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0278-0046 (c) 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TIE.2016.2523440, IEEETransactions on Industrial Electronics

IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS 1

Energy-Conscious Warm-Up of Li-ion Cells FromSub-Zero Temperatures

Shankar Mohan, Youngki Kim, Member, IEEE, and Anna G. Stefanopoulou, Fellow, IEEE

Abstract—Lithium (Li) ion battery cells suffer from significantperformance degradation at sub-zero temperatures. This paperpresents a Predictive Control based technique that exploitsthe increased internal resistance of Li-ion cells at sub-zerotemperatures to increase the cell’s temperature until the desiredpower can be delivered. Specifically, the magnitude of a sequenceof bidirectional currents is optimized such as to minimize totalenergy discharged. The magnitude of current is determinedby solving an optimization problem that satisfies the batterymanufacturer’s voltage and current constraints. Drawing bi-directional currents necessitates that a temporary energy reser-voir for energy shuttling, such as an ultra-capacitor or anotherbattery, be available. When compared with the case when nopenalty on energy withdrawn is imposed, simulations indicatethat reductions of up to 20% in energy dispensed as heat in thebattery as well as in the size of external storage elements can beachieved at the expense of longer warm-up operation time.

Index Terms—Lithium-ion batteries, battery management, low-temperature operation, warm-up, model predictive control, Hy-brid Electric Vehicle

I. INTRODUCTION

THE use of Li-ion battery technologies has recently enjoyedwidespread adoption in consumer electronics and automo-

tive/aerospace applications [1]. The archetype of rechargeabletechnology, Li-ion batteries, has over the last decade benefitedfrom improvements in material science through increasedenergy and power density [2]. Although widely adopted, thesebatteries suffer from significant performance degradation at lowtemperatures (≤ −10oC) posing a challenge for automotiveapplications (refer [3] and [4] for electrochemistry basedreasons).

Although improvement to sub-zero performance throughchanges in design and construction of cells have been pursued,the need for fast warm-up is relevant for existing equipments.Battery warm-up techniques can be broadly classified as – (1)jacket/resistive/external heat-up; (2) internal heating using high-frequency bidirectional currents. In [5], various methods to useconvective heating of cells/packs using air and phase change

Manuscript received October 12, 2014; revised March 16, 2015 and July 1,2015; accepted September 9, 2015.

Copyright (c) 2016 IEEE. Personal use of this material is permitted. However,permission to use this material for any other purposes must be obtained fromthe IEEE by sending a request to [email protected].

This work was supported by the Automotive Research Center through theU.S. Army Tank Automotive Research, Development and Engineering Center,Warren, MI, USA, under Grant W56HZV-04-2-0001.

S. Mohan is with the Department of Electrical Engineering and ComputerScience, University of Michigan, Ann Arbor, MI 48109 USA (e-mail:[email protected]).

Y. Kim and A. G. Stefanopoulou are with the Department of MechanicalEngineering, University of Michigan, Ann Arbor, MI 48109 USA (e-mail:[email protected]; [email protected]).

materials is surveyed. More recently, in [6], the authors studythe use of phase change slurry to externally warm the pack andin [7] the authors use power electronics to place air heatersin parallel (power) to the motors in Electric Vehicles (EVs) towarm the battery pack. Pesaran et.al. in [8] and Stuart et.al. in[9] studied the use of bi-directional sinusoidal high frequencycurrent to increase the cell’s temperature through internalheating. In Ji et al. in [10] compare different heating strategiesand conclude that for Li-ion cells, internal heating is moreeffective than using external heating elements if no externalpower source is utilized, a scenario we term as standalone andis of consideration in this paper.

Most techniques discussed in literature strive to warm the celluntil a certain pre-specified cell temperature is reached. Sincein most applications, the cell serves as a source of power, weuse the cell’s pulse power capability, instead of the temperature,as a condition to terminate the warm-up operation. In addition,we seek to investigate the feasibility of reaching the necessarypower capability in an energy efficient manner.

Pulse power capability or state-of-power (SOP) is anestimated quantity whose accuracy is determined by the fidelityof the model that captures the electrical dynamics of thecell [11]. Modeling the electrical behavior of Li-ion cellsat sub-zero temperatures, particularly at high current rates,is more challenging than emulating its thermal dynamics[4]. Thus, owing to the inherent relation between operatingtemperature and power capability, in this paper, temperature riseis taken as a measurable surrogate. Then, the stated objectiveof increasing power capability can be re-written as one ofeffecting temperature rise in an energy conscious manner untilthe desired power can be delivered.

