applicable to automotive battery management systems (BMSs). Due to the parameter estimator, the battery model can maintain reliability over the wider and longer use of the battery. To this end, the electrochemical model is used, which can reflect the aging-induced physicochemical changes in the battery to the aging-relevant parameters within the model. To update the effective kinetic and transport parameters using a computationally light BMS, the parameter estimator is built based on a covariance matrix adaptation evolution strategy (CMA-ES) that can function without the need for complex Jacobian matrix calculations. The existing CMA-ES implementation is modified primarily by region-based memory management such that it satisfies the memory constraints of the BMS. Among the several aging-relevant parameters, only the liquid-phase diffusivity of Li-ion is chosen to be estimated. This also facilitates integrating the parameter estimator into the BMS because a smaller number of parameter estimates yields the fewer number of iterations, thus, the greater computational efficiency of the parameter estimator. Consequently, the BMS-integrated parameter estimator enables the voltage to be predicted and the capacity retention to be estimated within 1 % error throughout the battery life-time.

Lithium-ion batteries are widely used as power sources in industrial applications. Electrochemical models and simulations are crucial to disclose many details that cannot be directly measured through experiments. Parameter identification of an accurate electrochemical model is much more cost-effective than direct and destructive measurement methods. However, the complex structure and strong nonlinearity of electrochemical models will make the parameter identification very difficult. Additionally, time-consuming electrochemical simulations can significantly limit the identification efficiency. This article proposes a surrogate-model-based scheme to achieve high-efficiency parameter identification of an electrochemical battery model. To be specific, the proposed method is implemented by the close integration of an evolutionary algorithm and a surrogate model. A sensitivity-based identification strategy is first designed to alleviate the difficulty of optimization. Then, a surrogate model is developed from historical data to gradually approach the objective function used for parameter evaluations. Finally, an evolutionary algorithm is employed to find promising solutions by minimizing the output of the surrogate model. Simulations and experimental studies demonstrate the effectiveness and high efficiency of the proposed method.

Starting from the inexact interior-point framework from Curtis, Schenk, and Wächter [SIAM J. Sci. Comput., 32 (2012), pp. 3447--3475], we propose an effective Schur-complement slack-control preconditioner for the full Lagrangian Hessian matrix needed at each Newton iteration. Together they yield a scalable, robust, and highly parallel method for the numerical solution of large-scale nonconvex PDE-constrained optimization problems with inequality constraints. Because it uses the full Hessian matrix, modifying it whenever needed, the method not only is globally convergent, but also converges fast locally. Our preconditioner is not tailored to any particular class of PDEs or constraints, but instead judiciously exploits the sparsity structure of the Hessian. Numerical examples from PDE-constrained optimal control, parameter estimation, and full-waveform inversion demonstrate the robustness and efficiency of the method, even in the presence of active inequality constraints.

This paper presents developments and validations of Lithium Nickel Manganese Cobalt Oxide (NMC) battery cell and battery pack models. A 7.4 kWh NMC battery pack is installed on an electric Tuk-Tuk equipped with a 3 kW motor. The battery pack is configured of four cells connected in parallel and 20 sets of the four cells connected in series. Based on the first-order and the second-order equivalent circuit models, battery cell and battery pack models are developed in Matlab/Simulink environment. For parameter identifications of the cell model, pulse discharge tests are performed at room temperature with 10% State of Charge (SOC) interval. The lumped heat capacitance method is applied to the battery models. The battery cell model is validated with SOC, cell center temperatures and terminal voltage using the experimental data of constant current discharge tests. With the battery pack configuration, the battery cell model is scaled up to match the battery pack voltage and its capacity. The dynamics of the battery pack models are validated with the vehicle testing data using the no-load, wide-open throttle and real road tests. The simulation results show good agreement with the test data. Using the developed battery models, vehicle performance and its energy consumption can be improved by optimizing battery cell and pack configurations.

