IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY 1 Electrochemical Model-Based State of Charge and Capacity Estimation for a Composite Electrode Lithium-Ion Battery Alexander Bartlett, James Marcicki, Simona Onori, Giorgio Rizzoni, Xiao Guang Yang, and Ted Miller Abstract—Increased demand for hybrid and electric vehicles consisting of various circuit elements arranged in series or has motivated research to improve onboard state of charge and parallel combinations such that the circuit output voltage state of health estimation. In particular, batteries with composite dynamics mimic that of the battery. Although ECMs are electrodes have become popular for automotive applications due to their ability to balance energy density, power density, computationally inexpensive, considerable effort must be made and cost by adjusting the amount of each material within to parameterize these models, as the parameters are typically the electrode. State of health estimation algorithms that do functions of SOC, temperature, current direction, and battery not use electrochemical based models may have more difficulty life [2], [3]. Additionally, ECM parameters have little physical maintaining an accurate battery model as the cell ages under meaning, and associated aging models must rely on empirical varying degradation modes, such as lithium consumption at the solid-electrolyte interface or active material dissolution. Fur- correlations between SOH and damage factors such as total thermore, efforts to validate electrochemical model-based state current throughput, operating temperature, and depth of dis- estimation algorithms with experimental aging data are limited, charge [4]. Nonetheless, ECMs have been used successfully particularly for composite electrode cells. In this paper we first for SOC and SOH estimation using extended Kalman filters present a reduced order electrochemical model for a composite (EKFs) and other methods [5]–[10]. LiMn2O4-LiNi1=3Mn1=3Co1=3O2 (LMO-NMC) electrode battery that predicts the surface and bulk lithium concentration of each Recently, increased attention has been given to electrochem- material in the composite electrode, as well as the current split ical models for SOC and SOH estimation. These models are between each material. The model is then used in dual nonlinear based on first principles in porous electrode theory [11] and observers to estimate the cell state of charge and loss of cyclable therefore have the potential to predict cell performance more lithium over time. Three different observer types are compared: accurately, as well as provide more information about the the extended Kalman filter, fixed interval Kalman smoother, and particle filter. Finally, an experimental aging campaign is used internal battery states such as lithium concentrations and re- to compare the estimated capacities for five different cells to the action overpotentials. However, uncertainty in electrochemical measured capacities over time. parameters can limit the accuracy of the model and resulting state estimates [12]. Furthermore, electrochemical models rely on partial differential equations (PDEs) to describe lithium I. INTRODUCTION diffusion and potential gradients throughout the electrode and EHICLE electrification continues to be a key topic electrolyte. Solution methods for PDEs can be computationally V of interest for automotive manufacturers, with lithium- expensive, making them ill-suited for most practical onboard ion batteries being the technology of choice for hybrid and control and estimation algorithms. A common approximation, electric vehicles. State of charge (SOC) and state of health known as the single particle model (SPM), is to neglect the (SOH) estimates are essential inputs to the vehicle’s battery spatial variation of concentration and potential throughout the management system (BMS) in order for the battery pack to cell and treat each electrode as a single, spherical particle, operate efficiently and safely [1]. Knowledge of SOC allows subject to an appropriately scaled, spatially constant current the BMS to predict the available instantaneous power, while flux. To further facilitate control and estimation, researchers ensuring the battery is operating within safe limits. SOH can have used various model order reduction techniques to reduce be defined in a variety of ways, but typically refers to the the PDEs governing diffusion in the spherical particles to degradation of capacity and increase in internal resistance as low order, ordinary differential equations (ODEs). A reduced the battery ages. order electrochemical model, linearized over a typical op- Traditionally, SOC and SOH estimates are made with the erating range was used in [13] in a Kalman filter to es- use of lookup tables or equivalent