Maximizing temperature rise while regulating energy lossprovides for certain desirable characteristics of the batterycurrent. Heat generated being proportional to the input current,it follows that the candidate current profile be bi-directionalto minimize cumulative discharge and achieve fast warm-up.Drawing bi-directional currents necessitates that a temporaryenergy reservoir for energy shuttling, such as an ultra-capacitoror another battery, be available. Since the bi-directional currentincludes a charging phase, it is important to note that chargingthe cell at low temperatures is challenging and imposes stringentcharging current constraints (see [12], [13] for challenges atroom temperature).

Charging Li-ion cells at sub-zero temperatures is difficultbecause of the reduced diffusivity in the anode that results inincreased polarization and a drop in electrode overpotential[14], [15]. From a control perspective, the propensity ofcharging currents to cause plating can be minimized by actively




regulating the electrode overpotential. Pulsed charging is one ofthe most widely adopted technique to slow down polarizationand allow for more even ion distribution [16]. In this paper, inaddition to using bi-directional pulses, anode polarization isindirectly controlled by enforcing the magnitude of chargingcurrents to be less than the discharging portion of the pulse.

This paper attempts to study the feasibility of using com-putationally efficient models to improve the power capabilityof Li-ion cells in an energy efficient manner. This paper isorganized as follows. The models that are used to mimic thecell’s electrical and thermal behavior are detailed in Section IIand their parametrization is discussed in Section III. The controlproblem is formulated in Section IV and an example simulationis studied in Section V. Conclusions and final remarks are madein Section VI.

II. MODELING

This section introduces the models of electrical and thermaldynamics adopted in this study. The dynamic behavior ofa cylindrical (26650) LFP cell is captured using simplereduced order models. The validity of the chosen models forthe application at hand is ascertained through experimentalvalidation.

A. Electrical Model

Over the decades, much effort has been expended indeveloping phenomenological models of the electrical dynamics.The more complex models are based on concentration theory,first proposed by Doyle, Fuller and Newman in [17]. Modelsso derived are hard to parameterize [18], have notable memoryrequirements and, are computationally intensive. On the otherhand, equivalent circuit models have been widely adopted inliterature and in practice, eg. [19] and references therein.

Small signal and local approximations of the dynamicbehavior of electrochemical studies can be obtained by usingimpedance measurements [20]. Results of the impedancespectroscopy study conducted in [21] suggest that at lowoperating temperatures, for high frequencies of current, the Li-ion cell’s electrical dynamics exhibits a first order characteristic.Thus, in this paper, an equivalent circuit model whose dynamicsis governed by Eqn. (1) is utilized to capture the electricaldynamics of the Li-ion cell. Note that the system Eqn. (1)describes is one of a Linear Parameter Varying system whereinthe parameters are scheduled based on the state of charge(SOC), z, and the cell temperature T

[z

V1

]︸︷︷︸xel

=

0 0

0 − 1

R1(·)C1(·)

︸︷︷︸

Ael

[zV1

]+

−1

3600 · Cb(T )

− 1

C1(·)

︸︷︷︸

Bel

vel,

Vt = VOCV (z, T )− V1 −Rs(·)vel. (1)

where vel = I (sign convention – charge : negative; discharge: positive), Cb is the temperature dependent capacity of thecell; Vt is the terminal voltage of the cell; VOCV is theOpen Circuit Voltage (OCV), a function of SOC and celltemperature; and Rs(·) is the series resistance. Figure 1 presents

+− Vocv

R sC1R1

+ −V1

Vt

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 22

2.5

3

3.5

Capacity Discharge (Ah)

Vol

tage

/V

25oC

−10oC−20oC

0oC

Fig. 1. Relation between temperature, OCV and capacity, (inset) single R-Cequivalent circuit representation of electrical dynamics

an electric equivalent of the dynamical system in Eqn (1)and the dependence of Cb on temperature. State V1 can beinterpreted as being indicative of the bulk polarization in thecell; its time constant is determined by the pair R1, C1which is assumed to be a function of SOC, cell temperatureand current direction. In the interest of notational simplicity, inthe remainder of the paper, the dependence of model parameterson dynamic states and input is not explicitly stated when thereis little room for confusion.

The power capability of a cell is defined as the product ofthe maximum continuous current that can be drawn over afixed time interval without violating current and or voltageconstraints. In this paper, estimates of power capability fora pulse duration of N samples are computed in discrete-timeusing expressions provided in [22]. In discrete-time domain,denoting the linearized system matrices of the electrical modelas Adel, B

del, C

del, D

del,

Pcap,k = Vmin

Vmin − VOCV (zk) + Cd

elLzk − Cdel(A

del)

Nxel,k

CdelM

del +Dd

el

,

(2)

where Vmin is the minimum permissable terminal voltage,L = [1, 0, 0]′, M =

∑N−1i=0 (Adel)

jBdel and N is the numberof samples in the constant discharge pulse.

