Estimation and Control of Battery Electrochemistry Models:A Tutorial

Scott J. Moura

Abstract— This paper presents a tutorial on estimation andcontrol problems for battery electrochemistry models. Wepresent a background on battery electrochemistry, along with acomprehensive electrochemical (EChem) model. EChem modelspresent a remarkably rich set of control-theoretic questionsinvolving model reduction, state & parameter estimation, andoptimal control. We discuss fundamental systems and controlschallenges, and then present opportunities for future research.

I. INTRODUCTION

Batteries are ubiquitous – they exist in our smart phones,laptops, electric vehicles (EVs), and electric grids. Energystorage is a critical enabling technology for designing sus-tainable energy systems. Although battery materials sci-ence has seen rapid advances [1], batteries are chronicallyunderutilized and conservatively designed [2]. Consumerspurchase batteries with 20-50% excess energy capacity, lead-ing to added weight, volume, and upfront cost. Intelligentbattery control can lead to faster charge times, increasedenergy and power capacity, as well as a longer life. The keyto realizing such advanced battery management systems iselectrochemical model based controls. These electrochemicalmodels, however, contain a multitude of fundamental control-theoretic challenges, yielding a rich opportunity for systemsand controls researchers.

A. Background & Fundamentals of Battery Electrochemistry

Italian physicist Alessandro Volta invented the first batterycell in 1800. The so-called voltaic pile consisted of twometals in series, zinc and copper, coupled by a sulphuricacid electrolyte. Volta constructing this system in responseto experiments performed by his colleague Luigi Galvani,who was fascinated by the interaction between electricity andbiological nervous systems. During an experiment, Galvanidiscovered that a dead frog’s legs would kick when connectedto two dissimilar metals. Galvani conjectured the energyoriginated within the animal, coining the phenomenon “an-imal electricity.” Volta believed the different metals causedthis behavior, and proved his hypothesis true with the voltaicpile [3]. Thus, the electrochemical battery cell was born.

A battery converts energy between the chemical andelectrical domains through oxidation-reduction reactions. Itconsists of two dissimilar metals (electrodes) immersed in an

This work was supported in part by the National Science Foundationunder Grant No. 1408107 and the Advanced Research Projects Agency-Energy, U.S. Department of Energy, under Award Number DE-AR0000278.

S. J. Moura is with the Energy, Controls, and Applications Lab (eCAL) inCivil and Environmental Engineering at the University of California, Berke-ley in Berkeley, California, 94720, USA smoura@berkeley.edu

Zn(s)   Cu(s)  

e- e-

ZnSO4(aq)   CuSO4(aq)  

Separator  

Zn+2(aq) Cu+2

(aq)

_   +  SO-2

4

Oxida7on   Reduc7on  

Ca7on   Ca7on  

Anion  

Zinc    Anode  

Copper    Cathode  

Fig. 1. An example zinc-copper Galvanic (or Voltaic) cell demonstratingthe principles of operation for an electrochemical cell.

electrolyte, exemplified by the zinc-copper Galvanic cell inFig. 1. The cathode and anode materials are jointly selectedto have a large electrochemical potential between each other.This creates the desired electrochemical energy storage prop-erty. The electrodes are electrically isolated by a separator.Hence, conservation of charge forces electrons through anexternal circuit, powering a connected device, while cationsflow between the electrodes within the electrolyte.

Electrode and electrolyte materials are selected for theirvoltage, charge capacity, weight, cost, manufacturability,etc. For example, lithium-ion cells are attractive in mobileapplications because lithium is the lightest (6.94 g/mol) andmost electropositive (-3.01V vs. standard hydrogen elec-trode) metal in the periodic table. Lead acid cells featureheavier electrodes (Pb and PbO2), yet provide high surgecurrents at cost effective prices. Lithium-air batteries featurecathodes that couple electrochemically with atmosphericoxygen, thus producing energy densities that rival gaso-line fuel. In battery energy management, we are interestedin maximizing performance and longevity. This requires adetailed understanding of the underlying electrochemistry.However, the electrochemical state variables are not directlymeasurable. At best, one can measure voltage, current, andtemperature only. Consequently, modeling and control arenecessary to extract the full potential from batteries.

B. Estimation and Control Problem Statements

Figure 2 outlines an electrochemical model based controlsystem, comprised of a state estimator, parameter estimator,and controller. Next, we provide precise problem statementsfor each control-theoretic task.

1) State-of-Charge (SOC) Estimation: SOC indicates thequantity of lithium within each electrode’s solid phase. Itis analogous to a fuel tank level, since it represents the

EChem-based State/Param Estimator

I(t)   V(t), T(t)  Battery Cell EChem-based

Controller Ir(t)  

V(t), T(t)  ^ ^

+ _

Innovations

Estimated States & Params

ElectroChemical Controller (ECC)  

Measurements

^ x(t), θ(t)  ^

Fig. 2. Block diagram for ElectroChemical Control (ECC) System,comprised of a battery cell, state/parameter estimator, and controller.

stored electrochemical energy. Unlike fuel tanks, SOC is notmeasurable and must be estimated. This can be directly castas a state estimation problem.The State Estimation Problem: Given measurements ofcurrent I(t), voltage V (t), and temperature T (t), estimatethe concentration of lithium in each electrode’s solid phasematerial. Optionally, one may estimate other electrochem-ical variables of interest, such as electrolyte concentrationand overpotentials (see Section II-B).

