Electrochemical Model-Based State of Charge and Capacity Estimation

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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 the composite cathode. The i The single particle approximation removes the spatial de- paper then presents a novel, numerically inexpensive method pendence of the current density, allowing it to be defined for solving for the current split and resulting SOC of each only as a function of time for a given particle. To form the cathode particle (part of preliminary work presented in [20]), current density, the applied current is multiplied by a current and a novel combined SOC/SOH estimation algorithm to split factor, β , defined as the fraction of current flowing to estimate capacity fade resulting from loss of cyclable Li. i a given particle within an electrode. For the single particle Finally, the estimation algorithm is validated with experimental negative electrode, β = 1, but for the multiple particle aging data, by comparing 3 different observer types (EKF, n positive electrode, β and β vary dynamically and Kalman smoother, and particle filter). LMO NMC are calculated at each time step, as shown in section III.A. II.MODEL DEVELOPMENT The current is normalized by the electrode volume, including a parameter, , representing the active material volume fraction The model used in this paper is based on the traditional i split. Again, for the single particle negative electrode, = 1; single particle model, with the addition of liquid phase n however for the multiple particle positive electrode, fitting diffusion dynamics to improve model accuracy under high experimental data gave a material split of = 0.64 and current. Similar models have been presented many times in LMO = 0.36 [22]. literature [2], [13], [14], [16], [18], [21], however in order to NMC The electrode potential (5) is calculated by subtracting the accommodate the composite LMO-NMC positive electrode, kinetic and ohmic overpotentials from the OCV, evaluated at the positive electrode is modeled as two particles acting in the particle surface concentration. parallel (shown in Fig. 1). Therefore the potentials of each positive electrode particle model are equal, but the current is split between each particle according to its dynamic, effective B. Electrolyte Sub-Model impedance. This idea of representing a composite electrode as Similar to the solid phase, the liquid phase contains sources two particles acting in parallel has been presented in previous of concentration and ohmic overpotential. The concentration work [20], [22]. of Li throughout the electrolyte is governed by Ficks law in IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY 3

TABLE I GOVERNING EQUATIONS FOR THE SOLID PARTICLE AND LIQUID SUB-MODELS.

Sub-Model Variable Governing Equation RT¯ −1 ji(t) Solid Particle* Kinetic overpotential, ηi ηi(t) = sinh (1) αF 2i0,i(t) p Exchange current density, i0,i i0,i(t) = F ki cs,i(t)ce,i(t)(cmax,i − cs,i(t)) (2) I(t)βi(t)Ri Averaged particle current density, ji ji(x, t) = ji(t) = (3) 3ALiam,ii ∂ci Di ∂ 2 ∂ci ∂ci ∂ci ji(t) Li concentration**, ci ∂t = r ∂r r ∂r , ∂r = 0 at r = 0,Di ∂r = ∓ F at r = Ri (4) Electrode potential**, φi φi(t) = Ui (cs,i(t)) ± ηi (cs,i(t), βi(t)I(t)) ± Rc,iβi(t)I(t) (5) 2 + ∂ce ∂ ce 3(1−t0 ) Liquid Li concentration**, ce e = De 2 ± ji(t) (6) ∂t ∂x FRi ¯ + ∂φe ie(t) 2RT (1−t0 ) ∂ ln(ce) Electrolyte potential, φe = + (1 + γ) (7) ∂x σe F ∂x 1 1 ¯ + I(t)( 2 Lp+Lsep+ 2 Ln) 2RT (1−t0 ) c¯e,n(t) Electrolyte voltage, Ve(t) = + (1 + γ) ln (8) σeA F c¯e,p(t) Ve = φe(x = 0) − φe(x = Lc) Cell Cell voltage, VV (t) = φp(t) − φn(t) − Ve(t) (9) *Note that subscript, i, refers to the negative particle, positive LMO particle, or positive NMC particle: n, LMO, or NMC, respectively. **Deﬁning discharge current to be positive, occurrences of ± or ∓ correspond to negative (top sign) and positive (bottom sign) electrodes.

  0 1 0  0 Cartesian coordinates (6), with a source term related to the 0 A = 0 0 1 Bi =   current density. Since ji(t) is considered piecewise constant i     0 1 b2,i ∓ βi(t)Ri 1 within a given electrode and zero in the separator, (6) is b3,i b3,i 3F ALiam,ii b3,i solved separately in each region, subject to coupling bound- cs,i a0,i a1,i a2,i = xi ary conditions that match the concentration and flux at the c¯i a0,i a0,ib2,i a0,ib3,i electrode/separator interfaces. To obtain the liquid phase voltage across the entire cell, (7) φi = Ui(cs,i) ± ηi(cs,i, βi(t)u) ± Rc,iβi(t)u (11) is integrated directly from x = 0 to x = Lc, where ie(t) is where both the bulk and surface concentrations are written as considered constant in the separator and linearly decreasing linear combinations of the states, xi, the input, u, is current, to zero from the electrode/separator boundaries to the current and the output electrode potential is a nonlinear function of collectors. the states and input. Although there are 2 separate transfer functions for bulk and surface concentration (1st order and III.MODEL SOLUTION METHOD 3rd order, respectively), both metrics can be written as linear combinations of the same 3 states, through some algebraic The concentrations of Li in the solid and liquid phases manipulation (as shown in [23]). In other words, there are are governed by PDEs, however this is undesirable if the 6 total states in the multiple particle positive electrode sub- model is to be used for onboard estimation. Instead, the Pade´ model, and 3 states in the single particle negative electrode approximation method for model order reduction proposed in sub-model. The states are updated in discrete time using a [21], [23] was leveraged here to approximate the PDE-based finite difference solution. Constants a0,i, a1,i, a2,i, b2,i, and model with a system of low order ODEs. b3,i are obtained from the third order Pade´ approximation, and are functions of the particle radius and diffusion coefficient [21], [23]. A. Solid Phase Solution Since the positive electrode consists of two particles acting In the solid phase, the important metrics for tracking cell in parallel, the potentials for each particle are equal and the SOC and available power are the particle bulk concentration current is multiplied by a current split factor, βi. For the and particle surface concentration, respectively. A first order positive electrode, βi must be solved at each time step to linear transfer function between the bulk concentration and determine how the current is allocated, with the constraint input current is obtained simply by integrating (4) in the that the currents going to each particle sum to the total Laplace domain from r = 0 to r = Ri. However in the case current (i.e. βLMO + βNMC = 1). The βi term appears of surface concentrations, a Pade´ approximation is used to in both the dynamics and output equation, so one possible obtain a low order, linear transfer function relating surface solution method is to guess the value of each βi, update concentration to input current for each particle [21], [23]. A the states, calculate the particle potentials, and iterate until third order approximation is used for the solid phase PDE, the two particle potentials converge to the same value. This since it gives good accuracy in the frequency range typically approach could be implemented using a standard nonlinear seen in a vehicle drive profile [21], while still mitigating the solver, however it may not be ideal for online estimation computational effort required for higher order models. since it requires iteration at each time step. Alternatively, The electrode particle sub-model is formulated in state an approximated solution is obtained more quickly using the space, as follows: following linearization procedure: 1) At each time step, k, assume that changes in diffusion x˙ i = Aixi + Bi(t)u (10) dynamics are much slower than changes in current flow. IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY 4