B. Thermal Model

The thermal model of a cylindrical battery developed in [23]is taken to represent the thermal dynamics in this study. Themodel of the thermal dynamics when expressed in terms ofthe core (Tc), surface (Ts), ambient (Tamb) temperatures andrate of heat generation (q) is represented as

xth =Athxth +Bthvth,

yth = Cthxth +Dthvth, (3)

TABLE ITHERMAL MODEL PARAMETERS

Parameter Symbol Value UnitDensity ρ 2047 kg/m3

Specific heat coeff. cp 1109 J/kgKThermal conductivity kt 0.610 W/mK

Radius r 12.9×10-3 mHeight L 65.15× 10-3 mVolume vb 3.421× 10-5 m3




where the states represent temperature gradient across theradius (γ) and average temperature (T ); xth = [T γ]T ,vth = [q Tamb]

T and yth = [Tc Ts]T . System matrices Ath,

Bth, Cth, and Dth are defined as follows:

Ath =

[ −48αhr(24kth+rh)

−15αh24kth+rh

−320αhr2(24kth+rh)

−120α(4kth+rh)r2(24kth+rh)

],

Bth =

[α

kthVb

48αhr(24kth+rh)

0 320αhr2(24kth+rh)

],

Cth =

[24kth−3rh24kth+rh

− 120rkth+15r2h8(24kth+rh)

24kth24kth+rh

15rkth48kth+2rh

],

Dth =

[0 4rh

24kth+rh

0 rh24kth+rh

], (4)

where kth, h and ρ are the thermal conductivity, convectioncoefficient and bulk density, α, the thermal diffusivity is definedas the ratio of kth to the heat capacity, cp. These parameters areassumed to independent of the cell and ambient temperatures.

The bulk of heat generation in electrochemical cells can beattributed to three components – Joule, entropic and heatingdue to polarization. Since the current in this application isbidirectional and is large in magnitude, Joule heating dominatesentropic heating. Further, the heat generated by polarization isaffected by the time constant of the R-C pair and the voltageacross them.

q =V 21

R1+ I2Rs (5)

III. MODEL PARAMETRIZATION & VALIDATION

The parameters of the thermal model, thermal properties ofthe cell and the environment, are not significantly influenced bytemperature variations. This affords us the option of adoptingvalues presented in [23] (reproduced in Table I) without change.However, a similar argument cannot be made for the electricalmodel.

Modeling the electrical dynamics of Li-ion cells as a linearparameter varying system has been extensively pursued inliterature eg. [19] and references therein. In this paper, the

−20 0 20 40

0.02

0.04

0.06

Rsch

g /Ω

−20 0 20 40

30

40

50

τ 1chg /s

−20 0 20 40

1000

1200

1400

1600

τ 2chg /s

−20 0 20 40

0.02

0.04

0.06

0.08

Rsdi

schg

/Ω

Temperature /oC

−20 0 20 40

30

40

50

τ 1disc

hg /s

Temperature /oC

−20 0 20 40

1000

1500

2000

2500

τ 2disc

hg /s

Temperature /oC

0.13 0.33 0.53 0.73 0.93

Fig. 2. Estimated SOC and temperature dependent parameters at differentSOCs during charge (chg) discharge (dischg)

0 20 40 60 80 100 120 140 160 180

0.8

0.85

Time /s

SO

C

50 50.5 51 51.5 52 52.5 53 53.5 54 54.52.62.8

33.23.4

Time /s

Ter

min

al V

olta

ge /V

0 20 40 60 80 100 120 140 160 180

−0.2

0

0.2

Time /s

Est

imat

ion

erro

r /V

MeasuredEstimated

50 50.5 51 51.5 52 52.5 53 53.5 54 54.5−10

−5

0

5

Time /s

Cur

rent

/A

0 20 40 60 80 100 120 140 160 180

0.8

0.85

Time /s

SO

C

50 50.5 51 51.5 52 52.5 53 53.5 54 54.52.62.8

33.23.4

Time /s

Ter

min

al V

olta

ge /V

0 20 40 60 80 100 120 140 160 180

−0.2

0

0.2

Time /s

Est

imat

ion

erro

r /V

MeasuredEstimated

0 20 40 60 80 100 120 140 160 180

0.8

0.85

Time /s

SO

C

50 50.5 51 51.5 52 52.5 53 53.5 54 54.52.62.8

33.23.4

Time /s

Ter

min

al V

olta

ge /V

0 20 40 60 80 100 120 140 160 180

−0.2

0

0.2

Time /s

Est

imat

ion

erro

r /V

MeasuredEstimated

0 20 40 60 80 100 120 140 160 180

−0.2

−0.1

0

V1

Time /s

Figure 1: Model validation – Predicting Terminal Voltage

1

Fig. 3. Model validation – Predicting Terminal Voltage

standard method utilized to parameterize equivalent circuitmodels and which is described in [19] is extended to sub-zerotemperatures.

