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Optimal Control of Film Growth in Lithium-IonBattery Packs via Relay Switches

Scott J. Moura, Student Member, IEEE, Joel C. Forman, Saeid Bashash, Member, IEEE,Jeffrey L. Stein, Hosam K. Fathy

Abstract—Recent advances in lithium ion battery modelingsuggest unequal but controlled and carefully timed chargingof individual cells by reduce degradation. This article com-pares anode-side film formation for a standard equalizationscheme versus unequal charging through switches controlled bydeterministic dynamic programming (DDP) and DDP-inspiredheuristic algorithms. A static map for film growth rate isderived from a first-principles battery model adopted fromthe electrochemical engineering literature. Using this map, weconsider two cells connected in parallel via relay switches. Thekey results are: (1) Optimal unequal and delayed charging indeedreduces film buildup; (2) A near-optimal state feedback controllercan be designed from the DDP solution and film growth rateconvexity properties. Simulation results indicate the heuristicstate-feedback controller achieves near optimal performancerelative to the DDP solution, with significant reduction in filmgrowth compared to charging both cells equally, for severalfilm growth models. Moreover, the algorithms achieve similarfilm reduction values on the full electrochemical model. Theseresults correlate with the convexity properties of the film growthmap. Hence, this article demonstrates that unequal charging mayindeed reduce film growth given certain convexity propertiesexist, lending promise to the concept for improving battery packlife.

Index Terms—Lithium-ion batteries, health management, bat-tery management systems, optimal control, dynamic program-ming.

I. INTRODUCTION

THIS article examines the problem of optimizing chargingfor individual modules in a battery pack such that the

pack’s overall health degradation is minimized. Such health-conscious battery pack management has the potential to in-crease the useful life and reduce the long-term replacementcosts of expensive high-capacity battery packs. This is im-portant for ensuring the financial feasibility of battery energystorage in systems such as electric vehicles and smart grids, es-pecially if such systems are able to share energy through, e.g.,

Manuscript received October 28, 2009. Revised July 22, 2010. Acceptedfor publication September 18, 2010. This work was supported in part bythe University of Michigan Rackham Merit Fellowship and National ScienceFoundation Graduate Research Fellowship.

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S. J. Moura, J. C. Forman, S. Bashash, and J. L. Stein are with theDepartment of Mechanical Engineering, University of Michigan, Ann Arbor,MI 48109-2133 USA (e-mail: sjmoura@umich.edu; jcforman@umich.edu;bashash@umich.edu; stein@umich.edu).

H. K. Fathy is with the Department of Mechanical and Nuclear En-gineering, Pennsylvania State University, University Park, PA 16802 USA(Address all correspondences to this author; phone: +1-814-867-4442; e-mail:hkf2@psu.edu).

vehicle-to-grid (V2G) integration [1]. The article’s overall goalis therefore to design battery pack management algorithms thatcontrol degradation in some optimal sense. We pursue this goalspecifically for a pack consisting of two modules connectedin parallel. Moreover, we specifically focus on lithium ionchemistries, which have been identified as a promising batterytechnology for achieving high energy and power densities,among other benefits [2]. Managing degradation is particularlychallenging because the associated mechanisms, includingresistive film growth at the anode, are typically simulated usingcomputationally intensive electrochemistry-based models thatmay not be directly conducive to control design. Furthermore,it is currently impractical to directly access and control themechanisms causing degradation inside the cell. To addressthese issues, we construct a simple map-based model of batterydegradation and use it to design a nearly optimal controller thatutilizes existing relays in high-energy capacity battery packs.

Previous research has examined at least three importantproblems related to health-conscious battery management.First, researchers have developed cell-to-cell charge equaliza-tion circuits that protect cells connected in series strings fromover-charging or over-discharging due to capacity imbalances[3]–[6]. This article proposes an additional battery healthmanagement algorithm at the cell module level. Specifically,we consider the potential advantages of allowing unequalcharge values across modules connected in parallel, and allowflexibility in determining the timing of the charge process.

Second, the literature on lithium ion batteries also addressesthe problem of modeling battery degradation, power fade, andcapacity fade. A popular model for capturing the lithium diffu-sion dynamics and intercalation phenomena was developed byDoyle, Fuller, and Newman in [7], [8]. This model is partic-ularly appealing because it accurately captures the diffusiondynamics and voltage response characteristics relevant for awide range of electrolyte materials and physical parameters.Ramadass et al. [9] extended this model to study capacity fadeby hypothesizing an irreversible solvent reduction reaction atthe anode-side electrode/electrolyte interface that generates aresistive film by consuming cyclable lithium. Because thisprocess is considered one of the chief contributors to capacityfade and power loss [10]–[12], this article uses the modelpresented in [9] to study battery health management.

Finally, although there have been few publications on con-trolling battery health degradation, the concept of modelingbattery degradation in terms of charge capacity fade andincreased internal resistance spawned a body of researchknown as state-of-health (SOH) estimation. Research on SOH

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estimation generally uses empirical equivalent circuit batterycell models to estimate charge capacity and internal resistance,using a variety of algorithms, such as batch data reconcilia-tion, moving-horizon parameter estimation [13], recursive leastsquares [14], and extended Kalman filtering [15]–[17]. Re-cently, Smith, Rahn, and Wang [18] and Di Domenico, Fiengo,and Stefanopoulou [19] respectively used linear and extendedKalman filters to estimate the internal spatial-temporal statesof reduced order electrochemical models derived from [7], [8].This work does not estimate SOH-related parameters, however.