With this assumption, the surface concentration in the 4 output equation (11) can be treated as constant for a

given time step. Since the surface concentrations at time 3.5 Predicted step k are not known a priori, the values at the previous Measured

time step are used to evaluate the output equation. The Cell Voltage (V) 3 output equation then becomes a nonlinear function of 0 500 1000 1500 2000 2500 3000 3500 a) current only: 1 i

φk ≈ Ui,k−1 − ηi,k(βi,kIk) − Riβi,kIk (12) β LMO 2) Now the nonlinear kinetic overpotential term in (12) can 0.5 NMC be linearized using a Taylor series expansion about its

value at the previous time step: Current Split, 0 ∂f 0 500 1000 1500 2000 2500 3000 3500 b) βi,kIk ≈ f(ηi,k−1) + (ηi,k − ηi,k−1) ∂η 1 η=ηi,k−1 (13)

= gi,k + hi,k(ηi,k − ηi,k−1) where the function f is the kinetic overpotential term 0.5 LMO (solved for the current, βi,kIk), and gi,k and hi,k are NMC constants. 0 3) Finally, the current splits, βi,k, can be written as the so- Normalized Concentration 0 500 1000 1500 2000 2500 3000 3500 Time (s) lution to a matrix inversion problem, with an additional c) equation requiring the current splits to sum to 1: 1 Fig. 2. a) Model prediction of the cell voltage compared to the measured Rc,LMO + 0 1   hLMO,k βLMO,kIk voltage, b) current split factor, βi for each material, and c) normalized 1 concentration for each material, for a 1C constant current discharge.  0 Rc,NMC + h 1 βNMC,kIk NMC,k ˜ 1 1 0 φp,k gLMO,k  ULMO,k + − ηLMO,k−1  hLMO,k βi, since it relies on the previous state estimate (as per step gNMC,k = UNMC,k + − ηNMC,k−1 1). This would in turn cause an error in the state estimate  hNMC,k  Ik calculated at the next time step. However, this does not (14) affect the stability of the state estimate. Even if there is an extreme error in βi, eventually the concentration estimate In addition to the current splits, solving (14) also gives of one of the materials will become saturated, which will φ˜ a value for the linearized output electrode potential, p,k; decrease β for that material, thereby driving the estimate however, this value is discarded at this point. Instead the i back towards the correct value. Instead, calculation of βi original nonlinear output equation (11) is used to calculate using an unconverged state estimate can be categorized as the electrode potential, thereby avoiding the voltage prediction a model error, which affects the estimate accuracy, but will inaccuracies seen with the linearization process. In this way, not cause the estimate to become unstable. Furthermore, in a the errors associated with the output linearization only affect real automotive application, there are periodic instances when β the calculation of i, and not the calculation of the output the current is near zero, and the influence of β is negligible. β i voltage (apart from the indirect effect from i). During these times the state estimate can converge accurately, Figure 2 shows the predicted current split and resulting SOC thereby minimizing the error in βi once the current resumes. for each cathode material for a constant 1C discharge. The To summarize the solution method for the composite posi- predicted voltage shows good agreement with the measured tive electrode: at each time step, k, the current split fractions, voltage (more rigorous validation of the model was done in βi,k, are first calculated using the linearization method outlined [22]). More LMO is utilized initially at high voltages, until above (14), the states are then updated in discrete time using it is nearly depleted, at which point NMC carries the bulk of the pre-calculated values for βi,k (10), and finally the output the current for the remainder of the discharge. This result is potentials are calculated using the updated states and pre- in situ consistent with an x-ray diffraction (XRD) study of a calculated values for β (11). composite LMO-NMC cathode [24]. i,k One drawback of this linearized solution method is that for instances where the current abruptly changes in magnitude B. Liquid Phase Solution and/or direction, the calculated βi for a given particle can Equation (8) governs the liquid phase voltage across the greatly exceed 1 (resulting in a large negative βi for the other cell and depends on the Li concentrations at each electrode. particle). While it is unclear if this occurrence makes physical Therefore the PDE governing Li concentration is solved at sense, it typically lasts for only a few time steps before the each electrode end (x = 0, and x = Lc) using a first βis return to more sensible values between 0 and 1. order Pade´ approximation. This results in two first order It should also be noted that if the state estimate has not transfer functions relating liquid phase concentration at each yet converged, there will be some error in the calculation of electrode to the current input. Only a first order approximation IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY 5 is needed for the liquid phase concentration since it shows Measurement: very good accuracy throughout the frequency range of interest Vexp(t) Pseudo- Estimates: [21]. In state space, the 2 states in xe represent the liquid measurement: cLMO(t) concentrations at each electrode, the input, u, is current, and e(t) (t) I(t) Open-Loop p cs,LMO(t) the output voltage is a nonlinear function of the states and (t) Observer Model n c (t) input. NMC cs,NMC(t) x˙ e = Aexe + Beu (15) " # " # 1 + 1 Fig. 3. Block diagram of the algorithm for estimating the positive electrode b 0 1 − t0 − b A = 1,e,p B = 2,e,p states. The negative electrode and liquid phase are simulated open-loop in e 0 1 e F AL 1 b1,e,n c e b2,e,n order to generate a pseudo-measurement of the positive electrode potential. Alternatively, the negative electrode states could be estimated by simulating c¯e,p the positive electrode open-loop. = 1 1 xe c¯e,n 1 1 ¯ + u( 2 Lp + Lsep + 2 Ln) 2RT (1 − t0 ) c¯e,n Ve = + (1+γ) ln derivatives easier computationally, a cubic spline is used to σeA F c¯e,p evaluate the OCV at a given concentration. The cubic spline (16) is continuously differentiable at the spline transitions up to the Again, the constants b , b , b , and b are ob- 1,e,p 1,e,n 2,e,p 2,e,n second derivative, so proving observability for a given spline tained from the first order Pade´ approximation [21]. segment guarantees observability for the entire OCV curve. Substituting an arbitrary cubic spline segment for the OCV IV. SYSTEM OBSERVABILITY term, the electrode sub-model becomes: In order to design an observer, the system must be shown to be observable (or at least detectable) to ensure that the x˙ i = f(xi, u) = Aixi + Bi(t)u (17) state estimates converge to the true states in a finite time [25]. Observability of a system requires that the state trajectory can be uniquely reconstructed based solely on knowledge 3 2 ! s3,ic s2,ic s c of the output measurement and input. However not all of y = g(x , u) = s,i + s,i + 1,i s,i + s i c3 c2 c 0,i the states of the complete battery model, including positive max,i max,i max,i (18) electrode, negative electrode, and liquid phase sub-models, − ηi(cs,i, βiu) − Rc,iβiu are observable (or detectable) since only the cell voltage can realistically be measured in an automotive application. In other where s0,i, s1,i, s2,i, and s3,i are the OCV cubic spline words, there is no way to uniquely identify the states of both coefficients, and cs,i is a linear function of the states, as shown electrodes if the only available measurement is the difference in (10). The Lie derivatives, Lf , for a nonlinear system with between positive and negative electrode potentials. To solve state dynamics x˙ = f(x, u) and output equation y = g(x, u) this observability problem, the states of only one electrode form the vector:      0  are estimated at a given time, while the liquid phase and y g Lf opposite electrode states are simulated open-loop. For example 1  y˙   g˙   Lf  if the positive electrode states are to be estimated, the negative l(x, u) =   =   =   (19)  .   .   .  electrode and liquid phase potentials are calculated using the  .   .   .  (n−1) (n−1) n−1 open-loop model. These potentials are subtracted from the y g Lf measured cell voltage to obtain a ”pseudo-measurement” of The observability matrix is obtained by taking the Jacobian of positive electrode potential. This pseudo-measurement is then the Lie derivate vector: used as feedback in the observer to estimate the positive  ∂L0 ∂L0  electrode states. The process is outlined in Fig. 3 for estimating f ... f ∂x1 ∂xn the positive electrode states. Clearly, since the liquid phase and  . .  O =  . .  (20) one electrode are simulated open-loop, any initial condition  . .  ∂Ln−1 ∂Ln−1 and model errors in these sub-models will be projected forward f ... f to adversely affect the opposite electrode estimate. ∂x1 ∂xn Even though the estimated states are limited to just one The result is too lengthy to show here, in particular due electrode, that electrode sub-model must be shown to be to differentiating the hyperbolic sine function in ηi, but the observable with respect to a pseudo-measurement of its output process is outlined in appendix B. The observability matrix potential. The electrode sub-model has linear state dynamics has full rank for non-zero parameter values, proving that the (10) with a nonlinear output equation (11). The goal is to show electrode particle sub-model is locally observable. Although that all 3 states in the sub-model are observable, allowing for the LMO-NMC composite positive electrode OCVs are fairly estimation of both the surface and bulk electrode concentra- steep, indicating strong observability, the graphite negative tions, cs,i and c¯i. Local observability of a nonlinear system can electrode OCV does have various flat regions (where s1,n, be shown with use of Lie derivatives [25]. The open circuit s2,n, and s3,n are close to zero), indicating poor observability voltage term, Ui, is often calculated via lookup table or by an over certain SOC ranges. This problem will be discussed in empirical functional fit; however to make calculating the Lie detail in section VII.B. IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY 6