The key challenges for state estimation are lack of completeobservability and model nonlinearities.

2) State-of-Health (SOH) Estimation: Battery SOH met-rics indicate a battery’s relative age. The two most commonSOH metrics are charge capacity fade and impedance rise(i.e. power fade). Charge capacity fade indicates how chargecapacity has decreased relative to its nameplate value, e.g.a 2 Ah cell may hold 1.6 Ah after two years of use. Powercapacity fade indicates how power capacity has decreasedrelative to its nameplate value, e.g. a fresh cell may provide360W of power for 10 seconds, but only 300W after twoyears of use. Gradual changes in SOH metrics are directlyrelated to changes in the mathematical model parameters.The Parameter Identification Problem: Given measure-ments of current I(t), voltage V (t), and temperature T (t),estimate uncertain parameters related to SOH, such ascyclable lithium, solid-electrolyte interface resistance, andvolume fractions.

The key challenges for electrochemical models includeparametric modeling, nonlinear parameter identifiability, andpersistent excitation.

3) Controlled Charging/Discharging: In many applica-tions, additional capacity is added to mitigate cell imbalance,capacity/power fade, thermal effects, and estimation errors.This leads to larger, heavier, and more costly batteries thannecessary. Electrochemical control alleviates oversizing bysafely operating batteries near their physical limits. Today,operational limits are defined by what can be measured– voltage, current, and temperature. Battery degradation,however, is more closely related to limits on immeasurableelectrochemical states, such as overpotentials and surfaceconcentrations [2]. We seek a paradigm-shifting architecturethat expands the operating envelope by constraining internalelectrochemical states instead of measured values such asvoltage, current, and temperature (see Fig. 3).

Cell Current Surface concentrationTer

min

al V

olt

age

Ov

erp

ote

nti

al

Traditional limits

of operation

Traditional limits of operation

Electrochemical model-based limits of operation

Fig. 3. An electrochemical model-based control architecture constrainselectrochemical state variables, as opposed to measurable outputs. Thisexpands the operational envelope and ensures safety.

The Constrained Control Problem: Given measurementsof current I(t), voltage V (t), and temperature T (t), controlcurrent such that critical electrochemical variables aremaintained within safe operating constraints.

II. MATHEMATICAL MODELING

A. Background

Electrochemical (EChem) models capture the spatiotem-poral dynamics of lithium-ion concentration, electric po-tential, and intercalation kinetics. Most models in the bat-tery controls literature are derived from the Doyle-Fuller-Newman (DFN) model [4], which is based upon porouselectrode and concentrated solutions theory. Figure 4 showsa cross section of the layers described in Fig. 1.

At full charge the majority of lithium exists within theanode solid phase particles, typically a lithiated carbonLixC6. These particles are idealized as spherically sym-metric. During discharge, lithium diffuses from the interiorto the surface of these porous spherical particles. At thesurface an electrochemical reaction separates lithium into apositive lithium ion and electron (1). Next, the lithium ionmigrates from the anode, through the separator, and into thecathode. Since the separator is an electrical insulator, thecorresponding electron travels through an external circuit,powering the connected device. The lithium ion and electronmeet at the cathode particle surface, typically a lithiummetal oxide LiMO2, and undergo the reverse electrochemicalreaction according to (2).

LixC6 ⇀↽ C6 + xLi+ + xe−, (1)Li(1−x)MO2 + xLi+ + xe− ⇀↽ LiMO2. (2)

The resultant lithium atom diffuses into the interior of thecathode’s spherical particle. This entire process is reversibleby applying sufficient potential across the current collectors– rendering an electrochemical storage device. In addition tolithium migration, this model captures the spatial-temporaldynamics of internal potentials, electrolyte current, and cur-rent density between the solid and electrolyte phases.

B. Doyle-Fuller-Newman Model

We consider the Doyle-Fuller-Newman (DFN) model inFig. 4 to predict the evolution of lithium concentration inthe solid c±s (x, r, t), lithium concentration in the electrolyte

Separator CathodeAnode

LixC6 Li1-xMO2

Li+cs-(r) cs

+(r)

css- css

+

Electrolyte

e- e-

ce(x)

0- L- L+ 0+0sep Lsep

r r

x

Fig. 4. Schematic of the Doyle-Fuller-Newman model [4]. The modelconsiders two phases: the solid and electrolyte. In the solid, states evolve inthe x and r dimensions. In the electrolyte, states evolve in the x dimensiononly. The cell is divided into three regions: anode, separator, and cathode.

ce(x, t), solid electric potential φ±s (x, t), electrolyte electricpotential φe(x, t), ionic current i±e (x, t), molar ion fluxesj±n (x, t), and bulk cell temperature T (t) [4]. The governingequations are given by