The main goal of this article is to extend the above re-search on battery health management by adding four importantand original contributions. First, we utilize a high-fidelityelectrochemistry-based model of film growth to generate areduced degradation model more suitable for control design.Second, we set up a technique for active control of film growththat uses existing relay switches in battery packs, typicallydesigned to prevent thermal runaway. Third, we formulate anoptimal control problem that seeks to minimize total batterypack film growth through appropriate relay switching se-quences. Fourth, we demonstrate that a nearly optimal controlpolicy can be implemented as a set of heuristic rules, designedfrom the optimal control results and convexity properties offilm growth rate. Portions of this work have been previouslypublished in non-archival form [20]; however this article addssignificant new material which validates the aforementionedcontrollers on a high-fidelity electrochemistry-based batterymodel and alternative cycles and film growth parameteriza-tions. In summary, the aim of this article is to extend dynamicsystems battery health research into the arena of lithium-ionfilm growth control. Namely, this article demonstrates that filmgrowth can be reduced through novel charge unequalizationschemes when certain convexity properties of the film growthmap exist.

The remainder of the article is organized as follows: Section2 reviews an electrochemical model for film growth developedby Ramadass, et al. [9], the reduced model utilized in thisarticle, and a simple battery pack design. Section 3 formulatesa deterministic dynamic programming (DDP) problem forminimizing total film growth in the battery pack under con-sideration. Section 4 analyzes the DDP results and interpretsthe solution via the convexity properties of film growth rate.We also consider the same optimization process for alternativefilm growth maps. This analysis motivates the design of asuboptimal heuristic feedback control law. Section 5 charac-terizes the performance of the heuristic algorithm vis-a-vis theDDP solution and standard charge equalization scheme, bothon the equivalent circuit model used for optimization and thefull electrochemical model. Finally, Section 6 summarizes thearticle’s main conclusions.

II. MODEL DEVELOPMENT

A. Electrochemical Capacity Fade Mechanics

In this article, a function mapping cell state of charge (SOC)and current to film growth rate is extracted from a first-principles electrochemical Li-ion battery cell model adoptedfrom [9]. This model simulates phenomena such as lithium

Film δ (x,t)

x

Anode

(Graphite)

Cathode

(LiCoO2)

J1(x,t)Negative

Terminal Js(x,t)

Film δfilm(x,t)

J1(x,t)

Positive

Terminal

Anode Separator Cathode

x

Fig. 1. Structure of the electrochemical lithium-ion battery cell model.

ion diffusion and intercalation to determine the potential andconcentration gradients in the solid and solution sections ofthe anode, cathode, and separator. A schematic of the cellmodel is provided in Fig. 1, where Ramadass et al. argue thata resistive film builds up on the anode electrode/electrolyteinterphase [9]. The exact chemical side reaction depends onthe chemistry of the electrode and electrolyte. Equations (1)-(6), developed by Ramadass et al. argue that a simple andgeneral method for modeling capacity loss is to assume anirreversible solvent reduction reaction of the following form

S + Li+ + e− → P (1)

where S denotes the solvent species and P is the product.As a result of this irreversible side reaction, the products

form a film at the electrode/electrolyte interface, which hasa time and spatially varying thickness δfilm(x, t). This irre-versibly formed film combines with the initial solid electrolyteinterphase (SEI) resistance RSEI to compose the total resis-tance at the electrode/electrolyte interface as follows

Rfilm(x, t) = RSEI +δfilm(x, t)

κP(2)

where κP , denotes the conductivity of the film, x is the spatialcoordinate, and t is time. The state equation correspondingto the growth of film thickness, due to the unwanted solventreduction described in Eq. (1), is given by

∂δfilm(x, t)∂t

= − MP

anρPFJs(x, t) (3)

In Eq. (3), MP , an, ρP , and F represent the product’smolecular weight, specific surface area, mass density, andFaraday’s constant, respectively. The term Js denotes thelocal volumetric current density for the side reaction, whichis governed by Butler-Volmer kinetics. If we assume thesolvent reduction reaction is irreversible and the variation ofLi-ion concentration in the solution is small, then we mayapproximate Js by the following Tafel equation [21].

Js(x, t) = −i0,sane(−0.5F

RT ηs(x,t)) (4)

In Eq. (4), i0,s, R, and T respectively denote the exchangecurrent density for the side reaction, universal gas constant,and cell temperature. The term ηs represents the side reactionoverpotential, which drives the solvent reduction reaction inEq. (1). This variable is expressed by the following equation,

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−10 −5 0 5 10 0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

SOCCurrent [A]

Ave

rage

Film

Gro

wth

Rat

e [ µ

Ω/m

2 /sec

]

Fig. 2. Static approximation of film growth rate vs. cell current and SOCfor fresh, relaxed cell (i.e. δfilm = 0 and concentration distributions areconstant). Positive current corresponds to discharge.

based on Kirchoff’s voltage law.

ηs(x, t) = ∆φ(x, t)− Us,ref −Jtot(x, t)

anRfilm(x, t) (5)

The variable ∆φ represents the difference in potentials be-tween the solid and solution. The symbol Us,ref denotes theequilibrium potential of the solvent reduction reaction, whichwe assume to be constant. The total intercalation current Jtotrepresents the flow of charge exchanged with the anode-sidesolution. Specifically, the total intercalation current Jtot isgiven by the sum of current between the solid and solution(J1), and the solvent reduction reaction and solution (Js), thatis

Jtot = J1 + Js (6)

Equations (2)-(6) encompass the film growth subsystem ofthe Li-ion battery cell model, adopted from [9]. This subsys-tem connects to the remainder of the battery model throughthe total intercalation current Jtot and potential difference ∆φ.Since these variables vary with respect to space (across theelectrodes and separator) and time, they are determined bysolving coupled partial differential equations and algebraicconstraints representing the concentration and potential dis-tributions in the solid and solution of the anode, cathode,and separator (see [7], [8] for details). For the lithium-ioncells studied in this paper the parameters pertaining to SOCdynamics have been fitted using a custom-built battery test rig.Details on the experimental setup and identification procedurecan be found in [22]. Although this model accounts for com-plex electrochemical phenomena such as diffusion dynamicsand film growth, its complexity makes control design forhealth management difficult. Therefore, the present researchseeks to use the high fidelity model to generate simpler modelsfor the purposes of control design.