circuit models (ECMs) timate lithium bulk and surface concentration. The authors This work was supported by the Ford University Research Program. showed that rather than enforcing strict limits on cell voltage A. Bartlett and G. Rizzoni are with the Center for Automotive Research, to avoid damaging the battery, enforcing limits on lithium The Ohio State University, Columbus, OH, 43212, USA e-mail: (bartlett.137, surface concentration and solid/electrolyte potential difference [email protected]). J. Marcicki, X.G. Yang, and T. Miller are with Ford Motor Company, Re- allowed the cell voltage limits to be exceeded without risking search and Innovation Center, Dearborn, MI, 48124, USA e-mail: (jmarcick, electrode saturation/depletion and lithium plating. In [14], xyang11, [email protected]) an averaged, single particle model with discretization of the S. Onori is with the Automotive Engineering Department, Clemson Uni- versity, Greenville, SC, 29607, USA e-mail: ([email protected]) diffusion PDEs was used in a Kalman filter to estimate the Manuscript received Aug XX, 2014; revised December XX, 2014. bulk and surface concentrations of the positive electrode. The IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY 2 authors recognized the unobservability of the complete cell LMOI Liquid model that includes both electrodes (since the only available Solid Charge diffusion LMO diffusion measurement depends on the difference between the electrode Contact transfer + I Li Charge Solid potentials), and therefore only applied the observer to the C Contact transfer diffusion positive electrode states. Single Particle NMC Loss of spatial information through-the-thickness of the Liquid Sub-Model Sub-Model cell is avoided in [2], [16], [17] by using observers capable NMCI Multiple Particle of handling PDE-based models, although these approaches Sub-Model add to the computational cost. The authors of [17] also Fig. 1. Schematic of the electrochemical model, with a single negative discussed the unobservability of the complete battery system, electrode particle and two positive electrode particles acting in parallel. and approximated the cathode states at their equilibrium, since the average positive concentration can be related to the average negative concentration via conservation of lithium. In the simplest case (at steady state), the measured cell There have been a few attempts to use electrochemical based voltage, V , is just the difference between each electrode open models for online SOH estimation using dual SOC/parameter circuit voltage (OCV), evaluated at their respective steady state estimation. In [18], the active material fractions of each concentrations, i.e. V = Up(¯cp) − Un(¯cn). However as the electrode are estimated along with the bulk and surface SOC, cell is charged or discharged, various sources of overpotential using an unscented Kalman filter (UKF). SOH estimation was cause the measured voltage to deviate from this steady state done in [15], [17], [19] by applying least squares techniques solution. The overpotentials can be categorized as acting in to estimate model parameters associated with aging; however either the solid (electrode) or liquid (electrolyte) phase, and the computational effort required by these approaches may sub-models for each phase can be developed. The dynamic limit their applicability to onboard estimation. Additionally, cell voltage (9) is then obtained by subtracting out the solid electrochemical model-based SOH estimation algorithms that and liquid phase overpotentials from the electrode OCVs. are validated with experimental aging data are scarce in the literature, and to the author’s knowledge, no such studies have A. Electrode Sub-Model been conducted on cells with composite electrodes. Composite The governing equations for Li concentration and electrode electrodes add to the complexity of the electrochemical model, potential are shown in (1-5) for a given particle, i. Subscript, particularly if it is desired to estimate the SOC and SOH of i, refers to the negative particle, positive LMO particle, or each electrode material individually. positive NMC particle: n, LMO, or NMC, respectively. Fick’s In this paper, a reduced order, electrochemical model is law of mass diffusion in spherical coordinates (4), governs the used for SOC/SOH estimation on a composite LiMn O - 2 4 concentration throughout a spherical particle. The boundary LiNi Mn Co O (LMO-NMC) cathode cell with a 1=3 1=3 1=3 2 conditions for this PDE are set so as to require symmetry about graphite anode. The model is based on the single particle the particle center and impose a flux at the particle surface, model with added liquid phase dynamics, but considers two dictated by the current density, j (t). particles in parallel to represent