Figure 2 presents some of the key characteristics of the rep-resentative sub-model; each line in every subplot correspondsto the trajectory of the variable as a temperature changes for aparticular SOC.

Based on the estimated values for model parameters, forlarge currents, it can be shown that the heat generated can beapproximated by Joule heating. Hence in the remainder of thepaper, the generated heat is computed as

q = I2Rs. (6)

To validate the models described in the sections afore,a 26650 LFP cell was instrumented with a thermocouplein its center cavity and placed in a Cincinnati Sub-ZeroZPHS16-3.5-SCT/AC temperature controlled chamber. Thechamber temperature was set to -20 C and the air-flow wasregulated to mimic natural convection (h = 5 W/m2K). Thiscell was excited with square current pulse-train provided by aBitrode FTV1-200/50/2-60 cycler. Each pulse in current wasset to have a duty-cycle of 50% and the magnitude of chargingand discharging currents were set at five and 10 amperes. Thefrequency of pulse-train was set to 1 Hz and measurements ofterminal voltage, current, surface and core temperature werecollected at the rate of 100 Hz. The measured current was fed toboth the electrical (single R-C model) and thermal models andthe estimated terminal voltage, surface and core temperaturesare plotted in Figs. 3 – 4.




0 20 40 60 80 100 120 140 160 180

−18

−16

−14

Time

Sur

face

Tem

pera

ture

0 20 40 60 80 100 120 140 160 180

−18

−16

−14

−12

−10

Time

Cor

e T

empe

ratu

re

SimulatedMeasured

Fig. 4. Model Validation – Predicting Surface and Core temperatures

From Fig. 3 it is noted that the root mean squared (rms) errorin estimating the terminal voltage is less than 50 mV. Muchof the large errors in estimation of terminal voltage is incidentwith changes in current direction. The most likely reason isthat while the model is able to capture the steady state values,it has deficiencies in capturing the very fast transients. Therelatively slower transients are captured by the R-C pair in themodel and the first subplot in Figure 3 traces the trajectory ofthe estimated bulk polarization. Observe that the polarizationvoltage is at-times almost 10% of the total voltage swing acrossthe entire SOC range and can can significantly the measuredterminal voltage.

In this work, we are interested in warming the cell. Sincemost of the heat is generated through Joule heating and giventhat the parameterized model is able to capture the steady-state voltage fairly accurately, the developed model is assumedadequate and is used in the remainder of the paper.

Figure 4 presents the outcome of simulating the thermalmodel. The input to the thermal model, namely Joule heating,was computed using the electrical model parameters and states.Upon inspection, it is possible to conclude that the thermalmodel is able to predict the surface and core temperatures towithin the accuracy of the T-junction thermocouples, 0.5C,for the critical range of cold conditions

IV. AUTOMATED OPTIMAL WARM-UP FORMULATION

The primary focus of this work is on warming the cell inan energy efficient manner until the desired power can bedrawn from the cell. To this end, based on electrochemicalconsiderations, the profile of input current is chosen asa sequence of bi-directional pulses recurring at a certainfrequency. To keep the problem formulation simple, each periodis stipulated to have just one sign change in current as shownin Figure 5. To completely characterize the current profile,one would require four control variables – frequency, duty-cycle, peaks of charge and discharge pulses. The frequency ofthe pulse train influences the rate of heat generation – fromEIS tests, increasing frequencies decreases the effective series

resistance while decreasing the reactive component of thetotal impedance [24]. In this study, the optimal frequency atwhich the resistance is large yet the reactive component issmall is assumed to be known. Since the frequency is pre-determined, the values of the remaining variables – duty-cycleand magnitudes, need to be determined.

The dynamic behavior of the electrical and thermal sub-systems of the cell are functions of its operating conditionsand internal states. Specifically, the optimal decision at thekth instance is influenced by the trajectory of states until then.As the model dynamics is affected by the value of its states,the problem of deciding the values of the control variables isformulated as a linearized receding finite horizon optimizationproblem and described in this section.

