Estimation and Control of Battery Electrochemistry Models: a Tutorial

Estimation and Control of Battery Electrochemistry Models: A Tutorial

Scott J. Moura

Abstract— This paper presents a tutorial on estimation and control problems for battery electrochemistry models. We present a background on battery electrochemistry, along with a e- e- comprehensive electrochemical (EChem) model. EChem models Zinc _ Copper present a remarkably rich set of control-theoretic questions Anode Separator Cathode involving model reduction, state & parameter estimation, and + optimal control. We discuss fundamental systems and controls Zn(s) Anion Cu(s) -2 challenges, and then present opportunities for future research. SO 4 Oxidaon Reducon

+2 +2 I.INTRODUCTION Zn (aq) Cu (aq) ZnSO4(aq) CuSO4(aq) Batteries are ubiquitous – they exist in our smart phones, Caon Caon laptops, electric vehicles (EVs), and electric grids. Energy Fig. 1. An example zinc-copper Galvanic (or Voltaic) cell demonstrating storage is a critical enabling technology for designing sus- the principles of operation for an electrochemical cell. tainable energy systems. Although battery materials science has seen rapid advances [1], batteries are chronically electrolyte, exemplified by the zinc-copper Galvanic cell in underutilized and conservatively designed [2]. Consumers Fig. 1. The cathode and anode materials are jointly selected purchase batteries with 20-50% excess energy capacity, lead- to have a large electrochemical potential between each other. ing to added weight, volume, and upfront cost. Intelligent This creates the desired electrochemical energy storage prop- battery control can lead to faster charge times, increased erty. The electrodes are electrically isolated by a separator. energy and power capacity, as well as a longer life. The key Hence, conservation of charge forces electrons through an to realizing such advanced battery management systems is external circuit, powering a connected device, while cations electrochemical model based controls. These electrochemical flow between the electrodes within the electrolyte. models, however, contain a multitude of fundamental control- Electrode and electrolyte materials are selected for their theoretic challenges, yielding a rich opportunity for systems voltage, charge capacity, weight, cost, manufacturability, and controls researchers. etc. For example, lithium-ion cells are attractive in mobile applications because lithium is the lightest (6.94 g/mol) and A. Background & Fundamentals of Battery Electrochemistry most electropositive (-3.01V vs. standard hydrogen elec- Italian physicist Alessandro Volta invented the first battery trode) metal in the periodic table. Lead acid cells feature cell in 1800. The so-called voltaic pile consisted of two heavier electrodes (Pb and PbO2), yet provide high surge metals in series, zinc and copper, coupled by a sulphuric currents at cost effective prices. Lithium-air batteries feature acid electrolyte. Volta constructing this system in response cathodes that couple electrochemically with atmospheric to experiments performed by his colleague Luigi Galvani, oxygen, thus producing energy densities that rival gaso- who was fascinated by the interaction between electricity and line fuel. In battery energy management, we are interested biological nervous systems. During an experiment, Galvani in maximizing performance and longevity. This requires a discovered that a dead frog’s legs would kick when connected detailed understanding of the underlying electrochemistry. to two dissimilar metals. Galvani conjectured the energy However, the electrochemical state variables are not directly originated within the animal, coining the phenomenon “an- measurable. At best, one can measure voltage, current, and imal electricity.” Volta believed the different metals caused temperature only. Consequently, modeling and control are this behavior, and proved his hypothesis true with the voltaic necessary to extract the full potential from batteries. pile [3]. Thus, the electrochemical battery cell was born. A battery converts energy between the chemical and B. Estimation and Control Problem Statements electrical domains through oxidation-reduction reactions. It Figure 2 outlines an electrochemical model based control consists of two dissimilar metals (electrodes) immersed in an system, comprised of a state estimator, parameter estimator, This work was supported in part by the National Science Foundation and controller. Next, we provide precise problem statements under Grant No. 1408107 and the Advanced Research Projects Agency- for each control-theoretic task. Energy, U.S. Department of Energy, under Award Number DE-AR0000278. 1) State-of-Charge (SOC) Estimation: SOC indicates the S. J. Moura is with the Energy, Controls, and Applications Lab (eCAL) in Civil and Environmental Engineering at the University of California, Berke- quantity of lithium within each electrode’s solid phase. It ley in Berkeley, California, 94720, USA [email protected] is analogous to a fuel tank level, since it represents the Electrochemical model- Measurements based limits of operation r I (t) EChem-based I(t) V(t), T(t) Traditional limits Traditional limits Battery Cell Controller of operation of operation