V. EXPERIMENTAL DATA COLLECTION TABLE II TUNING PARAMETERS FOR EACH OBSERVER TYPE An aging campaign was conducted using 15 Ah (nominal) automotive pouch cells with a composite LMO-NMC cathode Positive electrode state estimation (Obsp) and graphite anode, cycled under different operating condi- Observer Parameter   tions. The cells were cycled using the charge depleting (CD) 1500 0 0 Q = 10−19   current profile defined by the United States Advanced Battery EKF*  0 1.5 0  Consortium (USABC) [26], which is representative of a plug- 0 0 0.0015 −4 in hybrid vehicle (PHEV) application. The CD current profile R = 2.24 × 10 is shown in Fig. 9, and was repeated until the cell reached 45, Negative electrode state estimation (Obs ) 35, or 25% SOC. Each CD cycle was followed by a constant n Observer Parameter current, constant voltage (CCCV) charge at a C/3, 3C/2, or   ◦ 2500 0 0 5C charge rate. All cells were maintained at a constant 30 C Q = 10−17   EKF  0 2.5 0  by Peltier junctions. Periodic capacity assessments were con- 0 0 0.0025 ducted at approximately every 2000-5000 Ah of throughput. R = 2.45 × 10−5   The capacity assessments involved an initial 1C CCCV charge 2500 0 0 Q = 10−17   to 4.15 V, followed by a 1C discharge to 2.8 V, followed by Kalman Smoother  0 2.5 0  a second 1C CCCV charge. The measured capacity was taken 0 0 0.0025 as the average of the discharge and charge capacities. The R = 2.45 × 10−5   electrode OCVs (shown in Figs. 4 and 7) and parameter values 1200 0 0 Q = 10−15   from [22] were used to calibrate the model. Particle Filter**  0 1.2 0  0 0 0.0012 R = 2.45 × 10−5 VI.OBSERVER DESIGN Np = 10 Three different observer types are used for state estimation *Σw and Σv are identical for LMO and NMC particles. **Gaussian sensor and process noise are assumed. in order to compare their estimation accuracy and computational effort. Specifically, the extended Kalman filter, fixed interval Kalman smoother, and particle filter are implemented and compared. The implementation of each observer can be be performed in real time. It is necessary to wait until all found in appendix C, but brief descriptions will be given here. the measurements within the interval become available before The EKF [5], [6], [29] is an extension of the Kalman filter, even the initial smoothed state estimate can be obtained. optimal for linear systems, to nonlinear systems where the However for SOH estimation this is not a significant problem, process and sensor noise are assumed to be uncorrelated, since in practice aging parameters change very slowly and white, and Gaussian. As is typical of model-based observers, need not be continuously updated in real time. the EKF involves both a prediction step and a correction The particle filter is a Monte Carlo method used for state step. In the prediction step, open-loop predictions of the state estimation by representing the posterior probability density vector and output are obtained from the model. The state function (PDF), i.e. the conditional PDF of the state given prediction is then adjusted in the correction step by utilizing a measurement, by a set of weighted random samples, or feedback from the measurement. The feedback gain (optimal particles [29]–[31]. Each particle is assigned a weight based on for a linear system) is calculated at each time step, based the likelihood that it represents the true state. The advantage on knowledge of the model and noise covariance matrices. of the particle filter is that it does not require linearization of For nonlinear systems, calculation of the observer gain relies the nonlinear model, nor does it require that the process and on a linearization of the nonlinear system dynamics and sensor noise be Gaussian. Choosing the number of particles to output equation. In practice, the process and sensor noise sample, Np, represents a tradeoff between estimate accuracy covariance matrices are not necessarily known, and there may and computational effort. be significant model errors. Therefore, the noise covariances The relevant parameters for each observer are outlined in are typically used as tuning factors to achieve the desired table II, and were tuned to achieve a desired compromise observer behavior for a particular system. between model error robustness, state convergence time, and The fixed interval Kalman smoother (or forward-backward computational effort. smoother) has the potential to improve upon the traditional EKF by essentially running the filter twice, once forward in VII.METHODSFOR SOC/SOHESTIMATION time and once backward in time, over a fixed interval of measurements [29], [34]. Once both forward and backward A. Estimating Cell SOC estimates are obtained, they are combined in an optimal way Estimating the states of one of the electrode particle sub- to minimize the estimation error covariance. The resulting models gives an estimate of both bulk and surface concentra- smoothed estimate has a smaller estimate error covariance than tions of the particle; however without the use of in situ XRD either of the forward or backward estimates individually. The or similar techniques, it is only possible to experimentally obvious disadvantage of this method, other than the added validate the bulk concentration estimate by comparing it computational effort, is that the estimation can no longer against Coulomb counting, or integrating the measured current IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY 7 over time: Cell SOC 1 0.8 0.6 0.4 0.2 0 Z tf 1 4.3 SOC = SOC − I(t)dt (21) y y y exp 0 3600C 0,NMC 0 0,LMO At upper Positive Electrode t0 voltage limit LMO 4.2 NMC The initial SOC0 is taken from a cell level OCV-SOC lookup table and the cell capacity, C, is obtained from the exper- 4.1 imental capacity in Ah measured between the two voltage 4 limits. In this paper, the positive electrode state estimates are validated with Coulomb counting because the positive 3.9 electrode is typically the power limiting electrode, and the positive electrode is more strongly observable. Since the 3.8 observer estimates electrode concentrations of each composite 3.7 material (ranging from 0 to the saturation concentration of the Positive Electrode Potential (V) electrode material), some effort must be made to convert the 3.6 y f,LMO concentration estimates to a cell level SOC (ranging from 0 y 3.5 At lower f,NMC to 1, defined at the cell voltage limits) in order to compare voltage limit y f the result with Coulomb counting. The positive electrode bulk 3.4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 concentration estimates for each of the composite materials Normalized Concentration are first converted to an overall positive electrode bulk SOC estimate via conservation of lithium within the electrode: Fig. 4. Initial and final concentrations are defined at the upper and lower cell voltage limits. The overall initial and final normalized concentrations, y and c¯ +c ¯ 0 LMO LMO NMC NMC yf correspond to a cell SOC (top axis) of 1 and 0, respectively. SOCˆ p = (22) cmax,LMOLMO + cmax,NMC NMC This overall positive electrode bulk SOC operates between an 4.5 Positive Electrode Shift initial and final value, y0 and yf , corresponding to the upper and lower voltage limits. The initial and final values y0 and yf 4 y y are functions of the initial and final values of each composite 0,aged 0,new (V)