This paper presents a procedure for deriving and tuning a compact and solvable differential-algebraic equations (DAE) model for the LiFePO4-graphite battery cell. Electrochemical models of the battery cell are typically represented by complex partial differential equations of state variables and parameters whose relationships can be highly nonlinear. Several software packages are available for numerically solving these equations for simulations. However, their long computing time due to the complexity of the model is a major bottleneck for control and monitoring applications. A reduced order model (ROM) can drastically decrease the simulation time with a minimal loss of the prediction accuracy. A ROM is constructed by simplifying or approximating parameters of the full-order model of the battery cell. For on-line diagnostics and real-time control purposes, a lower-order compact representation of complicated transport and diffusion phenomena of a lithium-ion battery is highly desirable. In this paper, the psedo-2D (P2D) electrochemical model of the battery cell is simplified based on polynomial representations of pore-wall flux and lithium concentrations in a similar manner to previous studies. However, instead of adopting the standard Galerkin method which tends to increase the number of equations, this paper adopts Subramanian et al.’s method which demonstrated a potential for producing a compact description of a lithium-ion battery cell. While a traditional Galerkin method requires selecting particular polynomial representations which satisfy the boundary conditions, Subramanian et al.’s method selects a simple polynomial which can be analytically integrated foregoing setting up separate polynomials for solid and electrolyte potentials. This technique, therefore, has a potential to drastically reduce the size of simplified system. However, since the boundary conditions should be satisfied by additional analytical equations, one requires judicious choice of Galerkin formulations for electrolyte concentration in order to build a solvable DAE system. This paper introduces a systematic way of assessing a Galerkin formulation for the electrolyte concentration based on the theorem of Weierstrass. It is shown that each different Galerkin formulation converts the governing equation and its boundary conditions for the electrolyte concentration into a different set of linear DAE’s. However, not all DAE systems are solvable and some may induce numerical instabilities. It is shown that a full rank of the matrix pencil can ensure that a certain Galerkin formulation produces a solvable DAE system. Usefulness of the proposed method is illustrated by comparing several different Galerkin formulations for the LiFePO4-graphite system. In addition, this paper improves the accuracy of the model by accounting for the hysteresis of open-circuit voltage in LiFePO4 using a differential equation model. When the order of the polynomials are chosen at fours, the P2D model of the cell is simplified into a system of 24 DAE’s. This size is significantly smaller than those of comparable ROMs in the literature. The developed DAE system is solved by using SUNDIAL’s IDA solver where a typical charge/discharge cycle can be simulated under 10 seconds on a regular personal computer. The parameters of the developed DAE model are tuned based on charge/discharge experimental data from a commercial LiFePO4-graphite battery. In this paper, a systematic tuning of the model parameters is investigated by exploiting the fast simulation capability of the developed model. Based on a sensitivity study, six model parameters are chosen for tuning. These tuning parameters, thus selected, include the solid phase diffusivity of the negative electrode, initial hysteresis parameters, reaction rate constants, and the contact resistance. The genetic algorithm is applied to simultaneously find the six model parameters that minimize the sum of squared errors in charge/discharge voltages. The tuned model shows a good agreement with the experimental data at rates upto 4C. Contributions of this paper can be summarized as follows. This paper proposes a new method to ensure a compact DAE system is solvable when it is driven from the original higher-order model of the Lithium-ion battery cell. Moreover, a systematic tuning of the model parameters using the global optimization method is demonstrated by exploiting the computational efficiency of the simplified model. When coupled with the model for describing the hysteresis of the battery cell, the tuned model shows a good agreement with the experimental data from a LiFePO4-graphite battery cell.

Development and parameterization of a control-oriented electrochemical model of lithium-ion batteries for battery-management-systems applications

Real-time estimation of battery internal states and physical parameters is of the utmost importance for intelligent battery management systems (BMS). Electrochemical models, derived from the principles of electrochemistry, are arguably more accurate in capturing the physical mechanism of the battery cells than their counterpart data-driven or equivalent circuit models (ECM). Moreover, the electrochemical phenomena inside the battery cells are coupled with the thermal dynamics of the cells. Therefore, consideration of the coupling between electrochemical and thermal dynamics inside the battery cell can be potentially advantageous for improving the accuracy of the estimation. In this paper, a nonlinear adaptive observer scheme is developed based on a coupled electrochemical–thermal model of a Li-ion battery cell. The proposed adaptive observer scheme estimates the distributed Li-ion concentration and temperature states inside the electrode, and some of the electrochemical model parameters, simultaneously. These states and parameters determine the state of charge (SOC) and state of health (SOH) of the battery cell. The adaptive scheme is split into two separate but coupled observers, which simplifies the design and gain tuning procedures. The design relies on a Lyapunov's stability analysis of the observers, which guarantees the convergence of the combined state-parameter estimates. To validate the effectiveness of the scheme, both simulation and experimental studies are performed. The results show that the adaptive scheme is able to estimate the desired variables with reasonable accuracy. Finally, some scenarios are described where the performance of the scheme degrades.

Fault detection of new and aged lithium-ion battery cells in electric vehicles

<div class="section abstract"><div class="htmlview paragraph">Lithium-ion batteries have been applied in the new energy vehicles more and more widely. The inconsistency of battery cells imposes a lot of difficulties in parameter and state estimations. This paper proposes a new algorithm which can online identify the parameters of each individual battery cell accurately with limited increase of computational cost. An equivalent circuit battery model is founded and based on the RLS (recursive least squares) algorithm, an optimization algorithm with the construction of weight vectors is proposed which can identify the parameters of lithium battery pack considering inconsistency of single battery cell. Firstly, the average value of the parameters of the battery pack is identified with the traditional RLS algorithm. Then the ratios between the parameters of each battery cell can be deduced from the mathematical model of battery. These ratios are used to determine the weight vector of each parameter of individual battery cells. Finally, with the average battery parameters and the weight vectors, we can obtain the parameters of each cell. The proposed weighted algorithm (WA) is verified with the bench test data. The results show that the WA can achieve a good accuracy of parameter identification with a low computation cost.</div></div>

As the core power unit of new energy vehicles, the accurate modeling of power batteries is of great significance for evaluating their operating status, diagnosing faults throughout their lifecycle, and ensuring safety control under multiple operating conditions. The electrochemical model represented by the P2D model serves as a mechanistic model that can characterize the internal electrochemical reaction process of batteries on a microscale. Its accurate description of the aging and heating behavior of power batteries is an important basis for evaluating the capacity degradation, increase in internal resistance, uneven heating, and inconsistent performance of battery modules. The paper summarizes the latest advances in electrochemical modeling of lithium-ion power batteries, analyzes the coupling methods and application status of electrochemical models with equivalent circuit models, aging models, and thermal models, and focuses on the problem of numerous parameters and difficult identification of electrochemical models. In this paper, the advantages and disadvantages of the single particle model, single particle model with electrolyte, electrochemical mean model, solid-liquid phase reconstruction model, one-dimensional electrochemical model and other methods are compared with each other and analyzed for reducing the order of power battery electrochemical models, the key difficulties in characterizing electrochemical model order reduction are pointed out, and the research trends of electrochemical model reduction order reconstruction methods are prospected, in order to provide direction for the research on electrochemical model reduction order reconstruction of power batteries.

Electrochemical aging model of lithium-ion battery with impedance output and its parameter sensitivity analysis and identification