Innovations + _

^ ^ Estimated EChem-based x(t), θ(t) States & Params State/Param Overpotential ^ Voltage Terminal Surface concentration Estimator V^( t), T(t) Cell Current ElectroChemical Controller (ECC) Fig. 3. An electrochemical model-based control architecture constrains Fig. 2. Block diagram for ElectroChemical Control (ECC) System, electrochemical state variables, as opposed to measurable outputs. This comprised of a battery cell, state/parameter estimator, and controller. expands the operational envelope and ensures safety. stored electrochemical energy. Unlike fuel tanks, SOC is not measurable and must be estimated. This can be directly cast The Constrained Control Problem: Given measurements as a state estimation problem. of current I(t), voltage V (t), and temperature T (t), control The State Estimation Problem: Given measurements of current such that critical electrochemical variables are current I(t), voltage V (t), and temperature T (t), estimate maintained within safe operating constraints. the concentration of lithium in each electrode’s solid phase material. Optionally, one may estimate other electrochem- II. MATHEMATICAL MODELING ical variables of interest, such as electrolyte concentration A. Background and overpotentials (see Section II-B). Electrochemical (EChem) models capture the spatiotem- The key challenges for state estimation are lack of complete poral dynamics of lithium-ion concentration, electric po- observability and model nonlinearities. tential, and intercalation kinetics. Most models in the bat- 2) State-of-Health (SOH) Estimation: Battery SOH met- tery controls literature are derived from the Doyle-Fuller- rics indicate a battery’s relative age. The two most common Newman (DFN) model [4], which is based upon porous SOH metrics are charge capacity fade and impedance rise electrode and concentrated solutions theory. Figure 4 shows (i.e. power fade). Charge capacity fade indicates how charge a cross section of the layers described in Fig. 1. capacity has decreased relative to its nameplate value, e.g. At full charge the majority of lithium exists within the a 2 Ah cell may hold 1.6 Ah after two years of use. Power anode solid phase particles, typically a lithiated carbon capacity fade indicates how power capacity has decreased LixC6. These particles are idealized as spherically sym- relative to its nameplate value, e.g. a fresh cell may provide metric. During discharge, lithium diffuses from the interior 360W of power for 10 seconds, but only 300W after two to the surface of these porous spherical particles. At the years of use. Gradual changes in SOH metrics are directly surface an electrochemical reaction separates lithium into a related to changes in the mathematical model parameters. positive lithium ion and electron (1). Next, the lithium ion The Parameter Identification Problem: Given measure- migrates from the anode, through the separator, and into the ments of current I(t), voltage V (t), and temperature T (t), cathode. Since the separator is an electrical insulator, the estimate uncertain parameters related to SOH, such as corresponding electron travels through an external circuit, cyclable lithium, solid-electrolyte interface resistance, and powering the connected device. The lithium ion and electron volume fractions. meet at the cathode particle surface, typically a lithium The key challenges for electrochemical models include metal oxide LiMO2, and undergo the reverse electrochemical parametric modeling, nonlinear parameter identifiability, and reaction according to (2). persistent excitation. + LixC6 )* C6 + xLi + xe−, (1) 3) Controlled Charging/Discharging: In many applica- + Li(1 x)MO2 + xLi + xe− )* LiMO2. (2) tions, additional capacity is added to mitigate cell imbalance, − capacity/power fade, thermal effects, and estimation errors. The resultant lithium atom diffuses into the interior of the This leads to larger, heavier, and more costly batteries than cathode’s spherical particle. This entire process is reversible necessary. Electrochemical control alleviates oversizing by by applying sufficient potential across the current collectors safely operating batteries near their physical limits. Today, – rendering an electrochemical storage device. In addition to operational limits are defined by what can be measured lithium migration, this model captures the spatial-temporal – voltage, current, and temperature. Battery degradation, dynamics of internal potentials, electrolyte current, and cur- however, is more closely related to limits on immeasurable rent density between the solid and electrolyte phases. electrochemical states, such as overpotentials and surface concentrations [2]. We seek a paradigm-shifting architecture B. Doyle-Fuller-Newman Model that expands the operating envelope by constraining internal We consider the Doyle-Fuller-Newman (DFN) model in electrochemical states instead of measured values such as Fig. 4 to predict the evolution of lithium concentration in voltage, current, and temperature (see Fig. 3). the solid cs±(x, r, t), lithium concentration in the electrolyte − − - - sep sep + + + + 0 L 0 0 cs (x, r, t)/cs,max cs (x, r, t)/cs,max x L L