∂c±s∂t

(x, r, t) =1

r2∂

∂r

[D±s r

2 ∂c±s

∂r(x, r, t)

], (3)

∂ce∂t

(x, t) =∂

∂x

[De(ce)

∂ce∂x

(x, t) +1− t0cεeF

i±e (x, t)

],

(4)∂φ±s∂x

(x, t) =i±e (x, t)− I(t)

σ±, (5)

∂φe∂x

(x, t) = − i±e (x, t)

κ(ce)+

2RT

F(1− t0c)

×(

1 +d ln fc/a

d ln ce(x, t)

)∂ ln ce∂x

(x, t), (6)

∂i±e∂x

(x, t) = asFj±n (x, t), (7)

j±n (x, t) =1

Fi±0 (x, t)

[eαaFRT η±(x,t) − e−αcFRT η±(x,t)

],

(8)

ρavgcPdT

dt(t) = hcell [Tamb(t)− T (t)] + I(t)V (t)

−∫ 0+

0−asFjn(x, t)∆T (x, t)dx, (9)

where De, κ, fc/a are functions of ce(x, t) and

i±0 (x, t) = k±√c±ss(x, t)ce(x, t)

(c±s,max − c±ss(x, t)

),

(10)η±(x, t) = φ±s (x, t)− φe(x, t)

− U±(c±ss(x, t))− FR±f j±n (x, t), (11)

c±ss(x, t) = c±s (x,R±s , t), (12)

∆T (x, t) = U±(c±s (x, t))− T (t)∂U±

∂T(c±s (x, t)), (13)

c±s (x, t) =3

(R±s )3

∫ R±s

0

r2c±s (x, r, t)dr. (14)

Space, r

c−s (x, r, t)/c−

s,max

Cente

rS

urf

ace

Space, r

c+s (x, r, t)/c+s,max

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

Space, x/L

c e(x

,t)

0.45

0.5

0.55

0.6

0.65

0.7

0.75

CathodeSeparatorAnode

Fig. 5. Solid c±s (x, r, t) and electrolyte ce(x, t) concentrations asfunctions of space, after 30sec of 5C discharge. Symbols {−,+} denotesthe negative electrode (anode) and positive electrode (cathode), respectively.In the solid phase concentrations, the top and bottom denote particle centerr = 0 and surface r = R±

s , respectively.

Along with these equations are corresponding boundaryand initial conditions. For brevity, we only summarize thedifferential equations here. Further details, including nota-tion definitions, can be found in [2], [4]. Parameters for aLiCoO2-C cell are publicly available from the DUALFOILmodel, developed by Newman and his collaborators [5]. Notethe mathematical structure, which contains PDEs (3)-(4),ODEs in time (9), ODEs in space (5)-(7), and nonlinearalgebraic constraints (8). This presents a formidable task withrespect to control-oriented modeling, state estimation, andcontrol design (see Section III and IV).

C. Electrochemical Model Simulations

Illustrative simulations of the electrochemical model areprovided in Fig. 5. Specifically, this figure depicts the solidphase c±s (x, r, t) and electrolyte ce(x, t) concentrations asfunctions of space, after 30sec of 5C1 discharge. The anodesolid phase concentrations c−s (x, r, t) exhibit a decreasinggradient as r increases toward the particle surface, sincelithium is exiting the anode during discharge. This createsa spatial gradient in the electrolyte phase ce(x, t) alongthe x-dimension. On the cathode, solid phase concentrationincreases c−s (x, r, t) w.r.t. r, especially near the particlesurface, since lithium is entering the cathode.

Note our ultimate goal: estimate these spatio-temporaldynamics and regulate their evolution, given only measure-ments of voltage, current, and temperature.

D. Model Reduction

A rapidly growing body of literature is establishing aspectrum of EChem models that achieve varying balancesof mathematical simplicity and accuracy. Concepts includespectral methods [6], [7], residue grouping [8], quasilin-earization & Pade approximation [9], principle orthogonal

1C-rate is a normalized measure of electric current that enables compar-isons between different sized batteries. Mathematically, the C-rate is definedas 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 of2.5 Ah, then C-rates of 2C, 1C, and C/2 correspond to 5 A, 2.5 A, and 1.25A, respectively. Note that C-rate has dimensions of [A] / [Ah] = [1/h].

decomposition [10], and single particle model variants [11],[12], [13]. Due to the richness of existing literature, we donot focus on model reduction in this paper. Instead, we focuson the state/parameter estimation and control challenges.