p y material: f,aged U y f,new y0,LMOcmax,LMOLMO + y0,NMC cmax,NMC NMC 3.5 y0 = cmax,LMOLMO + cmax,NMC NMC (23) yf,LMOcmax,LMOLMO + yf,NMC cmax,NMC NMC 3 yf = 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 cmax,LMOLMO + cmax,NMC NMC (24) 1.5 Finally, the estimated cell level SOCˆ is calculated by: Negative Electrode Shift SOCˆ − y 1 x SOCˆ = 1 − p 0 (25) f,aged

yf − y0 (V) x

n f,new An example of this procedure is given in Fig. 4 for a U 0.5 x x constant current discharge. Once a cell level SOC estimate is 0,new 0,aged calculated, it can be validated against the experimental SOC 0 from Coulomb counting. 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 Normalized Concentration

B. Estimating Capacity Fig. 5. Example of a shift in the electrode operating range from a new to an aged cell, as cyclable lithium is lost to SEI growth. In each electrode, the Unlike with SOC estimation, capacity estimates do not operating range tends to shift to a less lithiated state. necessarily need to be made in real time, since capacity only changes significantly over months or years of usage. This gives some flexibility in making capacity estimates, and it is possible a shift in the normalized concentration operating ranges of the to wait for ideal conditions before making an estimate. positive (y0 to yf ) and negative (x0 to xf ) electrodes, shown For the composite LMO-NMC cells tested, previous work in Fig. 5, as the electrodes becomes less lithiated overall. used differential capacity analysis to show the dominant aging The normalized concentration operating range of either mechanism to be solid-electrolyte interface (SEI) layer growth electrode can be converted to a capacity in Ah by: at the negative electrode [22]. Although other mechanisms, such as loss of active material, may become significant during C = (x0 − xf )cmax,nF ALnam,nn/3600 later stages of life [32], [33] or at high temperatures, only loss F AL = p am,p ((y − y )c + of cyclable lithium is considered in this work. The SEI layer 3600 f,LMO 0,LMO max,LMO LMO grows as a result of a lithium-consuming side reaction between (yf,NMC − y0,NMC )cmax,NMC NMC ) the electrolyte solvent and the electrode material. This causes (26) IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY 8

Therefore an estimate of the normalized concentration range 1 Estimated of a given electrode can be used to obtain an estimate of cell 0.9 "Real" capacity. First it should be noted that a direct estimate of the lower concentration limit, x or y cannot be relied upon, 0.8 f f Flat since a complete discharge of the battery is not expected in OCV 0.7 practice. The upper concentration limit x0 or y0 however can Steep OCV be expected to be reached during regular charging. Again, 0.6 since estimation can only be performed on one electrode at a time, determining which electrode normalized concentration 0.5 Flat OCV range to estimate presents an interesting dilemma. The positive 0.4 electrode is strongly observable due to its steep OCV curve, Normalized Concentration 0.3 but the shift in y0 for a given δC is very small and difficult to Steep OCV detect. Conversely, the negative electrode has poor observability over certain ranges due to its flat OCV curve, but the shift 0.2 in x0 is large and easier to detect. Through experimentation it 0.1 0 500 1000 1500 2000 2500 3000 was determined that the latter approach gave the best results, Time (s) as the problems of poor observability can be mitigated with careful selection of when to make an estimate. Fig. 6. Demonstration of the increased negative electrode estimation error in regions where the OCV is flat. Perfect model is assumed in order to generate 1) Estimating Capacity from an Estimate of x0: Estimating pseudo-data. A constant 10 mV voltage measurement bias is applied to the x0 and xf is not as straight forward as simply estimating the pseudo-data voltage. The pseudo-data is then used as the ”measurement” in electrode concentration at a given point in time. First, it is the EKF to estimate the negative electrode states. unlikely that the cell will be fully discharged during regular 1 automotive use; so estimating xf directly is not possible. Instead, xf is related to x0 in a lookup table, via offline 0.9 simulation of the model during a low rate, constant current 0.8 discharge. Second, as noted previously, the negative electrode OCV has 0.7 regions that are relatively flat, which implies weak observabil- 0.6 ity. This can be demonstrated by estimating the negative elec-