Center 0.75 0.7 Anode Separator Cathode 0.65 0.6

Space, r 0.55 Space, r 0.5 0.45

- + Surface - cs (r) + cs (r) - 2 e Li e Anode Separator Cathode ) 1.5 - +

css css x, t 1 ( e r r c 0.5 0 LixC6 ce(x) Li1-xMO2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Electrolyte Space, x/L ± Fig. 5. Solid cs (x, r, t) and electrolyte ce(x, t) concentrations as Fig. 4. Schematic of the Doyle-Fuller-Newman model [4]. The model functions of space, after 30sec of 5C discharge. Symbols {−, +} denotes considers two phases: the solid and electrolyte. In the solid, states evolve in the negative electrode (anode) and positive electrode (cathode), respectively. the x and r dimensions. In the electrolyte, states evolve in the x dimension In the solid phase concentrations, the top and bottom denote particle center ± only. The cell is divided into three regions: anode, separator, and cathode. r = 0 and surface r = Rs , respectively. Along with these equations are corresponding boundary ce(x, t), solid electric potential φs±(x, t), electrolyte electric and initial conditions. For brevity, we only summarize the potential φe(x, t), ionic current ie±(x, t), molar ion fluxes differential equations here. Further details, including nota- jn±(x, t), and bulk cell temperature T (t) [4]. The governing equations are given by tion definitions, can be found in [2], [4]. Parameters for a LiCoO -C cell are publicly available from the DUALFOIL ∂c 1 ∂ ∂c 2 s± 2 s± model, developed by Newman and his collaborators [5]. Note (x, r, t) = 2 Ds±r (x, r, t) , (3) ∂t r ∂r ∂r the mathematical structure, which contains PDEs (3)-(4), 0 ∂ce ∂ ∂ce 1 tc ODEs in time (9), ODEs in space (5)-(7), and nonlinear (x, t) = De(ce) (x, t) + − i±(x, t) , ∂t ∂x ∂x ε F e algebraic constraints (8). This presents a formidable task with e (4) respect to control-oriented modeling, state estimation, and ∂φ± i±(x, t) I(t) control design (see Section III and IV). s (x, t) = e − , (5) ∂x σ± C. Electrochemical Model Simulations ∂φe i±(x, t) 2RT 0 (x, t) = e + (1 t ) Illustrative simulations of the electrochemical model are ∂x − κ(c ) F − c e provided in Fig. 5. Specifically, this figure depicts the solid d ln fc/a ∂ ln ce 1 + (x, t) (x, t), (6) phase cs±(x, r, t) and electrolyte ce(x, t) concentrations as × d ln c ∂x 1 e functions of space, after 30sec of 5C discharge. The anode ∂ie± solid phase concentrations c−(x, r, t) exhibit a decreasing (x, t) = asF j±(x, t), (7) s ∂x n gradient as r increases toward the particle surface, since 1 αaF ± αcF ± RT η (x,t) RT η (x,t) lithium is exiting the anode during discharge. This creates jn±(x, t) = i0±(x, t) e e− , F − a spatial gradient in the electrolyte phase c (x, t) along (8) e h i the x-dimension. On the cathode, solid phase concentration avg dT increases c (x, r, t) w.r.t. r, especially near the particle ρ cP (t) = hcell [Tamb(t) T (t)] + I(t)V (t) s− dt − surface, since lithium is entering the cathode. + 0 Note our ultimate goal: estimate these spatio-temporal asF jn(x, t)∆T (x, t)dx, (9) − 0− dynamics and regulate their evolution, given only measure- Z ments of voltage, current, and temperature. where De, κ, fc/a are functions of ce(x, t) and D. Model Reduction i±(x, t) = k± css± (x, t)c (x, t) cs,±max css± (x, t) , 0 e − A rapidly growing body of literature is establishing a q (10) spectrum of EChem models that achieve varying balances of mathematical simplicity and accuracy. Concepts include η±(x, t) = φs±(x, t) φe(x, t) − spectral methods [6], [7], residue grouping [8], quasilin- U ±(c± (x, t)) FR±j±(x, t), (11) − ss − f n earization & Pade´ approximation [9], principle orthogonal css± (x, t) = cs±(x, Rs±, t), (12) 1C-rate is a normalized measure of electric current that enables compar- ∂U ± ∆T (x, t) = U ±(c±(x, t)) T (t) (c±(x, t)), (13) isons between different sized batteries. Mathematically, the C-rate is defined s − ∂T s as the ratio of current, I, in Amperes [A] to a cell’s nominal capacity, Q, ± in Ampere-hours [Ah]. For example, if a battery has a nominal capacity of 3 Rs 2 (14) 2.5 Ah, then C-rates of 2C, 1C, and C/2 correspond to 5 A, 2.5 A, and 1.25 cs±(x, t) = 3 r cs±(x, r, t)dr. (R±) A, respectively. Note that C-rate has dimensions of [A] / [Ah] = [1/h]. s Z0 + decomposition [10], and single particle model variants [11], V (t) = h(css− , css,I; θ), (18) [12], [13]. Due to the richness of existing literature, we do where the j is the exchange current density and j not focus on model reduction in this paper. Instead, we focus i0( ) css(t) = j j is the· surface concentration for electrode on the state/parameter estimation and control challenges. cs(Rs, t) j +, . The functions U j( ) are the equilibrium potentials∈ III. STATE & PARAMETER ESTIMATION of{ each−} electrode material,· given the surface concentration. A. Survey of SOC/SOH Estimation Literature Mathematically, the functions U j( ) are strictly monotonically decreasing functions of their· input. This fact implies Over the past decade, the SOC/SOH estimation literature the inverse of its derivative is