B. Static Approximation of Film Growth Rate

To acquire insight on the relationship between battery cellSOC, current, and film growth rate, consider an ideal fresh

Fig. 3. Circuit diagram of battery pack.

cell, that is δfilm(x, 0) = 0. Suppose all the intercalationcurrents, overpotentials, and concentration profiles are constantwith respect to space and correspond to zero initial appliedcurrent. Starting from these initial conditions, we simulate theelectrochemical battery cell model for different initial SOCand applied current levels and measure the instantaneous filmgrowth rate. From these data we produce a static relationshipmapping cell SOC and applied current to the spatially averagedfilm growth rate δfilm, shown in Fig. 2.

The map indicates that film growth rate increases with cellSOC. The film growth rate also increases as the dischargecurrent becomes increasingly negative, i.e. for increasingcharge current. Finally, film grows when zero current isapplied, indicating that aging occurs even when the cells arenot in use. The authors of [9] assume film growth occursonly during charging. However, we simulate the side reactionequations under all applied current conditions and allow theside reaction overpotential to govern film growth. In SectionV-E we consider alternative film growth model realizations by(1) obeying the assumption in [9] and (2) using a specific set ofalternative model parameter values. A key question we revisitafter obtaining the optimal control solution is what insight canbe extracted from this map to design controllers that reducefilm formation in battery packs?

C. Battery Pack Model

Switched capacitor circuits [5], [6] are typically appliedto equalize individual SOC levels for cells connected inseries. In this article, we examine the potential advantages ofallowing unequal charge levels for battery modules connectedin parallel. A simple method to independently control modulecharge levels uses switches in protection circuits [23] (e.g.solid state relays or contactors). These devices are primarilydesigned to disconnect the battery in case of imminent catas-trophic behavior, such as thermal runaway [24]. When multiplemodules are arranged in parallel, individual solid state relayscan be connected in series with each parallel branch. Theserelays may serve as one potential opportunity for individuallycontrolling battery module SOC, and will be the topology weconsider henceforth.

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Consider a battery pack architecture consisting of twomodules connected in parallel through two switches, whereeach module contains one cell for simplicity (Fig. 3). The goalis to determine the optimal switching strategy that minimizesthe total film growth of both cells, given an exogenous currenttrajectory i0. Due to the computational complexity of thedistributed parameter electrochemical cell model describedin Section 2.1, and the curse of dimensionality imposed bydynamic programming [25], we require a simplified model forcontrol design. As such, we utilize an equivalent circuit model[16], [26], written in discrete time, with a ten second time step(∆T = 10 sec). This equivalent circuit model consists of anopen circuit voltage source OCV in series with an internalresistor Rint. Open circuit voltage and internal resistance arenonlinear functions of SOC, that is OCV (zi) and Rint(zi)where i = 1, 2. The state variables z1 and z2 represent the SOCof battery cells 1 and 2 respectively. The dynamic equationsfor each cell are based on integrating current i1, i2 to obtaincharge, and then dividing by the total charge capacity of thecell Q.

z1,k+1 = z1,k −i1,kQ

∆T (7)

z2,k+1 = z2,k −i2,kQ

∆T (8)

The currents i1, i2 are determined by the configuration of theswitches and exogenous current demand on the battery packi0. The currents are given by Kirchoff’s current law, wherethe switching signals q1 and q2 equal zero and one when thecorresponding switch is respectively open or closed:

i1,k = q1,k(1− q2,k)i0,k (9)

+OCV (z1,k)−OCV (z2,k) + i0,kRint(z2,k)

Rint(z1,k) +Rint(z2,k)q1,kq2,k

i2,k = (1− q1,k)q2,ki0,k (10)

+OCV (z2,k)−OCV (z1,k) + i0,kRint(z1,k)

Rint(z1,k) +Rint(z2,k)q1,kq2,k

The first terms on the right-hand sides of (9) and (10) modelone cell connected at a time. The second terms model whenboth cells are connected. When both q1 and q2 equal zeroneither cell charges (i.e. both cells experience zero current).

The parameters OCV and Rint for the equivalent cir-cuit model are identified from experimental characteriza-tion of commercial lithium-ion cells with LiFePO4 cathodechemistries. The measured values are provided in Fig. 4. Theopen circuit voltage is determined by charging and dischargingthe cells at a C/10 rate across the entire voltage range. Thenwe average the measured terminal voltage for each SOC value.Internal resistance is determined by applying step changes incurrent and measuring the associated jump in terminal voltage,for each SOC value. This is done for both charging anddischarging, rendering internal resistance as a function of SOCand direction of current flow.

III. OPTIMAL CONTROL PROBLEM FORMULATION

The control objective is to determine the optimal switchingsequence that minimizes the total resistive film growth in the

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 12

2.5

3

3.5

4

Vol

tage

[V]

SOC

Open Circuit Voltage

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.02

0.04

0.06

0.08

0.1

SOC

Res

ista

nce

[Ohm

s]

Internal Resistance

Charge ResistanceDischarge Resistance

Fig. 4. Parameterization of equivalent circuit battery model identified fromcommercial lithium-ion cells with LiFePO4 cathode chemistries. [Top] Opencircuit voltage and [Bottom] internal resistance.

battery pack described in Section 2, given a current trajectory,i0, known a priori. We formulate this as a finite horizonconstrained optimal control problem

min(q1,q2)

J =N∑k=0

2∑j=1

δfilm(zj,k, ij,k) + gz(zk)

+αN‖zN − 0.95‖22 (11)

subject to

zk+1 = f(zk, ik) (12)ik = h(qk, i0,k) (13)z0 = zic (14)

where

(q1, q2) ∈ 0, 1 × 0, 1 (15)

gz(zk) = αz

∑i=1,2

max 0.05− zi,k, 0, zi,k − 0.98

2

+

αv

∑i=1,2

max 2.0− vi,k, 0, vi,k − 3.6

2

(16)

zk = [z1,k z2,k]T (17)ik = [i1,k i2,k]T (18)

where the function δfilm maps SOC and current to averagefilm growth rate according to the relationship depicted in Fig.2. The function gz(zk) denotes soft constraints that limit cellSOC and cell voltage to protect against over-charging andover-discharging. However, for the simulation described inthis article, these constraints never become active due to themodest charging rate employed. A terminal constraint withweighting αN is provided to ensure the battery pack chargesto the SOC corresponding to the desired final voltage. Thefunction f(zk, ik) represents the dynamic equation in (7)-(8).The function h(qk, i0,k) maps the switch position and battery

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0 50 100 150−0.5

0

0.5

1

1.5S

witc

h P

ositi

on

Cell 1Cell 2

0 50 100 1500

0.5

1

Cel

l SO

C

0 50 100 1503.1

3.2

3.3

3.4

Pac

k V

olta

ge [V

]

Time [min]

Fig. 5. Time responses for optimal charging pattern identified by DDP, givena 1C battery pack charge rate.

pack current to cell current in (9)-(10). Finally, we impose afixed initial condition zic.