(V) 0.5 trode states for a CD cycle, where a perfect model is assumed n U but a 10 mV voltage measurement bias is introduced (Fig. 6). 0.4 Flat Steep Flat Steep The negative electrode normalized concentration estimate is OCV OCV OCV OCV poor in regions where the OCV is flat (Fig. 7), but converges 0.3 towards the true value in regions where the OCV is steeper. 0.2 The steep OCV regions are around normalized concentrations 0.1 of 0.58–0.65 and 0.22–0.35, with the latter region being the steeper of the two. However the goal is to estimate x 0 0 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 (essentially the negative electrode initial condition, if the cell Normalized Concentration begins at a fully charged state), which is typically greater than Fig. 7. Negative electrode OCV (measured from half cell experiment), 0.65, and lies in a flat region with poor observability. This showing regions where the OCV is flat or steep. can be seen in Fig. 6, as the estimate is unable to accurately converge from an initial condition error at the beginning of the discharge. Therefore, to obtain an estimate of x0, the strategy 4.15 V and 2.8 V throughout the battery life. is to wait until the estimate is likely converged (i.e. wait until 4) Calculate y0,LMO, yf,LMO, y0,NMC , and yf,NMC via a steep OCV region is reached), and then calculate the initial the corresponding positive OCV lookup. state by Coulomb counting backwards in time, starting from 5) Calculate y0 and yf using (23) and (24). the converged state. The complete SOC/SOH estimation procedure is outlined Further complicating the problem, any estimation of the in Fig. 8. An important point to discuss is that, ultimately, negative electrode requires simulating the positive electrode both positive and negative electrode states are being estimated model open loop. This requires knowledge of the positive (positive electrode for cell SOC, and negative electrode for ca- electrode range, y0 and yf , which also change as the cell ages. pacity), despite the fact that the complete model that includes However, y0 and yf at a given stage of life are directly related both electrodes is unobservable. This is possible only because to x0 and xf , according to the following procedure: the exchange of information between the two estimators is 1) Estimate x0 and xf . limited to the parameters x0 and xf , that vary slowly over 2) Calculate U0,n and Uf,n at the initial and final state via time. negative OCV lookup. Essentially there are dual observers running in parallel, 3) Calculate U0,p = 4.15 + U0,n and Uf,p = 2.8 + Uf,n, one that estimates the positive electrode states to obtain cell since the upper and lower cell voltage limits remain at SOC, Obsp, (simulating the negative electrode and liquid phase IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY 9

150 Initialize x0, xf, y0, yf at 100 BOL Cell SOC Estimation 50

Simulate n, Estimate xp Convert to cell 0

e open loop states via Obsp SOC Current (A) 50 x0, xf 100 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 Capacity Estimation a) 4.2 Simulate p, Estimate xn Estimate x0, Update y0, e open loop states via Obsn xf,Q yf 4 y0, yf 3.8

3.6 Fig. 8. Block diagram of the combined cell SOC/SOH estimation algorithm. Voltage (V) Measured 3.4 Predicted

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 open-loop) and one that estimates the negative electrode states b) 1 to obtain capacity, Obsn, (simulating the positive electrode and liquid phase open-loop). The states of each observer 0.8 are locally observable with respect to a pseudo-measurement 0.6 of the corresponding electrode voltage. However, exchange 1 Cell SOC of information between the two observers could potentially 0.4 0.95 Measured 0.9 Estimated violate the assumptions required for observability. For exam- 0 5 10 0.2 ple, attempting to feed in the closed-loop negative electrode 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 c) states estimated in Obsn as a replacement for the open-loop 0.04 negative electrode state prediction required in Obsp would result in unstable estimates. Therefore, the only exchange of 0.02 information between the two observers comes from the update 0 of x0 and xf , that are fed into the open-loop simulation of 0.02 Cell SOC Error the negative electrode used by Obsp. However, the parameters 0.04 x0 and xf change only very slowly over hundreds of Ahs and 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 they have a negligible effect on the stability of the cell SOC Time (s) estimation, which operates over a much faster time scale. d) Another important topic to study is the propagation of error Fig. 9. Estimation results for cell 2 over a CD, CCCV charge profile. a) in the combined SOC/SOH estimation algorithm. For example Imposed current profile, b) predicted and measured cell voltage, c) cell SOC in the SOH estimation block in Fig. 8, the estimated parameter estimate and SOC from Coulomb counting, showing convergence from an initial condition error, and d) cell SOC estimate error. y0 is fed into the next open-loop prediction of the positive electrode. It is conceivable that a small error in a given y0 estimate could contribute to additional error in the next y0 requires the least computational effort of the three observer estimate, and so forth, resulting in a positive feedback that types considered. The cell SOC estimate converges quickly would greatly amplify the initial error. An initial investigation from an imposed initial condition error, and shows good into this problem has shown the x0/y0 estimates to be robust agreement with the SOC from Coulomb counting, with a enough that this positive feedback is avoided; however, this maximum error of less than 2% during the CD portion of topic will be studied further in future work. Particularly, the the cycle. Increased estimate error up to 4% is seen during error in the SOC/SOH estimates should be quantified, or at the constant voltage charge due to poor open-loop model least bounded. agreement during this portion of the charge. This level of SOC estimate error is similar among all of the cells tested. VIII.RESULTS Fig. 10 shows the estimated surface and bulk concentrations To validate the combined SOC/SOH estimation algorithm, of each individual composite electrode material over the same the estimation was performed on 5 different automotive pouch PHEV cycle, as well as the current split factor; although again, cells, aged under different conditions. The 5 cells differ in the these cannot be easily validated experimentally. final SOC that was reached in CD mode and in the charge rate. B. Capacity Estimate Validation

For capacity estimation, the x0 estimate is obtained by A. SOC Estimate Validation waiting for state convergence in a steep OCV region be- The cell SOC estimate is compared to SOC from Coulomb fore Coulomb counting backwards to the beginning of the counting in Fig. 9 for cell 2 under a charge depleting PHEV discharge. Therefore, a single capacity estimate is obtained cycle taken at BOL. Cell SOC must be estimated continuously after each CD cycle is complete. As noted previously, the in real time, so only the traditional EKF was used, since it low-SOC region from 0.22–0.35 is the preferred region to IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY 10

1 SOC region. Also, the mid-SOC range is relatively small LMO Bulk 0.9 LMO Surface (only spans a normalized concentration of 0.07), so it becomes 0.8 NMC Bulk more difﬁcult to decide when the estimate is truly converged. NMC Surface 0.7 Essentially there is a chance of ”missing” this mid-SOC region, and taking the estimate too early in the discharge 0.6 (when the estimate has not yet converged) or too late (when 0.5 the estimate is starting to diverge again). At BOL, cells 4 0.4 and 5 are not discharged sufﬁciently to reach the low-SOC

Normalized Concentration 0.3 region, so the initial estimates for these cells are taken in the

0.2 mid-SOC region. However as the estimated x0 decreases over time, the end of the CD cycles begin to cross the threshold into 0.1 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 the low-SOC region, at which point the algorithm switches to a) estimate in the low-SOC region. An important point here is 2 that although the estimated concentration may cross into the low-SOC region, triggering the switch, the actual cell may not have done so. This is most likely the case for cell 5, which 1.5 is only discharged to 45% SOC. Right before the switch to LMO