always finite, a property that is has grown considerably rich with various algorithms, models, critically important for assessing observability. Symbol n and applications. Most existing literature to date uses equiv- Li represents the total moles of cyclable lithium. Finally, param- alent circuit models, but electrochemical model-based esti- + + + 1 1 T mation algorithms have also emerged. These studies develop eter vector θ = nLi, (a AL k )− , (a−AL−k−)− ,Rf represents uncertain parameters. estimators for reduced-order models. The model reduction and observer design process are intimately intertwined, as Remark 1 (SOH Metrics): Coincidently, the parameters simpler models ease estimation design at the expense of nLi and Rf represent capacity and impedance rise, fidelity. Ideally, one seeks to derive a provably stable esti- respectively. Identification of nLi and Rf provides a direct mator for the highest fidelity electrochemical battery model system-level estimate of SOH. possible. The first wave of studies utilized the “single particle Combined SOC/SOH Problem: Given measurements model” (SPM) for estimator design [11], [12], [14], [15]. of current I(t), voltage V (t), and the SPM equations, The SPM idealizes each electrode as a single spherical simultaneously estimate the lithium concentration states porous particle by neglecting the electrolyte dynamics. This c (r, t), c+(r, t) and parameters θ. model is suitable for low C-rates, but degrades at C-rates s− s above C/2. Recently, researchers have extended the single C. Model Properties particle model to include electrolyte dynamics [13], [16], although there have been limited estimation studies [17], Before embarking on an estimation design, we first study + [18]. Nonetheless, state estimation designs have emerged relevant mathematical properties of the SPM. The cs , cs− for other electrochemical models that incorporate electrolyte subsystems are mutually independent of each other. More- dynamics. Examples include spectral methods with output over, they are governed by linear PDEs. Also note that the + injection [19], residue grouping with Kalman filtering [20], PDE subsystems produce boundary values css(t), css− (t) that and composite electrodes with nonlinear filters [21]. Unfor- feed into the nonlinear output function (17). The SPM is tunately, it becomes more difficult to prove estimation error also characterized by the following dynamical properties, stability as model complexity increases. The core difficulty is which present notable challenges for state estimation. We lack of complete observability from voltage measurements. present the following propositions, whose proofs are straight- forward, non-insightful, and omitted for brevity. B. Single Particle Model (SPM) Proposition 1 (Marginal Stability): Each individual sub- + To illustrate the control-theoretic challenges of SOC/SOH system in (15)-(16) governing states cs (r, t), cs−(r, t) is estimation, we consider the SPM – the simplest form of marginally stable. In particular, each subsystem contains one electrochemical model. The SPM idealizes each electrode eigenvalue at the origin, and the remaining eigenvalues lie as a single aggregate spherical particle. One can derive this on the negative real axis of the complex plane. model by assuming the electrolyte Li concentration ce(x, t) Proposition 2 (Conservation of Lithium): The moles of from (4) is constant in space and time. Mathematically, the lithium in the solid phase are conserved [19]. Mathemati- d model consists of two diffusion PDEs, cally, dt (nLi(t)) = 0 where 2 j ∂cs± 2 ∂cs± ∂ cs± j j Rs (r, t) = Ds± (r, t) + (r, t) , (15) sL A 2 j 2 nLi(t) = 4πr c (r, t)dr (19) ∂t r ∂r ∂r 4 j 3 s π(R ) 0 j +, 3 s Z ∂cs± ∂cs± I(t) ∈{X−} (0, t) = 0, (Rs±, t) = ± . (16) Invertability Analysis: Next, we study invertability of the ∂t ∂t Ds±F a±AL± output function (17) w.r.t. boundary state variables css± (t). + The boundary conditions at r = Rs and r = Rs− signify Figure 6(a) provides the open circuit potential (OCP) func- + that flux is proportional to input current I(t). Output voltage tions U −( ), U ( ) as functions of the normalized anode · · is given by a nonlinear function of the state values at the concentration θ± = css± /cs,±max. The near zero gradient + boundary css(t), css− (t) and the input current I(t) as follows implies the voltage measurement is weakly sensitive to perturbations in the surface concentrations c± . This is further RT 1 I(t) ss − V (t) = sinh + +−+ + confirmed by Fig. 6(b), which depicts the output function’s αF 2a AL i0 (css(t); nLi) + partial derivatives w.r.t. css− and css at equilibrium conditions, RT 1 I(t) sinh− for currents ranging from -5C to +5C. It is important to note − αF 2a AL i−(c− (t); n ) + − − 0 ss Li that h is strictly monotonically decreasing w.r.t. css over a + + +U (c (t)) U −(c− (t)) + R I(t), (17) larger range than h is strictly monotonically increasing w.r.t. ss − ss f 2 5 3.5 − − (a) U (θ ) I V 3.4 Vˆ U +(θ +) 10 3.3 1 4 0 Voltage [V] Current [A] 3.2