To solve the optimization problem in (11)-(18), we re-express the equations as a dynamic programming problem bydefining a value function as follows [25]: Let Vk(zk) representthe minimum total film growth from discrete time k to the endof the time horizon, given that the cell SOC in the present timestep k is given by the vector zk. Then the optimization problemcan be written as the following recursive Bellman optimalityequation and boundary condition.

Vk(zk) = min(q1,q2)

∑2j=1 δfilm(zj,k, ij,k)

+gz(zk) + Vk+1(zk+1)

(19)

VN (zN−1) = min(q1,q2)

αN‖zN − 0.95‖22

(20)

The above dynamic programming problem is solved via afull enumeration algorithm. That is, we compute a family ofoptimal trajectories for a set of fixed initial conditions. Thisapproach enables us to analyze an ensemble of trajectories togain insight on how DDP minimizes total film growth.

IV. SOLUTION ANALYSIS

A. Analysis of Optimal Trajectories

To acquire insight on the optimal switching sequence forminimizing resistive film growth, we consider a constant1C (2.3 A) charge rate applied to the battery pack. Notethat while the battery pack experiences a constant currentcharge rate, the individual cells will have time-varying chargerates. Time responses for an initial SOC of 0.1 for each cellare provided in Fig. 5. Figure 6 demonstrates the optimaltrajectories for a set of initial battery cell SOC conditions.These figures indicate that the optimal switching sequencefollows a consistent pattern:

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

z1

z 2

TrajectoryInitial Condition

Fig. 6. Optimal trajectories for various initial conditions, given a 1C batterypack charge rate.

1) Leave the battery pack uncharged for as long as possible.This minimizes the duration of time over which thepack’s cells have large SOC values and, consequently,large film buildup rates.

2) Charge the cell with greater initial SOC.3) Charge the cell with less SOC until both cells approxi-

mately equalize.4) Charge both cells together, at approximately equal cur-

rent values, until the final state is reached.The key question is why does DDP identify the pattern in Steps2-4 above as the optimal switching sequence for minimizingfilm growth?

B. The Energy Storage-Film Growth Tradeoff

First, consider the result that film growth is minimized byleaving the battery pack uncharged for as long as possible.This is, film growth is minimized if battery packs are chargedonly immediately before use. This result was also found ina recent study on charge trajectory optimization for plug-inhybrid electric vehicles [22]. The reason for this result can beseen by observing that the film growth rate increases withSOC in Fig. 2. Therefore, maintaining each cell in a lowSOC reduces the overall film buildup. However, this requiresa priori knowledge of when the battery pack will be used.Moreover, if the battery is discharged sooner than expected,only a fraction of the total energy capacity is available for use.This suggests a fundamental tradeoff between electric energystorage and reducing anode-side film growth.

C. Convexity Analysis of Film Growth Rate

To answer the fundamental question of why DDP identifiesthe particular charging pattern described above, let us focus onthe switching pattern exhibited by the optimal solution whencharging does occur. Consider the film growth rate for varying

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4A

vg. F

ilm G

row

th R

ate

[ µΩ

/m2 /s

ec]

SOC

Reduced FilmGrowth

IncreasedFilm Growth

Fig. 7. Convexity analysis of spatially-averaged film growth rate for zeroapplied current.

SOC and zero current input, as portrayed in Fig. 7. For smallSOC values, δfilm is concave. Along this portion of the curve,the total film growth rate for two cells at different SOC valuesis less than the total film growth rate for two cells at the sameSOC value. However, for large SOC values δfilm is convex.This implies that the total film growth rate for two cells atdifferent SOC values is greater than the total film growth ratefor two cells at equal SOC values. If one assumes the solutionis infinitely greedy, these observations for reducing film growthcan be applied as follows:

1) In the concave region of δfilm, drive the individual SOCvalues apart.

2) In the convex region of δfilm, equalize the individualSOC values.

In other words, charge each module one-by-one through theconcave region and then charge them all simultaneously.

These results indicate that a reduction in total film growthcan be achieved by allowing individual modules to haveunequal SOC values - particularly within concave regions offilm growth. This result motivates the health reduction oppor-tunities for battery management systems if certain convexityproperties can be identified for lithium battery degradationmechanisms. Additionally, the optimal policy follows a con-sistent pattern that may be closely approximated by a heuristicfeedback control law, which leaves the battery discharged forthe maximum allowable time.

D. DDP-inspired Heuristic Control

Inspired by these results, and the convexity analysis pre-sented in Section IV-C, we examine a heuristic control schemefor minimizing film growth, depicted in Fig. 8. The advantageof a heuristic control scheme over the optimal trajectoriescomputed by DDP is that the former can be implementedin a feedback loop. Additionally, one expects the heuristicscheme to achieve nearly optimal performance, due to theconsistent pattern exhibited by the DDP solutions. The processof converting optimal trajectories into an explicit feedbackmap has been studied in model predictive control theory [27].These concepts are potentially applicable here, but a simplerless formal approach is used in this initial study. Note that

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

z1

z 2

Optimal Trajectories

q = [1 0]T

q = [0 1]T

q = [1 1]T

Fig. 8. DDP-inspired heuristic rule for charging, with optimal statetrajectories superimposed.

the switching pattern defined by the heuristic rule should notbe initiated until the last possible opportunity. In this example,each cell has a 1.8 A-h charge capacity and thus the total packcharge capacity is 3.6 A-h. Therefore charging both cells from0.1 SOC to 0.95 SOC at a 1C (1.8 A) rate requires about 100minutes. As a result, we initiate the heuristic charging scheme100 minutes prior to the final time.