β estimating at low-SOC is made near the end of the test, the

1 estimated capacities happen to be lower than the measured capacity (at least for the EKF and smoother). This means that the switch is made prematurely, resulting in additional Current Split, 0.5 inaccuracies in the estimate after the switch is made. That said, the EKF and smoother eventually switch back to estimating in the mid-SOC region, tending to oscillate between the two 0 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 regions. These problems illustrate the difficulty in relying on Time (s) an imperfect estimate to determine if thresholds have been b) crossed. Fig. 10. Estimation results for cell 2 over a CD, CCCV charge profile. a) In comparing the three observer types, there are not large Estimated bulk and surface normalized concentrations for each composite differences in the estimate error mean or variance among the material, and b) current split factor, βLMO (βNMC is simply equal to 1 - observers. The particle filter tends to produce slightly more βLMO). accurate but noisy estimates than the other observers, particularly when estimating in the mid-SOC region. Overall, the EKF performs very well without the additional computational record the converged state since it is has stronger observability effort required for the smoother and particle filter. than the mid-SOC region between 0.58–0.65; however some 1) Maintaining SOC Accuracy: In addition to simply es- cells are not discharged sufficiently to reach the low-SOC timating cell capacity, another important objective in esti- region. In these cases, the converged state estimate is taken mating the changes in model parameters corresponding to in the mid-SOC region (specifically at 0.6). Otherwise if the the dominant degradation mode within a cell (e.g. x0 as it cell does reach the low-SOC region, the estimate at mid- relates to loss of cyclable Li) is to maintain an accurate model SOC is discarded and the converged state estimate is taken at over time, thereby maintaining an accurate SOC estimate over the midway point between 0.35 and the lowest concentration time. Figure 16 shows how the accuracy of the SOC estimate reached (assuming the value is not less than 0.22). Capacity changes over time for cell 2 (the other cells showed a similar estimates for the 5 cells are shown in Figs. 11 - 15, and are result). The estimated x0 is used to update the model after each compared to the measured capacities at periodic assessments. CD cycle, and as described previously, the SOC estimation is For each capacity estimate it is noted whether the estimate done using the EKF on the positive electrode sub-model. The relied on an converged state estimate taken in the low- capacity used to calculate the measured SOC by Coulomb SOC region or mid-SOC region. Additionally, for each cell, counting is obtained by interpolating between each periodic three different estimation methods (EKF, Kalman smoother, capacity assessment. and particle filter) are compared. The estimation results are The SOC estimate error over a given CD-CCCV profile re- summarized in Table III. mains fairly constant throughout the life of the cell, indicating The first thing to note is that after each capacity assessment, that the model is being updated correctly with estimates of the estimates show some small amount of capacity recovery. x0. Note that the outliers in the plot are instances where the This is expected since the cells would typically rest for profile had two CCCV charge portions, over which the model around a day while the capacity assessments were carried out, is less accurate. allowing some capacity recovery. Overall the capacity estimates are more accurate for cells IX.CONCLUSIONS that discharged to the low-SOC region (Cells 1-3). This result In this paper, a reduced order electrochemical model for a is expected, since there is stronger observability in the low- composite electrode battery was applied in a dual observer al- IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY 11

TABLE III CAPACITY ESTIMATION RESULTS

Cell Test Condition Observer Estimate Error* Estimate Error* Normalized Computation Mean (Ah) Variance (Ah2) Time 1 CD to 25% SOC, C/3 Charge EKF -0.08 0.020 1.00 Kalman Smoother 0.18 0.023 1.14 Particle Filter -0.04 0.022 1.81 2 CD to 25% SOC, 3C/2 Charge EKF -0.01 0.026 1.00 Kalman Smoother 0.15 0.031 1.15 Particle Filter 0.05 0.027 1.82 3 CD to 25% SOC, 5C Charge EKF -0.47 0.012 1.00 Kalman Smoother -0.32 0.019 1.13 Particle Filter -0.17 0.028 1.78 4 CD to 35% SOC, C/3 Charge EKF -0.37 0.081 1.00 Kalman Smoother -0.16 0.019 1.13 Particle Filter -0.07 0.076 1.79 5 CD to 45% SOC, C/3 Charge EKF -0.48 0.073 1.00 Kalman Smoother -0.38 0.024 1.13 Particle Filter -0.00 0.081 1.78 *Estimate error deﬁned as the difference between each estimate and a capacity interpolated between the periodic measured capacity assessments.

15 15

14.5 14.5

14 14 Capacity (Ah) Particle Filter (Estimate at Low SOC) Capacity (Ah) Particle Filter (Estimate at Mid SOC) 13.5 13.5 Kalman Smoother (Estimate at Low SOC) Kalman Smoother (Estimate at Mid SOC) EKF (Estimate at Low SOC) 13 EKF (Estimate at Mid SOC) 13 Measured

0 2 4 6 8 10 12 14 0 5 10 15 20 25 30 35 kAh Throughput kAh Throughput

Fig. 11. Capacity estimation results for Cell 1 aged, with a CD proﬁle to Fig. 13. Capacity estimation results for Cell 3, aged with a CD proﬁle to 25% SOC, charged at C/3. 25% SOC, charged at 5C.

15 15

14.5 14.5

14 14 Capacity (Ah) Capacity (Ah)

13.5 13.5

13 13

0 5 10 15 20 25 30 0 5 10 15 kAh Throughput kAh Throughput

Fig. 12. Capacity estimation results for Cell 2, aged with a CD proﬁle to Fig. 14. Capacity estimation results for Cell 4, aged with a CD proﬁle to 25% SOC, charged at 3C/2. 35% SOC, charged at C/3. IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY 12

APPENDIX A 15 NOMENCLATURE A Current collector area (m2) 2 14.5 De Liquid diffusion coefficient (m /s) 2 Di Solid diffusion coefficient (m /s) F Faraday’s constant (C/mol) 14 I Current (A) Lc Total cell thickness (m) Capacity (Ah) Li Electrode thickness (m) 13.5 Lsep Separator thickness (m) Np Number of particles in the particle filter Q Process noise covariance 13 R Sensor noise covariance R¯ Universal gas constant (J/K/mol) 0 2 4 6 8 10 12 14 Rc,i Ohmic resistance (Ω) kAh Throughput Ri Particle radius (m) Fig. 15. Capacity estimation results for Cell 5, aged with a CD profile to T Cell temperature (K) 45% SOC, charged at C/3. Ui Electrode open circuit voltage (V) V Cell voltage (V) 0.08 Ve Liquid phase voltage across the cell (V) RMS Error 3 c¯i Bulk particle concentration (mol/m ) 0.07 Max Error 3 ce,i Electrolyte concentration (mol/m ) 3 cmax,i Electrode saturation concentration (mol/m ) 0.06 3 cs,i Concentration at the particle surface (mol/m ) 2 0.05 i0,i Exchange current density (A/m ) 2 ie Liquid phase current density (A/m ) 2 0.04 ji Solid phase current density (A/m ) 2.5 0.5