Anode OCP [V] −10 3.1 Cathode OCP [V] 0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40

0 3 Measures of SOC 60 Measures of SOH ˆ 1.5 − − 0.8 15 V V ∂ h/∂ c (θ ) SOCb− u lk − (b) s s 40 SOC − εˆ 1 + θ + 0.8 b u lk ∂ h/∂ cs s( ) − 0.6 SOCb u lk qˆ 6] 0.7 − 2010 − 0.5 SOCb u lk Bulk SOC True Value

10 d 0.60.4

× 0 [ 0 Voltage Error [mV] d Bulk SOC 0.5 5 0 5 10 15 20 25 30 35 40 PDE Params 0 5 10 15 20 25 30 35 40 −0.5 LOW 0.4 SENSITIVITY 15 0.3 −1 0.3 − 0 εˆ 0 0.2 0.4 0.6 0.8 1 SOC − SOCb u lk θ ± ± ± 0 5 10 15 20 25 30b u lk− 35 40 0 5 10 15 20 25 30 35 40 Normalized Surface Concentration, = c s s/c s , m ax 0.2 10 qˆ Measures of SOC Measures of SOH 300.1 d True Value − − + + U (θ ) U (θ ) 15 c − 53 Fig. 6. (a) Open circuit potential functions , ; and (b) ss nˆ L i/n L i + − SOC − 0 εˆ PDE Params Gradients0.8 of h(css, css,I) in (18), as functions of normalized surfaceb u lk Bulk SOC Error 25 cˆss− ± ± ± 2.5 ˆ concentration θ = css/cs,max, for currents ranging from -5C to +5C. −0.1 qˆ cˇss− 0 R f/R f 0.7 − 10 0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 SOCb u lk 2 20 True Value True Value I(t) 0.6 10 V(t) 3 c − cˆ− 1.5 Battery Cell d 15 ss ss nˆ L i/n L i Bulk SOC 0.5 5 − PDE Params c ss− cˇss−