The design of the heuristic control law follows two steps:First, we simulate the optimal trajectories from a family ofinitial conditions, such as shown in Fig. 6. Second, we identifyregions of the state-space corresponding to a certain switchconfiguration. For regions in which the optimal state trajecto-ries do not enter, we select a switch configuration that steersthe state toward an optimal trajectory. This step is required,because for the 1C charge rate input studied here, feasibletrajectories do not cover the entire state-space. The final resultof this procedure is depicted in Fig. 8, where several optimalstate trajectories are superimposed on the proposed heuristicrule. Note how the heuristic controller follows the generalguidelines of SOC separation and equalization in the respectiveconcave and convex regions of Fig. 7.

V. COMPARATIVE ANALYSIS AND SENSITIVITY STUDIES

To evaluate the performance of the proposed heuristiccontroller, we compare it to the optimal DDP-based and stan-dard equalization schemes (i.e. both switches closed duringcharging). We perform this study by simulating the closedloop battery pack degradation control system for a 1C (2.3A) constant current charge rate cycle. This study is performedon both the equivalent circuit model and static map of filmgrowth rate (which was used for optimization, and henceforthis referred to as the “Control” model) and the full electro-chemical model. In both cases the initial cell SOC values are0.1 each. In practice, the standard charge method is to applyconstant current to every cell in the pack until the voltage

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0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

z1

z 2

q = [1 0]T

q = [0 1]T

q = [1 1]T

StandardDDPHeuristic

Fig. 9. SOC trajectories for each control scheme, superimposed on theheuristic control map.

reaches a maximum value, then the voltage is held constantat this maximum value until the applied current reaches someminimal level. This is known as a constant current, constantvoltage (CCCV) charge cycle [9]. Here, we only investigatethe potential improvements incurred during the period whenthe cells charge up to a maximum voltage limit, correspondingto 0.95 SOC in our simulations. Subsequently, we report onsimilar results obtained for constant current discharge inputs.Finally, we analyze optimal switch patterns for alternative filmgrowth maps, created using different assumption sets or modelparameters. The latter analysis is motivated by the fact thataccurately modeling aging in lithium batteries is extremelydifficult - due to the extensive array of degradation mech-anisms and materials within lithium-ion batteries. Regardlessof these differences, the link between convexity properties andunbalanced cells remains in our studies.

A. Control Model Charge Cycle Simulation

The cell SOC trajectories for each control scheme simulatedon the Control model are provided in Fig. 9, superimposed onthe heuristic rule. Observe that the standard charging schememaintains each cell at equal SOC values as the battery packcharges. In contrast, the trajectories corresponding to DDP andthe heuristic rule follow trajectories similar to Fig. 6 and 8.Namely, both methods charge one cell at a time in the concaveregion of δfilm, and then apply charge equalization in theconvex region of δfilm. Also observe that trajectories for DDPand the heuristic controller match closely, indicating that theproposed heuristic controller closely approximates the optimalsolution for the trajectory shown here.

Time responses for the cell SOC, current, and battery packvoltage are provided in Fig. 10. Here we see that the heuristicrule is initiated approximately 50 minutes into the simula-tion, allowing 100 minutes of charging time. Figure 10(a)-(c)

0 50 100 1500

0.5

1

Cel

l SO

C

StandardDDPHeuristic

0 50 100 1503.1

3.2

3.3

3.4

Cel

l Vol

tage

[V]

0 50 100 1500

1

2

3

4

Film

Bui

ldup

[mΩ

/m2 ]

Time [min]

Cell 1

Cell 2

Cell 1Cell 2

(c)

(b)

(a)

Fig. 10. Time responses for each control scheme.

0 50 100 1500

0.5

1

1.5

2

2.5

3

3.5

Film

Bui

ldup

[m Ω

/m2 ]

Time [min]

Full Model | StandardFull Model | DDPFull Model | HeuristicControl Model | StandardControl Model | DDPControl Model | Heuristic

Fig. 11. Film Buildup for each control scheme, simulated on the controlmodel and full electrochemical model.

further demonstrate how closely the heuristic controller andDDP solution match, with respect to time. Since the standardmethod initiates charging immediately, the cells remain idleat 0.95 SOC once charging is complete. This is importantbecause film builds up at a faster rate for high SOC relative tolow SOC, which is the intuitive reason why delayed chargingsignificantly reduces total film buildup. The impact of thisproperty can be seen in Fig. 10(c). Figure 10(b) demonstrateseach cell’s voltage, which increases only when that particularcell is charging. Note that all schemes maintain the cell voltagewithin the safety range of 2.0V to 3.6V.

B. Film Buildup Validation on Full Electrochemical Model

To this point, all simulation results have been performed ona reduced equivalent circuit model and static film growth ratemap in Fig. 2 used for control optimization. In this subsection

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we study (1) if optimal switching indeed reduces film buildupfor a high-fidelity electrochemical battery model, and (2) ifthe static approximation of film growth reasonably matchesthe film model prediction. Towards this goal, we apply allthree controllers (standard, DDP, and heuristic) on the fullelectrochemical model (Full).

The aggregated film buildup for the Control and Fullmodels, simulated using each control scheme, are providedin Fig. 11. This figure indicates that the DDP and heuristiccontrol schemes indeed reduce film buildup on the full elec-trochemical model, despite being synthesized for the Controlmodel. Specifically, the open-loop DDP control and closed-loop heuristic controller reduced buildup by 49.5% and 48.7%,respectively. Moreover, the total film growth predicted by theControl model differ from the Full model by less than 10% forall control schemes. Therefore, we conclude that the reducedorder model, taking the form of a static nonlinear map, enablesthe accurate minimization of film growth for the charge cyclesstudied here.