|SOC Error| ki Reaction rate constant (m /mol /s) 0.03 r Coordinate along the particle radius, originating at the particle center 0.02 t Time (s) t+ Transference number () 0.01 0 x Coordinate through-the-thickness of the cell, 0 originating at the anode current collector 0 5 10 15 20 25 30 kAh Throughput α Symmetry factor () βi Current split factor () Fig. 16. Root mean square and maximum SOC estimate error for cell 2. Each γ Activity coefficient in electrolyte () point indicates the error over a single CD-CCCV profile. am,i Active material volume fraction () e Liquid volume fraction () i Active material volume split () gorithm to estimate SOC and capacity. The estimates were val- ηi Kinetic overpotential (V) idated against experimentally measured cell SOC and capacity σe Electrolyte conductivity (S/m) over the course of an aging campaign. The capacity estimation φe Electrolyte potential (V) algorithm performs well as long as the cell is discharged to φi Electrode potential (V) a sufficiently low SOC, where the negative electrode sub- Subscript i Refers to electrode p, n, LMO, or NMC model is more strongly observable. Estimates taken in the mid- SOC range produced less reliable capacity estimates despite the small window of relatively strong observability in this APPENDIX B range. There was not a significant improvement in the capacity MODEL OBSERVABILITY CALCULATIONS estimation accuracy when using the more computationally The nonlinear output equation for the single particle system expensive Kalman smoother and particle filter, over the simpler makes calculation of the Lie derivatives and observability EKF. Further estimation accuracy could be achieved with matrix tedious. An attempt will be made here to at least outline greater knowledge of the model parameters, but this work the process of calculating each element in the observability demonstrates the EKF approach to be an attractive option for matrix, even though the entire matrix cannot be shown. Zero onboard SOH estimation. Future work in this area will focus current will be assumed to reduce the length of the expressions, on capacity and available power estimation in the presence of thereby eliminating the charge transfer and ohmic overpoten- multiple degradation modes, i.e. if both cyclable lithium and tial terms. This results in the following simplification of the active material are lost simultaneously. dynamics and output equation (dropping the subscript i): IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY 13

TABLE IV EXPRESSIONSFOR LIE DERIVATIVES AND OBSERVABILITY MATRIX

 3  s3c(x) s2c(x) s1c(x)   3 + 2 + + s0 g cmax cmax cmax  2 x2 b2x3 x2 b2x3 x2 b2x3  l(x) = g˙ =  3s0c(x) a0x2+a1x3+a2 + 2s1c(x) a0x2+a1x3+a2 + s2 a0x2+a1x3+a2 +     b3 b3 b3 b3 b3 b3  3 + 2 + g¨  cmax cmax cmax  g¨

 2 2 2  3s0c(x) a0 + 2s1c(x)a0 + s2a0 3s0c(x) a1 + 2s1c(x)a1 + s2a1 3s0c(x) a2 + 2s1c(x)a2 + s2a2 c3 c2 c c3 c2 c c3 c2 c  max max max max max max max max max  O =  ∂g˙ ∂g˙ ∂g˙   ∂x1 ∂x2 ∂x3  ∂g¨ ∂g¨ ∂g¨ ∂x1 ∂x2 ∂x3

2) Prediction Step: The predicted state vector is advanced       x˙ 1 0 1 0 x1 in time according to the model dynamics: 0 0 1 x˙ 2 = f(x) =   x2 (27) − + x˙ 0 1 b2 x xˆk+1 = fk(ˆxk , uk) (30) 3 b3 b3 3 To advance the estimate covariance, P , in time, the state s c(x)3 s c(x) s c(x) g(x) = 3 + 2 + 1 + s (28) dynamics must be linearized: c3 c2 c 0 max max max ∂f(x , u ) F = k k (31) where the substitution c(x) = a x + a x + a x has k 0 1 1 2 2 3 ∂xk x =ˆx+ been made. The first Lie derivative and first row of the k k − + T observability matrix are shown in table IV, without expanding Pk+1 = FkPk Fk + Q (32) the other matrix elements. Again, it should be made clear that Note that for the single particle system in (10 - 11), the state the calculation of the complete observability matrix, without dynamics are already linear so F = A¯ . assuming zero current, was done using symbolic manipulation k k 3) Correction Step: The predicted state estimate is then software, resulting in a matrix with full rank. corrected using feedback from the measurement, yk+1. xˆ+ =x ˆ− + K [y − g(ˆx− , u )] (33) APPENDIX C k+1 k+1 k+1 k+1 k+1 k+1 IMPLEMENTATION OF OBSERVER ALGORITHMS The Kalman gain, K, is a function of the linearized output Note that some variables that were used previously in matrix, G. this paper are redefined in this section, in order to be more − T − T −1 Kk+1 = Pk+1Gk+1[Gk+1Pk+1Gk+1 + R] (34) consistent with conventional nomenclature. For each observer type, the following nonlinear single electrode particle system ∂g(xk+1, uk+1) Gk+1 = (35) (described in (10 - 11)) in discrete time is assumed: ∂xk+1 − xk+1=ˆxk+1

xk+1 = fk(xk, uk, wk) = (I + ∆tAk)xk + ∆tBkuk + wk Finally, the corrected estimate covariance is updated:

= A¯kxk + B¯kuk + wk + − Pk+1 = (I − Kk+1Gk+1)Pk+1 (36) yk = gk(xk, uk, vk) = U(xk) + η(xk, βuk) + Rcβuk + vk (29) B. Fixed Interval Kalman Smoother where x is the state vector, y is the measurement, u is the The traditional Kalman smoother involves optimally com- input, w is the process noise, v is the sensor noise, A¯ and bining state estimates from two filters, one running forward B¯ are the state dynamics matrices in discrete time, k is the in time and one running backward in time. The forward current time step, and ∆t is the sampling interval (fixed at and backward estimates are combined so as to minimize the 0.1 s). Q and R are defined as the covariance matrices of the covariance of the smoothed estimate [34]. There are several process noise and sensor noise, respectively. The subscript i algorithms that reduce the computation time of the smoother, is dropped in this section for clarity. which can also be extended for nonlinear systems. The Rauch, Tung, and Striebel (RTS) smoother avoids separate computation of the backward estimate by computing the smoothed A. Extended Kalman Filter estimate directly from the forward estimate. The modified The EKF is a common observer for nonlinear systems, Bryson-Frazier smoother developed by Bierman [34] avoids involving a prediction and correction step. calculation of the estimate covariance matrix inverse at each + + 1) Initialization: For k = 0, set xˆ0 = E[x0] and P0 = time step. The modified Bryson-Frazier smoother for nonlinear + + T E[(x0 − xˆ0 )(x0 − xˆ0 ) ]. systems (used in this work) is outlined below. IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY 14