III. STATE & PARAMETER ESTIMATION

A. Survey of SOC/SOH Estimation Literature

Over the past decade, the SOC/SOH estimation literaturehas grown considerably rich with various algorithms, models,and applications. Most existing literature to date uses equiv-alent circuit models, but electrochemical model-based esti-mation algorithms have also emerged. These studies developestimators for reduced-order models. The model reductionand observer design process are intimately intertwined, assimpler models ease estimation design at the expense offidelity. Ideally, one seeks to derive a provably stable esti-mator for the highest fidelity electrochemical battery modelpossible. The first wave of studies utilized the “single particlemodel” (SPM) for estimator design [11], [12], [14], [15].The SPM idealizes each electrode as a single sphericalporous particle by neglecting the electrolyte dynamics. Thismodel is suitable for low C-rates, but degrades at C-ratesabove C/2. Recently, researchers have extended the singleparticle model to include electrolyte dynamics [13], [16],although there have been limited estimation studies [17],[18]. Nonetheless, state estimation designs have emergedfor other electrochemical models that incorporate electrolytedynamics. Examples include spectral methods with outputinjection [19], residue grouping with Kalman filtering [20],and composite electrodes with nonlinear filters [21]. Unfor-tunately, it becomes more difficult to prove estimation errorstability as model complexity increases. The core difficulty islack of complete observability from voltage measurements.

B. Single Particle Model (SPM)

To illustrate the control-theoretic challenges of SOC/SOHestimation, we consider the SPM – the simplest form ofelectrochemical model. The SPM idealizes each electrodeas a single aggregate spherical particle. One can derive thismodel by assuming the electrolyte Li concentration ce(x, t)from (4) is constant in space and time. Mathematically, themodel consists of two diffusion PDEs,

∂c±s∂t

(r, t) = D±s

[2

r

∂c±s∂r

(r, t) +∂2c±s∂r2

(r, t)

], (15)

∂c±s∂t

(0, t) = 0,∂c±s∂t

(R±s , t) =±I(t)

D±s Fa±AL±. (16)

The boundary conditions at r = R+s and r = R−s signify

that flux is proportional to input current I(t). Output voltageis given by a nonlinear function of the state values at theboundary c+ss(t), c

−ss(t) and the input current I(t) as follows

V (t) =RT

αFsinh−1

( −I(t)

2a+AL+i+0 (c+ss(t);nLi)

)

−RTαF

sinh−1(

I(t)

2a−AL−i−0 (c−ss(t);nLi)

)

+U+(c+ss(t))− U−(c−ss(t)) +RfI(t), (17)

V (t) = h(c−ss, c+ss, I; θ), (18)

where the ij0(·) is the exchange current density and cjss(t) =cjs(R

js, t) is the surface concentration for electrode j ∈

{+,−}. The functions U j(·) are the equilibrium potentialsof each electrode material, given the surface concentration.Mathematically, the functions U j(·) are strictly monotoni-cally decreasing functions of their input. This fact impliesthe inverse of its derivative is always finite, a property that iscritically important for assessing observability. Symbol nLirepresents the total moles of cyclable lithium. Finally, param-eter vector θ =

[nLi, (a

+AL+k+)−1, (a−AL−k−)−1, Rf]T

represents uncertain parameters.Remark 1 (SOH Metrics): Coincidently, the parameters

nLi and Rf represent capacity and impedance rise,respectively. Identification of nLi and Rf provides a directsystem-level estimate of SOH.

Combined SOC/SOH Problem: Given measurementsof current I(t), voltage V (t), and the SPM equations,simultaneously estimate the lithium concentration statesc−s (r, t), c+s (r, t) and parameters θ.

C. Model Properties

Before embarking on an estimation design, we first studyrelevant mathematical properties of the SPM. The c+s , c

−s

subsystems are mutually independent of each other. More-over, they are governed by linear PDEs. Also note that thePDE subsystems produce boundary values c+ss(t), c

−ss(t) that

feed into the nonlinear output function (17). The SPM isalso characterized by the following dynamical properties,which present notable challenges for state estimation. Wepresent the following propositions, whose proofs are straight-forward, non-insightful, and omitted for brevity.

Proposition 1 (Marginal Stability): Each individual sub-system in (15)-(16) governing states c+s (r, t), c−s (r, t) ismarginally stable. In particular, each subsystem contains oneeigenvalue at the origin, and the remaining eigenvalues lieon the negative real axis of the complex plane.

Proposition 2 (Conservation of Lithium): The moles oflithium in the solid phase are conserved [19]. Mathemati-cally, d

dt (nLi(t)) = 0 where

nLi(t) =∑

j∈{+,−}

εjsLjA

43π(Rjs)3

∫ Rjs

0

4πr2cjs(r, t)dr (19)

Invertability Analysis: Next, we study invertability of theoutput function (17) w.r.t. boundary state variables c±ss(t).Figure 6(a) provides the open circuit potential (OCP) func-tions U−(·), U+(·) as functions of the normalized anodeconcentration θ± = c±ss/c

±s,max. The near zero gradient

implies the voltage measurement is weakly sensitive toperturbations in the surface concentrations c±ss. This is furtherconfirmed by Fig. 6(b), which depicts the output function’spartial derivatives w.r.t. c−ss and c+ss at equilibrium conditions,for currents ranging from -5C to +5C. It is important to notethat h is strictly monotonically decreasing w.r.t. c+ss over alarger range than h is strictly monotonically increasing w.r.t.