C. Performance Results

A comparison of the performance for each control schemeis provided in Table I. For the 2.3A rate charge cycle studied inthis article, the heuristic controller produces an additional 20µΩ/m2 (0.8%) of resistive film buildup over the DDP solutionon the full electrochemical model. Hence, the heuristic schemeexhibits nearly identical performance to the optimal controldesign. Both DDP and the heuristic controller reduce filmbuildup by about 50%, for this charge cycle. It is importantto note that the reduction in film buildup is a function of theparticular charge cycle and time horizon. That is, cycles thatremain within the concave region of δfilm may experiencegreater improvement, because the switched scheme proposedin this article has the greatest advantage in this domain.Moreover, the bulk of film reduction occurs due to delayingthe charging process to the end of the time horizon. For theexample studied here, 48% of film buildup reduction is due todelaying charging until the final 100 minutes.

D. Optimal Trajectories for Discharge

Throughout this article, we have consider optimally con-necting battery cells in parallel with relay switches to minimizetotal film growth - under charging events only. Here weconsider constant current discharge events. The problem isformulated exactly as before, except now we apply a 2.3Adischarge current input. The optimal switch, SOC, and voltagetrajectories are provided in Fig. 12, with the battery packinitialized at 95% SOC for each cell. Optimal SOC trajectoriesfor various initial conditions are provided in Fig. 13. Undera discharging scenario, these results indicate the optimalconstant current discharging trajectories follow a consistentpattern.

1) Discharge the pack immediately. This moves the systemaway from regions of fast film growth - so less interfacialfilm accumulates over time.

2) Equalize both cells as they discharge.

0 50 100 150−0.5

0

0.5

1

1.5

Sw

itch

Pos

ition

(a)

Cell 1Cell 2

0 50 100 1500

0.5

1

Cel

l SO

C

(b)

0 50 100 1502.8

3

3.2

3.4

Pac

k V

olta

ge [V

]

Time [min]

(c)

Fig. 12. Time responses for optimal discharging pattern identified by DDP,given a 1C battery pack discharge rate.

3) Continue to discharge both cells at equal charge levels,until a certain point.

4) Discharge each cell individually until the battery packis fully discharged.

In essence, these discharge trajectories follow the optimalcharge trajectories backwards. Moreover, the breakpoint be-tween charge equalization and unequalization is approximatelythe same - 60% for both cells. Convexity arguments for in-finitely greedy trajectory optimization solutions can be applied,once again, to interpret these results. Hence, allowing unequalcharge levels in battery management systems may providelong-term health benefits when concavity properties exist inthe aging mechanics.

E. Sensitivity to Alternative Film Growth Parameterizations

Anode-side film growth has been recognized as a significantcontributor to lithium-ion battery health degradation [12].However a plethora of other difficult-to-model aging mech-anisms can contribute to capacity and power fade. Moreover,modeling and accurately parameterizing these models across awide range of lithium-ion cell chemistries and manufacturerscan be difficult. This motivates the sensitivity analysis pre-sented here. Specifically, we consider alternative film growthmaps to evaluate the generality of unbalanced charging tovarying model assumptions and parameterizations.

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TABLE ICONTROLLER PERFORMANCE COMPARISON ON CONTROL MODEL AND FULL ELECTROCHEMICAL MODEL.

Control Model Full Model

Control Scheme Resistance of TotalFilm Buildup

Reduction inFilm Buildup

Resistance of TotalFilm Buildup

Reduction inFilm Buildup

Control vs.Full Error

Standard 3.20 mΩ/m2 0% 2.95 mΩ/m2 0% 8.47%DDP 1.55 mΩ/m2 51.8% 1.49 mΩ/m2 49.5% 4.03%Heuristic 1.56 mΩ/m2 51.2% 1.51 mΩ/m2 48.7% 3.31%

−10−5

05

100

0.20.4

0.60.8

1

0

0.1

0.2

0.3

0.4

0.5

0.6

SOCCurrent [A]

(a)

Ave

rage

Film

Gro

wth

Rat

e [ µ

Ω/m

2 /sec

]

00.2

0.40.6

0.81

−5

0

50

0.01

0.02

0.03

0.04

0.05

SOCCurrent [A]

(b)

Ave

rage

Film

Gro

wth

Rat

e [ µ

Ω/m

2 /sec

]

Fig. 14. Film growth maps for alternative electrochemical model parameterizations: (a) No film growth occurs during discharge or rest conditions, whichfollows Assumption 2 of [9]; (b) Preliminary parameterization to match the manufacturer’s cycling and storage performance data.

0 50 100 150−0.5

0

0.5

1

1.5

Sw

itch

Pos

ition

(a)

Cell 1Cell 2

0 50 100 1500

0.5

1

Cel

l SO

C

0 50 100 1503

3.5

4

Pac

k V

olta

ge [V

]

Time [min]

0 50 100 150−0.5

0

0.5

1

1.5

Sw

itch

Pos

ition

(b)

Cell 1Cell 2

0 50 100 1500

0.5

1

Cel

l SO

C

0 50 100 1503.2

3.3

3.4

3.5

Pac

k V

olta

ge [V

]

Time [min]

Fig. 15. Time responses of optimal charging trajectories for the alternative film growth maps. (a) Response for map in Fig. 14(a). This map suggests chargingthe battery pack one cell at a time. (b) Response for map in Fig. 14(b). This map suggests charge equalization is optimal.

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0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

z1

z 2

TrajectoryInitial Condition

Fig. 13. Optimal trajectories for various initial conditions, given a 1C batterypack discharge rate.