5 − Output Fcn Params 1 Rˆ f/R f 0.4 Surface Concentrations 10 2 0.5 I(t) True Value Padé0.3-based č - Adaptive 00 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 0 5 10 15 20ss 25 30 35 40 Error [kmol] 0 5 10 15 20 25 30 35 40 PDE Param ID εˆ,qˆ Measures of SOCOutput Fcn MeasuresTime [min] of SOH 1 Time [min] Output Fcn Params + Surface Concentration 15−5 30 _ Inversion 0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 c − 3 εˆ 0.8 SOCb− u lkss Time [min] nˆ L i/n L i Time [min] cˆ I(t)25 Adaptive PDE ĉ - ss− 2.5 qˆ 0.7 ss Output Fcn − 10 ˆ SOC ˆcˇss− R f/R f Backstepping bθ uh lk Nonlinear Param ID 2 True Value 0.620 Observer True Value I(t) V(t) d Bulk SOC 0.5 1.55 15 PDE Params - 0.4 ĉs (r,t) Output Fcn Params 1

Surface Concentrations 10 0.3 0.50 0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 Fig. 7. Block diagram of Adaptive PDETime Observer [min] for SOC/SOH Estima- Time [min] tion. See30 [14] for details. c − 3 ss Fig. 8. Evolution of state and parameter estimates fromnˆ AdaptiveL i/n L i PDE c . This25 property is critical, since it demonstrates that volt-cˆss− ss− observer2.5 [14] for UDDSx2 charge/discharge cycle. Zero mean Gaussian cˇ− Rˆ f/R f age is generally more sensitive to perturbations in cathodess noise with a 10 mV variance was added to the voltage measurement. The 20 2 surface concentration than anode surface concentration. Note EChem model (3)-(14) provides the “measured” plant data.True Value + + 1.5 ∂h/∂c 15 0 for 0.8 θ 0.9. This region is a “blind ss ≈ ≤ ≤ exploiting output function invertibility and lithium conser- spot” with respect to output inversion. Output Fcn Params 1

Surface Concentrations 10 vation. Figure 7 summarizes this scheme. The algorithm Nonlinear Parameter Sensitivity Analysis: Since the pa- 0.5 0 5 10 15 20 25 30 35 40is composed0 5 of an10 Adaptive15 20 PDE backstepping25 30 35 observer40 rameter vector θ enters nonlinearlyTime [min] in the output function Time [min] (18), we must first perform a sensitivity analysis to assess [23], Pade-based´ parameter identifier for PDE parameters [9], linear dependence. Define the sensitivity vector S = ∂h/∂θ. nonlinear recursive least squares identifier for output function Applying the ranking procedure outlined in [22] to ST S parameters, and an adaptive output function inversion scheme [24]. A simulation example is provided in Fig. 8, which reveals strong linear dependence exists between θ2, θ3, θ4, demonstrates SOC and SOH convergence given incorrect where θ4 exhibits the greatest sensitivity magnitude. As a result, we pursue a parameter identification scheme for the initial estimates. We remark that the fundamental challenges for estimation are significant, even for the simplest of EChem parameter subset (θ1, θ4). Coincidentally, these two parame- models – the SPM. ters are exactly nLi,Rf , corresponding to capacity fade and impedance rise. Refer to [14] for complete details. IV.CONSTRAINEDOPTIMALCONTROL D. Adaptive PDE Observer Approach A. Survey of Optimal Battery Charge/Discharge Literature The previous subsection described several fundamental A rich body of literature exists on battery charg- estimation challenges for the SPM, including marginal sta- ing/discharging performance. Interestingly, the bulk of re- bility, poor output sensitivity to states, and linear dependence search involves experimental studies and non-model based between parameters. An adaptive PDE observer was designed heuristic charging protocols (see [25] for a survey). Rel- for this model in [14], which achieves stable estimates by atively speaking, research on electrochemical model-based TABLE I 320 ELECTROCHEMICALVARIABLES y TO REGULATE WITHIN BOX 315 CONSTRAINTS, I.E. ymin ≤ y ≤ ymax. 310 Variable, y Definition Constraint 305