0

1

2

An

od

e O

CP

[V

]

3

4

5

Ca

tho

de

OC

P [

V]U −(θ−)

U +(θ+)

0 0.2 0.4 0.6 0.8 1−1

−0.5

0

0.5

1

1.5

Normalized Surface Concentrat ion, θ± = c±s s/c

±

s ,max

[×10−6]

∂h/∂ c−s s(θ−)

∂h/∂ c+s s(θ+)

(a)

(b)

LOW SENSITIVITY

Fig. 6. (a) Open circuit potential functions U−(θ−), U+(θ+); and (b)Gradients of h(c+ss, c

−ss, I) in (18), as functions of normalized surface

concentration θ± = c±ss/c±s,max, for currents ranging from -5C to +5C.

Adaptive Output Fcn Inversion

Adaptive PDE Backstepping

Observer

I(t)   V(t)  

ĉss-  

ĉs-(r,t)  

+_

Padé-based PDE Param ID

čss-  

Output Fcn Nonlinear Param ID θh

I(t)  

I(t)  

I(t)  ε, q

V(t)  

Battery Cell

Fig. 7. Block diagram of Adaptive PDE Observer for SOC/SOH Estima-tion. See [14] for details.

c−ss. This property is critical, since it demonstrates that volt-age is generally more sensitive to perturbations in cathodesurface concentration than anode surface concentration. Note∂h/∂c+ss ≈ 0 for 0.8 ≤ θ+ ≤ 0.9. This region is a “blindspot” with respect to output inversion.

Nonlinear Parameter Sensitivity Analysis: Since the pa-rameter vector θ enters nonlinearly in the output function(18), we must first perform a sensitivity analysis to assesslinear dependence. Define the sensitivity vector S = ∂h/∂θ.Applying the ranking procedure outlined in [22] to STSreveals strong linear dependence exists between θ2, θ3, θ4,where θ4 exhibits the greatest sensitivity magnitude. As aresult, we pursue a parameter identification scheme for theparameter subset (θ1, θ4). Coincidentally, these two parame-ters are exactly nLi, Rf , corresponding to capacity fade andimpedance rise. Refer to [14] for complete details.

D. Adaptive PDE Observer Approach

The previous subsection described several fundamentalestimation challenges for the SPM, including marginal sta-bility, poor output sensitivity to states, and linear dependencebetween parameters. An adaptive PDE observer was designedfor this model in [14], which achieves stable estimates by

0 5 10 15 20 25 30 35 40−10

0

10

Curr

ent

[A]

I

0 5 10 15 20 25 30 35 40

0.4

0.6

0.8

Bulk

SO

C

SOC−bu lk

S OC−bu lk

0 5 10 15 20 25 30 35 40−0.1

0

0.1

0.2

0.3

Bulk

SO

C E

rror

SOC−bu lk− S OC

−bu lk

0 5 10 15 20 25 30 35 40−5

0

5

10

Surf

ace

Conce

ntr

atio

n

Err

or

[km

ol]

Time [min]

c−ss− c−ssc−ss− c−ss

0 5 10 15 20 25 30 35 403.1

3.2

3.3

3.4

3.5

Volt

age

[V]

VV

0 5 10 15 20 25 30 35 40

0

20

40

60

Volt

age

Err

or

[mV

]

V − V

0 5 10 15 20 25 30 35 400

5

10

15

PD

E P

aram

s

ε

q

True Value

0 5 10 15 20 25 30 35 40

1

2

3

Time [min]

Outp

ut

Fcn

Par

ams

nLi/nLi

Rf/Rf

True Value

0 5 10 15 20 25 30 35 400.3

0.4

0.5

0.6

0.7

0.8

Bu

lk S

OC

Measures of SOC

SOC−bu lk

S OC−bu lk

0 5 10 15 20 25 30 35 40

10

15

20

25

30

Su

rfac

e C

on

cen

trat

ion

s

Time [min]

c−ssc−ssc−ss

0 5 10 15 20 25 30 35 400

5

10

15

PD

E P

aram

s

Measures of SOH

ε

q

True Value

0 5 10 15 20 25 30 35 400.5

1

1.5

2

2.5

3

Time [min]

Ou

tpu

t F

cn P

aram

s

nLi/nLi

Rf/Rf

True Value

0 5 10 15 20 25 30 35 400.3

0.4

0.5

0.6

0.7

0.8

Bu

lk S

OC

Measures of SOC

SOC−bu lk

S OC−bu lk

0 5 10 15 20 25 30 35 40

10

15

20

25

30

Su

rfac

e C

on

cen

trat

ion

s

Time [min]

c−ssc−ssc−ss

0 5 10 15 20 25 30 35 400

5

10

15

PD

E P

aram

s

Measures of SOH

ε

q

True Value

0 5 10 15 20 25 30 35 400.5

1

1.5

2

2.5

3

Time [min]

Ou

tpu

t F

cn P

aram

s

nLi/nLi

Rf/Rf

True Value

0 5 10 15 20 25 30 35 400.3

0.4

0.5

0.6

0.7

0.8

Bu

lk S

OC

Measures of SOC

SOC−bu lk

S OC−bu lk

0 5 10 15 20 25 30 35 40

10

15

20

25

30

Su

rfac

e C

on

cen

trat

ion

s

Time [min]

c−ssc−ssc−ss

0 5 10 15 20 25 30 35 400

5

10

15

PD

E P

aram

s

Measures of SOH

ε

q

True Value

0 5 10 15 20 25 30 35 400.5

1

1.5

2

2.5

3

Time [min]