The first map, shown in Fig. 14(a), is equivalent to Fig.2 except we enforce Assumption 2 of [9] which states thatfilm growth occurs only during charging events, i.e. zerogrowth during rest and discharge conditions. The optimalcharge trajectories for a 1C constant current rate are shown inFig. 15(a). In this case the optimal solution charges each cellindividually and in succession. This result can be understoodby noting that one cell charged at 1C and the other at rest (nogrowth) produces less total film growth than two cells chargedsimultaneously at 0.5C. Also note that although Fig. 15(a)shows a delayed charging strategy, delaying charging providesequivalent film growth as immediate charging since zerofilm growth occurs during rest. Therefore when Assumption2 of [9] holds true, unbalanced charging provides a 53%reduction in total film growth, which is a greater reductionfrom unbalanced charging when using the original film growthmap.

The second map, shown in Fig. 14(b), is based upon thesame model equations used for Fig. 2 but with an alterna-tive parameter set. This parameter set has been identified toproduce capacity fade trends that match the manufacturer’scycling and storage data [22]. The two parameter sets usedfor each map are provided in Appendix B. The optimal chargetrajectories for a 1C constant current rate are shown in Fig.15(b). Unlike the previous two cases the optimal solutiondoes not unbalance the cells’ charge levels. This result can beinterpreted through the convexity arguments of Section IV-Cby observing that Fig. 15(b) contains no concave regions. Thuscharge balancing minimizes film growth. However, delayedcharging still provides benefits by maximizing the time spentto low SOC levels, where film growth is slow.

VI. CONCLUSIONS

This research investigates battery health management inlithium ion battery packs using relay switches for modules

connected in parallel. To facilitate control design and analy-sis, we consider an electrochemical battery cell model withirreversible solvent reduction reaction dynamics at the anode,developed by Ramadass et. al. [9]. From this high fidelitymodel, we approximate film growth rate as a static mapthat functionally depends on cell SOC and applied current.Using this map, we formulate an optimal control problem tominimize total battery pack film growth for a constant currentcharge trajectory. Inspired by the optimal trajectories, and theconvexity properties of the film growth map, we design aheuristic rule base that produces nearly optimal performance.Further optimization results for constant current dischargetrajectories and alternative film growth models demonstrate thegenerality of charge unequalization to varying input profiles,model assumptions, and parameterizations.

The key result demonstrated by this work is that healthdegradation due to film growth can be reduced by: (1) Al-lowing battery modules connected in parallel to attain unequalSOC values when concavity features exist; and (2) Delayingcharging until immediately before discharging. Indeed, theoptimal solution approximately separates SOC in the concaveregion and equalizes SOC in the convex region of filmgrowth rate at the end of the time horizon. This process canbe implemented using a heuristic static feedback controllerdesigned from optimal trajectories computed via dynamicprogramming. Individual control of module SOC is achievedvia relay switches typically used for safety precautions. Withineach module, individual cell SOC may be equalized viatraditional switched capacitor circuits [5], [6] to protect againstover-charging or over-discharging. Simulation results indicatethis approach may significantly reduce total battery pack filmgrowth, if one can identify concavity features in the degrada-tion performance map. This motivates future work focusedin two directions. First, experimentally identifying a data-driven degradation map similar to Fig. 2 may enable significantimprovements in lithium ion battery lifetime through chargeunbalancing schemes. Second, experimental verification ofthese algorithms designed from data-driven degradation mod-els will provide the ultimate proof-of-concept.

APPENDIX ANOMENCLATURE

Symbol Description UnitsF Faraday’s constant [C/mol]i0 Battery pack current [A]

i0,sExchange current density [A/m2]for side reaction

i1, i2 Cell current [A]hpenalty Quadratic penalty function [pm/m2]J Cost functional [pm/m2]

J1Intercalation current between [A/m3]solid and solution

Jp Terminal state penalty function [pm/m2]Jtot Total intercalation current [A/m3]Js Side rxn. current density [A/m3]

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Symbol Description Units

MPMolecular weight of product [mol/kg]from side reaction

Q Battery cell charge capacity [A·h]q1, q2 Contactor switch positionR Universal gas constant [J/K/mol]

RfilmTotal film resistance at [Ω·m2]electrode/electrolyte interface

Rint Battery cell internal resistance [Ω]

RSEIResistance of solid electrolyte [Ω/m2]interphase (SEI)

Us,refEquilibrium potential of [V]side reaction

V Value function [pm/m2]v Battery pack voltage [V]x Spatial coordinate [m/m]z Battery cell state of charge [C/C]

∆φ Local potential difference b/w [V]solid and solution at anodeδfilm Resistive film thickness [pm/m2]ηs Overpotential driving side rxn. [V]κP Side rxn. product conductivity [1/m/Ω]ρP Side rxn. product density [kg/m2]

APPENDIX BFILM GROWTH MODEL PARAMETERS

Values for map depicted inSymbol Fig. 2 & 14(a) Fig. 15(b)i0,s 1.5× 10−6 A/m2 4× 10−8 A/m2

MP 73000 mol/kg 73000 mol/kgRSEI 7.4 mΩ·m2 7.4 mΩ·m2

Us,ref 0.4 V 0.4 VκP 1 (m·Ω)−1 1 (m·Ω)−1

ρP 2100 kg/m2 2100 kg/m2

Uref,n(θn) Adopted from [9] Adopted from [22]

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[1] B. K. Sovacool and R. F. Hirsh, “Beyond batteries: An examination ofthe benefits and barriers to plug-in hybrid electric vehicles (PHEVs) anda vehicle-to-grid (V2G) transition,” Energy Policy, vol. 37, no. 3, pp.1095–1103, 3 2009.

[2] M. Alamgir and A. M. Sastry, “Efficient batteries for transportationapplications,” SAE Paper 2008-21-0017, 2008.

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[5] J. W. Kimball, B. T. Kuhn, and P. T. Krein, “Increased performanceof battery packs by active equalization,” 2007 IEEE Vehicle Power andPropulsion Conference, Sept. 9-12 2007 2007.

[6] A. Baughman and M. Ferdowsi, “Double-tiered switched-capacitorbattery charge equalization technique,” IEEE Trans. on Industrial Elec-tronics, vol. 55, no. 6, pp. 2277–85, 2008.