I(t) Current Power electronics limit Temp [K] Expanded ± operating c (x, r, t) Li concentration in solid Material 300 s MRG range saturation/depletion VO (a) ∂c± s (x, r, t) Li concentration gradient Diffusion-induced 295 ∂r 320 in solid stress/strain c (x, t) Li concentration in Material e 315 electrolyte saturation/depletion T (t) Temperature High/low temperatures 310 accelerate aging ηs(x, t) Side reaction Li-plating, SEI-layer 305 overpotential growth Temp [K] Expanded operating 300 MRG range VO (b) 295 control is nascent. Existing optimal control studies include 320 constrained control [20], [26] and open-loop control [27], [28], [29]. A recent study by the author performs a compre- 315 hensive assessment of performance improvements enabled 310 by reference governors and electrochemical models [30]. 305

Temp [K] Expanded operating B. A Reference Governor Approach 300 MRG range VO (c) Ensuring safe operating constraints is a basic requirement 295 3 3.2 3.4 3.6 3.8 4 4.2 for batteries. Mathematically, this can be abstracted as a Volt [V] constrained control problem for which reference governors provide one effective solution. The problem statement is: Fig. 9. Temperature vs. Voltage operating points for (a) 1.0I, (b) 1.2I, and Constrained Control Problem: Given accurate (c) 1.4I over US06x3 cycle. state/parameter estimates (ˆx, θˆ), regulate input current I(t) such that the EChem constraints in Table I are enforced The VO governor is structurally equivalent to the MRG, pointwise in time (see Fig. 2). except the constraints in Table I are replaced by voltage limits of 2.8V and 3.9V. Various automotive-relevant We seek to maintain operation subject to electrochemical charge/discharge cycles cases were tested. To explore state state constraints. This protects the battery against catas- constraint management, reference current was scaled by trophic failure and maintains longevity. A list of relevant factors of 1.0, 1.2, 1.4 (1.0I, 1.2I, 1.4I). Due to space state constraints is provided in Table I. These limits are asso- constraints,× we only× provide× examples with three concate- ciated with material saturation/depletion, mechanical stress, nated US06 drive cycles (US06x3). extreme temperatures, and harmful side reactions, such as Figure 9 depicts the Temperature vs. Voltage operational lithium plating and solid/electrolyte interphase film growth. points for the MRG vs. VO controllers for the US06x3 A reference governor is an add-on algorithm that guaran- 1.0I, 1.2I, and 1.4I current profiles. The VO controller upper tees constraint satisfaction pointwise-in-time while tracking voltage limit becomes more constrictive as the current mul- a desired reference input [31], [32]. In our “modified” ref- tiplier increases. The MRG safely exceeds the VO voltage erence governor (MRG) implementation, the applied current r limits without violating the electrochemical constraints. In I[k] and reference current I [k] are related by automotive applications, this ultimately means the MRG I[k + 1] = β[k]Ir[k], β [0, 1], (20) recuperates more energy (i.e. from regenerate braking) than ∈ the VO controller. We numerically compared the MRG and where I[k] = I(t) for t [k∆t, (k + 1)∆t), k Z. The VO across five other automotive drive cycles from [33]. ∈ ∈ goal is to maximize β such that the state stays within an In the most aggressive drive cycle (US06x3) the MRG admissible set over some future time horizon, achieves 11.03% and 150.61% more discharge and charge O power, respectively, over the VO controller (1.4I case). The β∗[k] = max β [0, 1] : x(t) . (21) { ∈ ∈ O} MRG also achieves a 22.99% net energy increase. Across Variable x(t) represents the electrochemical model state at all six simulated drive cycles, the MRG achieves average time t and is the set of initial conditions that maintain the increases in discharge power, charge power, and net energy state withinO the constraints listed in Table I, over a future of 4.92%, 57.15%, and 10.04%, respectively (1.4I case). time horizon τ [t, t + Ts] [30]. See [30] for complete details. Consequently, we conclude We evaluate∈ the power and energy capacity benefits of that electrochemical model based control enables substantial the MRG versus an industry standard Voltage-Only (VO) performance improvements, provided the electrochemical controller on electric vehicle-like charge/discharge cycles. states and parameters can be accurately estimated. V. CONCLUSIONS [14] S. J. Moura, N. Chaturvedi, and M. 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