Ou

tpu

t F

cn P

aram

s

nLi/nLi

Rf/Rf

True Value

Fig. 8. Evolution of state and parameter estimates from Adaptive PDEobserver [14] for UDDSx2 charge/discharge cycle. Zero mean Gaussiannoise with a 10 mV variance was added to the voltage measurement. TheEChem model (3)-(14) provides the “measured” plant data.

exploiting output function invertibility and lithium conser-vation. Figure 7 summarizes this scheme. The algorithmis composed of an Adaptive PDE backstepping observer[23], Pade-based parameter identifier for PDE parameters [9],nonlinear recursive least squares identifier for output functionparameters, and an adaptive output function inversion scheme[24]. A simulation example is provided in Fig. 8, whichdemonstrates SOC and SOH convergence given incorrectinitial estimates. We remark that the fundamental challengesfor estimation are significant, even for the simplest of EChemmodels – the SPM.

IV. CONSTRAINED OPTIMAL CONTROL

A. Survey of Optimal Battery Charge/Discharge Literature

A rich body of literature exists on battery charg-ing/discharging performance. Interestingly, the bulk of re-search involves experimental studies and non-model basedheuristic charging protocols (see [25] for a survey). Rel-atively speaking, research on electrochemical model-based

TABLE IELECTROCHEMICAL VARIABLES y TO REGULATE WITHIN BOX

CONSTRAINTS, I.E. ymin ≤ y ≤ ymax .

Variable, y Definition ConstraintI(t) Current Power electronics limitc±s (x, r, t) Li concentration in solid Material

saturation/depletion∂c±s∂r

(x, r, t) Li concentration gradientin solid

Diffusion-inducedstress/strain

ce(x, t) Li concentration inelectrolyte

Materialsaturation/depletion

T (t) Temperature High/low temperaturesaccelerate aging

ηs(x, t) Side reactionoverpotential

Li-plating, SEI-layergrowth

control is nascent. Existing optimal control studies includeconstrained control [20], [26] and open-loop control [27],[28], [29]. A recent study by the author performs a compre-hensive assessment of performance improvements enabledby reference governors and electrochemical models [30].

B. A Reference Governor Approach

Ensuring safe operating constraints is a basic requirementfor batteries. Mathematically, this can be abstracted as aconstrained control problem for which reference governorsprovide one effective solution. The problem statement is:Constrained Control Problem: Given accuratestate/parameter estimates (x, θ), regulate input current I(t)such that the EChem constraints in Table I are enforcedpointwise in time (see Fig. 2).

We seek to maintain operation subject to electrochemicalstate constraints. This protects the battery against catas-trophic failure and maintains longevity. A list of relevantstate constraints is provided in Table I. These limits are asso-ciated with material saturation/depletion, mechanical stress,extreme temperatures, and harmful side reactions, such aslithium plating and solid/electrolyte interphase film growth.

A reference governor is an add-on algorithm that guaran-tees constraint satisfaction pointwise-in-time while trackinga desired reference input [31], [32]. In our “modified” ref-erence governor (MRG) implementation, the applied currentI[k] and reference current Ir[k] are related by

I[k + 1] = β[k]Ir[k], β ∈ [0, 1], (20)

where I[k] = I(t) for t ∈ [k∆t, (k + 1)∆t), k ∈ Z. Thegoal is to maximize β such that the state stays within anadmissible set O over some future time horizon,

β∗[k] = max {β ∈ [0, 1] : x(t) ∈ O} . (21)

Variable x(t) represents the electrochemical model state attime t and O is the set of initial conditions that maintain thestate within the constraints listed in Table I, over a futuretime horizon τ ∈ [t, t+ Ts] [30].

We evaluate the power and energy capacity benefits ofthe MRG versus an industry standard Voltage-Only (VO)controller on electric vehicle-like charge/discharge cycles.

295

300

305

310

315

320

Tem

p [K

]

MRGVO

295

300

305

310

315

320

Tem

p [K

]

MRGVO

3 3.2 3.4 3.6 3.8 4 4.2295

300

305

310

315

320

Volt [V]

Tem

p [K

]

MRGVO

(a)

(b)

(c)

Expandedoperatingrange

Expandedoperatingrange

Expandedoperatingrange

Fig. 9. Temperature vs. Voltage operating points for (a) 1.0I, (b) 1.2I, and(c) 1.4I over US06x3 cycle.

The VO governor is structurally equivalent to the MRG,except the constraints in Table I are replaced by volt-age limits of 2.8V and 3.9V. Various automotive-relevantcharge/discharge cycles cases were tested. To explore stateconstraint management, reference current was scaled byfactors of ×1.0,×1.2,×1.4 (1.0I, 1.2I, 1.4I). Due to spaceconstraints, we only provide examples with three concate-nated US06 drive cycles (US06x3).