[7] M. Doyle, T. Fuller, and J. Newman, “Modeling of galvanostatic chargeand discharge of the lithium/polymer/insertion cell,” Journal of theElectrochemical Society, vol. 140, no. 6, pp. 1526 – 33, 1993.

[8] T. Fuller, M. Doyle, and J. Newman, “Simulation and optimization of thedual lithium ion insertion cell,” Journal of the Electrochemical Society,vol. 141, no. 1, pp. 1 – 10, 1994.

[9] P. Ramadass, B. Haran, P. Gomadam, R. White, and B. Popov, “Devel-opment of first principles capacity fade model for Li-ion cells,” Journalof the Electrochemical Society, vol. 151, no. 2, pp. 196 – 203, 2004.

[10] L. Kanevskii and V. Dubasova, “Degradation of lithium-ion batteries andhow to fight it: A review,” Russian Journal of Electrochemistry, vol. 41,no. 1, pp. 3 – 19, 2005.

[11] P. Arora, R. E. White, and M. Doyle, “Capacity fade mechanisms andside reactions in lithium-ion batteries,” Journal of the ElectrochemicalSociety, vol. 145, no. 10, pp. 3647 – 3667, 1998.

[12] D. Aurbach, “Review of selected electrode-solution interactions whichdetermine the performance of Li and Li ion batteries,” Journal of PowerSources, vol. 89, no. 2, pp. 206 – 18, 2000.

[13] E. Gatzke, A. Stamps, C. Holland, and R. White, “Analysis of capacityfade in a lithium ion battery,” Journal of Power Sources, vol. 150, pp.229–39, 10/04 2005.

[14] M. Verbrugge and E. Tate, “Adaptive state of charge algorithm fornickel metal hydride batteries including hysteresis phenomena,” Journalof Power Sources, vol. 126, no. 1-2, pp. 236–249, 2004.

[15] G. L. Plett, “Extended Kalman filtering for battery management systemsof LiPB-based HEV battery packs. Part 1. Background,” Journal ofPower Sources, vol. 134, no. 2, pp. 252–61, 08/12 2004.

[16] G. Plett, “Extended Kalman filtering for battery management systemsof LiPB-based HEV battery packs. Part 2. Modeling and identification,”Journal of Power Sources, vol. 134, no. 2, pp. 262 – 76, 2004.

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[20] S. J. Moura, J. L. Stein, and H. K. Fathy, “Control of film growth inlithium ion battery packs via switches,” ASME Dynamic Systems andControl Conference, Oct. 2009.

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Scott J. Moura (S’09) received the B.S. degree fromthe University of California, Berkeley, in 2006 andthe M.S. degree from the University of Michigan,Ann Arbor, in 2008, both in mechanical engineering,where he is currently pursuing the Ph.D. degree.

His research interests include optimal control,energy conversion systems, and batteries.

Mr. Moura is a recipient of the National ScienceFoundation Graduate Research Fellowship, Univer-sity of Michigan Rackham Merit Fellowship, andCollege of Engineering Distinguished Leadership

Award. He has also been nominated for the Best Student Paper Award atthe 2009 ASME Dynamic Systems and Control Conference.

Joel C. Forman has received B.S. degrees inboth Mechanical Engineering and Applied Mathe-matics from the Rochester Institute of Technology(2007)and an M.S. degree in Mechanical Engineer-ing from the University of Michigan, Ann Arbor in(2010) where he is currently pursuing a Ph.D. de-gree. His research interests include model reduction,parameter identification, and batteries.

Saeid Bashash (M’09) is currently a postdoctoralresearch fellow at the University of Michigan, AnnArbor. He received his B.Sc. degree from SharifUniversity of Technology, Tehran, in 2004, andM.Sc. and Ph.D. degrees from Clemson University,South Carolina, in 2005 and 2008, respectively, allin Mechanical Engineering.

His Ph.D. research focused on the modeling andcontrol of piezoactive micro and nano systems,such as piezoelectric actuators and scanning probemicroscopy systems. He is currently working on

electrochemical battery systems modeling and simulation, vehicle-to-gridintegration, and demand-side energy management. His main research interestsinclude modeling, optimization, and control of complex dynamical systems.

Jeff L. Stein is a Full Professor with the Depart-ment of Mechanical Engineering, The University ofMichigan, Ann Arbor, and is the former Chair of theDynamic Systems and Control Division of ASME.His discipline expertise is in the use of computerbased modeling and simulation tools for systemdesign and control. He has particular interest inalgorithms for automating the development of properdynamic mathematical models (minimum yet suffi-cient complexity models with physical parameters).His application expertise is the areas of automotive

engineering including green energy transportation, machine tool design andlower leg prosthetics. In the area of green transportation he is working todevelop the techniques for the design and control of plug-in hybrid vehiclesand smart grid technologies that creates a more efficient use of energy,particular that generated from renewable resources, for a lower carbon andother emissions footprint. He has authored or coauthored over 150 articles injournals and conference proceedings.

Hosam K. Fathy received his B.Sc. (Summa cumLaude), M.S., and Ph.D. degrees - all in Mechani-cal Engineering - from the American University inCairo, Egypt (1997), Kansas State University (1999),and The University of Michigan, Ann Arbor (2003),respectively.

He has worked as a Control Systems Engineerat Emmeskay, Inc. (2003-2004), a postdoctoral Re-search Fellow at The University of Michigan (2004-2006), and an Assistant Research Scientist at TheUniversity of Michigan (2006-2010), where he won

the 2009 Michigan College of Engineering Outstanding Research ScientistAward. He is now an Assistant Professor in the Mechanical and NuclearEngineering Department at the Pennsylvania State University. He is thefounder and director of the Control Optimization Laboratory, which currentlyemploys graduate students and postdocs at both Michigan and Penn State.His research interests including optimal control and model reduction theoriesand their application to hybrid vehicles, the smart grid, battery modeling andcontrol, and networked hardware-in-the-loop simulation.