1) Forward Filter: The forward predicted state estimate, 2) Prediction Step: For each particle, i, the states, xî, and − − xˆf,k, and forward predicted estimate covariance, Pf,k are output, yî, at the next time step are predicted from the model: obtained using the standard EKF described in the previous xˆ− = f (ˆx+ , u , w ) (43) section. i,k k−1 i,k−1 k−1 i,k−1 2) Backward Filter: The backward filter uses an adjoint − yî,k = gk(ˆxi,k, uk) (44) state vector, λ, and its corresponding covariance matrix, Λ. Once the forward estimate is calculated, the adjoint states where wi,k−1 is a sample drawn from the PDF of the process and covariance are predicted and corrected starting from the noise, assumed here to be zero-mean Gaussian with covariance final time step k = N, progressing backwards in time. The matrix Q. predicted adjoint variables are advanced backward in time 3) Correction Step: Now the measurement at time step k is using the same linearized F matrix as in (31): introduced as yk. Since there is noise in the measurement, in theory, any of the predicted states and output could represent + T − λk = Fk λk+1 (37) reality, even if the measured yk is not close to the predicted yˆ . Therefore the goal of the update step is to decide which + T − − i,k Λk = (Fk − I) Λk+1 + Λk+1Fk (38) particles are more likely to represent reality, with knowledge of the actual measurement. This translates to evaluating, for each The adjoint states and covariance are updated taking into particle, the conditional probability of seeing the measurement, account the measurement, using the same linearized G matrix − yk, given the predicted states and output, xî,k and yî,k, are the and Kalman gain as in (35) and (34). − true values, i.e. p(yk|xî,k). The conditional probability is used − T T + λ =[I − G K ]λ to assign a normalized weight to each particle, qi, indicating k k k k (39) T − T −1 − the likelihood that it represents reality: − Gk [GkPf,kGk + R] [yk − g(ˆxk , uk)] − − T + p(yk|xî,k) Λ =[I − KkGk] Λ [I − KkGk] qi = (45) k k PNp − T − T −1 (40) i=1 p(yk|xî,k) + Gk [GkPf,kGk + R] Gk For the assumed Gaussian distribution of sensor noise with The adjoint variables are initialized at time step k = N zero mean and variance R, the conditional probability for each − T − T −1 − as: λN = −GN [GN Pf,N GN + R] [yN − g(ˆxN , uN )] and particle evaluates to: Λ− = GT [G P − GT + R]−1G . N N N f,N N N 2 1 −(yk − yî,k) Finally, the smoothed states, xˆs, and covariance, Ps, are − p(yk|xî,k) = √ exp (46) calculated as a function of the forward and backward adjoint 2πR 2R variables. The weighting process defines a new discrete probability mass − − − − xˆs,k =x ˆf,k − Pf,kλk (41) function over the particles, {xî,k : i = 1, ..., Np}, where element i has a probability mass of qi. − − − − Ps,k = Pf,k − Pf,kΛk Pf,k (42) 4) Re-Sampling: Now new particles are generated by re- sampling Np times from the weighted discrete probability + C. Particle Filter mass function. Thus, corrected state estimates, xî,k are gener- + − ated so that for any j, P r{xˆ =x ˆ } = qi. The distribution Rather than estimate a single state vector at each time step, j,k i,k of the corrected state estimates xˆ+ is approximately equal to such as in the EKF, the particle filter estimates a distribution i,k the posterior PDF, p(x |y ). 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Simona Onori is currently an Assistant Professor at Ted Miller is Ford’s Senior Manager of Energy the Automotive Engineering Department, Clemson Storage and Materials Strategy and Research. His University. She received her Laurea degree (summa team is responsible for energy storage strategy, re- cum laude) in computer engineering from University search, development and implementation for all Ford of Rome Tor Vergata, Rome, Italy, in 2003; her M.S. hybrid, plug-in hybrid, battery electric, and fuel cell degree in electrical and computer engineering from vehicles. Mr. Miller’s team supports global prototype University of New Mexico, Albuquerque, NM, USA, and production vehicle development programs. They in 2004; and her Ph.D. degree in control engineering are involved in every aspect of energy storage design from the University of Rome Tor Vergata, in 2007. and use from raw materials to end-of-life recycling. She was with IBM (2000-2002) and Thales-Alenia His team also sponsors collaborative research pro- Space (2007), both in Rome, Italy. From 2007-2013, grams at the University of Michigan, where they she was with the Center for Automotive Research and from 2010-2013 will soon be opening a new joint battery fabrication and evaluation lab, lecturer at the MAE Dept. at the Ohio State University. Her background Stanford, MIT, Ohio State University, and a number of other major universities is in control system theory, and her current research interests are in ground worldwide. Mr. Miller is Chairman of the United States Advanced Battery vehicle propulsion systems, including electric and hybrid electric powertrains, Consortium (USABC) Management Committee. He holds a number of energy electrochemical energy storage systems and aftertreatment systems with an storage technology patents, is the author of many published papers in the emphasis on modeling, simulation, optimization, and feedback control design. ﬁeld and an experienced speaker on advanced energy storage technology and Her research has been funded by, among others, by Ford, General Motors, materials. Cummins, the National Science Foundation, and the U.S. Department of Energy. Dr. Onori is a member of the ASME, IEEE and SAE. She is currently associate editor for ASME American Control Conference and Dynamic Sys- tems and Control Conference, and IEEE CSS Conference Editorial Board. She is associate Editor of SAE International Journal of Advanced Powertrains. She received the 2012 Lumley Interdisciplinary Research Award from the OSU College of Engineering and the TechColumbus 2011 Outstanding Technology Team Award.

Giorgio Rizzoni received his BS, MS and PhD in Electrical and Computer Engineering in 1980, 1982 and 1986, from the University of Michigan. Between 1986 and 1990 he was a post-doctoral fellow, and then a lecturer and assistant research scientist at the University of Michigan. In 1990 he joined the Department of Mechanical (now Mechan- ical and Aerospace) Engineering at Ohio State as an Assistant Professor. He was promoted to Associate Professor in 1995 and to Professor in 2000. In 1999 he was appointed director of the Center for Automotive Research. Since 2002 he has held the Ford Motor Company Chair in Electromechanical Systems, and has also been appointed Professor in the Department of Electrical and Computer Engineering. Prof. Rizzonis specialization is in dynamic systems and control, and his research activities are related to sustainable and safe mobility. He is a Fellow of the Institute of Electrical and Electronic Engineers and Fellow of the Society of Automotive Engineers, and has received numerous teaching and research awards, including the Stanley Harrison Award for Excellence in Engineering Education and the NSF Presidential Young Investigator Award.

Xiao Guang Yang received his BS, MS and PhD in Materials Sciences & Engineering in 1988, 1990 and 1995, from Zhejiang University, China. He worked as a Research Associate at the University Hawaii from 1999 - 2000, and at Pennsylvania State University from 2002 -2005. In 2005, Xiao Guang joined Ford as a Product Engineer for HEV programs. In 2010, he became a Technical Expert in Advanced Battery Technologies in the Energy Stor- age and Materials Research Department, Research and Advanced Engineering at Ford Motor Company. Yang currently works on battery development, benchmark testing, model simulation and validation. He has been a technical advisory member of USABC (Advanced Battery Consortium Technical Committee) since 2008.