Figure 9 depicts the Temperature vs. Voltage operationalpoints for the MRG vs. VO controllers for the US06x31.0I, 1.2I, and 1.4I current profiles. The VO controller uppervoltage limit becomes more constrictive as the current mul-tiplier increases. The MRG safely exceeds the VO voltagelimits without violating the electrochemical constraints. Inautomotive applications, this ultimately means the MRGrecuperates more energy (i.e. from regenerate braking) thanthe VO controller. We numerically compared the MRG andVO across five other automotive drive cycles from [33].In the most aggressive drive cycle (US06x3) the MRGachieves 11.03% and 150.61% more discharge and chargepower, respectively, over the VO controller (1.4I case). TheMRG also achieves a 22.99% net energy increase. Acrossall six simulated drive cycles, the MRG achieves averageincreases in discharge power, charge power, and net energyof 4.92%, 57.15%, and 10.04%, respectively (1.4I case).See [30] for complete details. Consequently, we concludethat electrochemical model based control enables substantialperformance improvements, provided the electrochemicalstates and parameters can be accurately estimated.

V. CONCLUSIONSThis paper provides a tutorial on estimation and control

challenges for electrochemical battery models. We first pro-vide a brief history of the electrochemical battery cell, andthen discuss the fundamental operating principles. Next, wepresent concrete statements for the state-of-charge (SOC),state-of-health (SOH), and charge/discharge control prob-lems. Mathematically, these are respectively cast as stateestimation, parameter estimation, and constrained controlproblems. We discuss combined SOC/SOH estimation andhighlight some solutions and several challenges. Finally, wediscuss constrained control via modified reference governors.

Electrochemical models provide an exceptionally rich ap-plication for challenging systems and controls problems. Thecombined state/parameter estimation problem is particularlydifficult. Ultimately, our objective is to derive provably stableestimators for increasingly complex electrochemical models.For example, recent work extends the SPM to includeelectrolyte dynamics [13], [18], temperature [16], and multi-ple material cathodes [21]. Fundamental challenges includemodel reduction, observability, identifiability, nonlinearity,spatio-temporal dynamics, and numerical implementations.

REFERENCES

[1] M. Armand and J.-M. Tarascon, “Building better batteries,” Nature,vol. 451, no. 7179, pp. 652–657, 2008.

[2] N. A. Chaturvedi, R. Klein, J. Christensen, J. Ahmed, and A. Kojic,“Algorithms for advanced battery-management systems,” IEEE Con-trol Systems Magazine, vol. 30, no. 3, pp. 49 – 68, 2010.

[3] G. Pancaldi, Volta: Science and culture in the age of enlightenment.Princeton University Press, 2005.

[4] K. Thomas, J. Newman, and R. Darling, Advances in Lithium-Ion Bat-teries. New York, NY USA: Kluwer Academic/Plenum Publishers,2002, ch. 12: Mathematical modeling of lithium batteries, pp. 345–392.

[5] J. Newman. (2008) Fortran programs for the simulation ofelectrochemical systems. [Online]. Available: http://www.cchem.berkeley.edu/jsngrp/fortran.html

[6] V. R. Subramanian, V. Boovaragavan, V. Ramadesigan, and M. Ara-bandi, “Mathematical model reformulation for lithium-ion batterysimulations: Galvanostatic boundary conditions,” Journal of the Elec-trochemical Society, vol. 156, no. 4, pp. A260 – A271, 2009.

[7] C. Mayhew, W. He, C. Kroener, R. Klein, N. Chaturvedi, and A. Ko-jic, “Investigation of projection-based model-reduction techniques forsolid-phase diffusion in li-ion batteries,” in 2014 American ControlConference (ACC), June 2014, pp. 123–128.

[8] K. A. Smith, C. D. Rahn, and C.-Y. Wang, “Model order reduction of1d diffusion systems via residue grouping,” ASME Journal of DynamicSystems, Measurement, and Control, vol. 130, no. 1, p. 011012, 2008.

[9] J. C. Forman, S. Bashash, J. L. Stein, and H. K. Fathy, “Reduction ofan electrochemistry-based li-ion battery model via quasi-linearizationand pade approximation,” Journal of the Electrochemical Society, vol.158, no. 2, pp. A93 – A101, 2011.

[10] L. Cai and R. E. White, “Reduction of model order based on properorthogonal decomposition for lithium-ion battery simulations,” Journalof The Electrochemical Society, vol. 156, no. 3, pp. A154–A161, 2009.

[11] S. Santhanagopalan and R. E. White, “Online estimation of the stateof charge of a lithium ion cell,” Journal of Power Sources, vol. 161,no. 2, pp. 1346 – 1355, 2006.

[12] D. D. Domenico, A. Stefanopoulou, and G. Fiengo, “Lithium-IonBattery State of Charge and Critical Surface Charge Estimation Usingan Electrochemical Model-Based Extended Kalman Filter,” Journalof Dynamic Systems, Measurement, and Control, vol. 132, no. 6, p.061302, 2010.

[13] S. K. Rahimian, S. Rayman, and R. E. White, “Extension of physics-based single particle model for higher charge-discharge rates,” Journalof Power Sources, vol. 224, no. 0, pp. 180 – 194, 2013.

[14] S. J. Moura, N. Chaturvedi, and M. Krstic, “Adaptive PDE Observerfor Battery SOC/SOH Estimation via an Electrochemical Model,”ASME Journal of Dynamic Systems, Measurement, and Control, vol.136, no. 1, pp. 011 015–011 026, Oct 2014.

[15] S. Dey and B. Ayalew, “Nonlinear observer designs for state-of-charge estimation of lithium-ion batteries,” in 2014 American ControlConference (ACC), June 2014, pp. 248–253.

[16] T. Tanim, C. Rahn, and C.-Y. Wang, “A reduced order electrolyteenhanced single particle lithium ion cell model for hybrid vehicleapplications,” in 2014 American Control Conference (ACC), June2014, pp. 141–146.

[17] T. R. Tanim, C. D. Rahn, and C.-Y. Wang, “State of charge estimationof a lithium ion cell based on a temperature dependent and electrolyteenhanced single particle model,” Energy, vol. 80, no. 0, pp. 731 – 739,2015.

[18] X. Han, M. Ouyang, L. Lu, and J. Li, “Simplification of physics-basedelectrochemical model for lithium ion battery on electric vehicle. PartII: Pseudo-two-dimensional model simplification and state of chargeestimation,” Journal of Power Sources, vol. 278, no. 0, pp. 814 – 825,2015.

[19] R. Klein, N. A. Chaturvedi, J. Christensen, J. Ahmed, R. Findeisen,and A. Kojic, “Electrochemical Model Based Observer Design for aLithium-Ion Battery,” IEEE Transactions on Control Systems Technol-ogy, vol. 21, no. 2, pp. 289–301, March 2013.

[20] K. A. Smith, C. D. Rahn, and C.-Y. Wang, “Model-based electro-chemical estimation and constraint management for pulse operationof lithium ion batteries,” IEEE Transactions on Control SystemsTechnology, vol. 18, no. 3, pp. 654 – 663, 2010.

[21] A. Bartlett, J. Marcicki, S. Onori, G. Rizzoni, X. Yang, and T. Miller,“Electrochemical model-based state of charge and capacity estimationfor a composite electrode lithium-ion battery,” IEEE Transactions onControl Systems Technology, vol. PP, no. 99, pp. 1–1, 2015.

[22] B. F. Lund and B. A. Foss, “Parameter ranking by orthogonalization -Applied to nonlinear mechanistic models,” Automatica, vol. 44, no. 1,pp. 278 – 281, 2008.

[23] A. Smyshlyaev and M. Krstic, “Backstepping observers for a classof parabolic PDEs,” Systems & Control Letters, vol. 54, no. 7, pp.613–625, 2005.

[24] P. Ioannou and J. Sun, Robust Adaptive Control. Prentice-Hall, 1996.[25] S. S. Zhang, “The effect of the charging protocol on the cycle life

of a li-ion battery,” Journal of power sources, vol. 161, no. 2, pp.1385–1391, 2006.

[26] R. Klein, N. A. Chaturvedi, J. Christensen, J. Ahmed, R. Findeisen,and A. Kojic, “Optimal charging strategies in lithium-ion battery,” inAmerican Control Conference, San Francisco, CA, United states, 2011,pp. 382 – 387.

[27] R. Methekar, V. Ramadesigan, R. D. Braatz, and V. R. Subramanian,“Optimum charging profile for lithium-ion batteries to maximizeenergy storage and utilization,” ECS Transactions, vol. 25, no. 35,pp. 139–146, 2010.

[28] S. Bashash, S. J. Moura, J. C. Forman, and H. K. Fathy, “Plug-inhybrid electric vehicle charge pattern optimization for energy cost andbattery longevity,” Journal of Power Sources, vol. 196, no. 1, pp. 541– 549, 2011.

[29] X. Hu, H. Perez, and S. J. Moura, “Battery charge control with anelectro-thermal-aging coupling,” in 2015 ASME Dynamic Systems andControl Conference, Columbus, OH, 2015.

[30] H. Perez, N. Shahmohammadhamedani, and S. Moura, “Enhancedperformance of li-ion batteries via modified reference governors andelectrochemical models,” IEEE/ASME Transactions on Mechatronics,vol. 20, no. 4, pp. 1511–1520, Aug 2015.

[31] E. Gilbert, I. Kolmanovsky, and K. Tan, “Discrete-time referencegovernors and the nonlinear control of systems with state and controlconstraints,” International Journal of Robust and Nonlinear Control,vol. 5, no. 5, pp. 487–504, 1995.

[32] I. Kolmanovsky, E. Garone, and S. Di Cairano, “Reference andcommand governors: A tutorial on their theory and automotive ap-plications,” in American Control Conference (ACC), 2014, June 2014,pp. 226–241.

[33] S. Moura, J. Stein, and H. Fathy, “Battery-Health Conscious PowerManagement in Plug-In Hybrid Electric Vehicles via ElectrochemicalModeling and Stochastic Control,” IEEE Transactions on ControlSystems Technology, vol. 21, no. 3, pp. 679–694, 2013.