The objective of the problem under consideration is toincrease the temperature of the cell while penalizing theeffective energy discharged (measured in terms of loss inSOC) from the cell. This objective can, in the general case, bemathematically formulated as

minU,D−[Tk+1+ns·N − Tk+1] + β

N∑j=1

(uc,j · dc,j + ud,j · dd,j),

(7)

where N is the number of periods in prediction horizon, β isthe relative penalty on energy loss, ns is the number of samplesper period of the pulse; in the jth period of the horizon, dc,jand dd,j are the durations of charge and discharge portion ofthe period as multiples of sampling period ∆T , and uc,j andud,j are the the charge and discharge currents. Then,

D =dc,j , dd,j | ∀j ∈ [1, N ] ∩ Z, dc,j + dd,j ≤ ns

,

U =uc,j , ud,j | ∀j ∈ [1, N ] ∩ Z, |uc,j | ≤ |ud,j |

.

Note that Eqn. (7) is, by virtue of the fact that the second termis non-convex and that the first and second terms do not haveterms in common, non-convex. The variables over which theproblem is optimized takes a mixture of integer and continuousvalues; the problem under consideration is a non-convex MixedNonlinear Integer Programming problem (MNIP). Non-convexMNIPs are NP-hard [25] and are not suitable for online control.In the interest of making the problem more tractable, in thispaper, the duty-cycle of both charge and discharge pulses areset to be equal; i.e. 50% duty-cycle; in so doing, the problem

Current/A

kk + dd

k + dd + dc

I = ud

I = uc

0

1

Fig. 5. Pulse current profile




devolves into a regular nonlinear programming problem (NLP)that could be solved online.

Having fixed the duty-cycle to be 50%, for simplicity ofexpressions, without loss of generality it is assumed that eachperiod of the current is spread over only two samples. Amore general case is easily derived by scaling the appropriatevariables.

A. Characterizing the Current Profile

At each instant l, for a prediction horizon of length 2Nsamples, the problem of deciding the magnitude of pulses toincrease cell temperature in an energy conscious manner iscomputed by solving the following problem P1:

minu− [Tl+2N+1 − Tl+1] + β|zl+2N+1 − zl+1|

s.t. : ∀k ∈ l + 0, . . . , l + 2Nxth,k+1 = Ad

thxth,k +Bdthvth,k

yth,k = Cdthxk +Dd

thvth,k

vth,k = [u2kRs,k;Tamb,k]

(8a)

xel,k+1 = Adelxel,k +Bd

eluk

yel,k = Cdelxel,k +Dd

eluk +Gdel

(8b)

|ui| ≤ |Id(Ti)|, ∀i ∈ l + 1, l + 3, . . . , l + 2N − 1|ui| ≤ |Ic(Ti)|, ∀i ∈ l + 2, l + 4, . . . , l + 2N|ui| ≥ |ui+1|, ∀i ∈ l + 1, l + 3, . . . , l + 2N − 1

(8c)

Vt,i ≤ Vmax, ∀i ∈ l + 1, . . . , l + 2N− Vt,i ≤ −Vmin, ∀i ∈ l + 1, . . . , l + 2N

(8d)

xel,k = xel,l, xth,k = xth,l

where zk = xel,k(1), Tk = xth,k(1), Gdel =Vocv(zk−1)Cdel(1)zk−1, u = [u1, . . . , u2N ]′, and β is a relativeweight that penalizes changes in SOC. In the above, thevector of control variables, u, is arranged such that odd andeven elements correspond to discharging and charging currentmagnitudes respectively.

The cost function of P1 strikes a compromise between totalincrease in the cell’s average temperature and penalized lossin state of charge over the entire prediction horizon. Eqns. (8a)and (8b) describe the equality constraints on the temperatureand electrical model dynamics in which a superscript ‘d’indicates the discrete version of the variable. Cell manufacturerstypically specify the voltage operating limit [Vmin, Vmax], andthe maximum charge and discharge current limits as a functionof temperature; Eqns. (8c) and (8d) enforce these constraints.

For ease of implementation, the optimal control problem inEqn. (8) is re-written as an optimization problem by recursivesubstitution of the dynamics as follows. Expressing the thermaldynamics, in discrete-time as

xth,k+1 =Adthxth,k +Bdth

[qk

Tamb,k

],

with qk = u2kRs, it can be seen that,

xth,k+2N+1 − xth,k+1 =

[(Adth)2N − In]xth,k+1 +2N∑j=1

(Adth)j−1Bdth

[Rsu

2p

Tamb,p

]

ud

uc

ν2

ν3ν4

(0, 0) e1 : Min. charge current

e4 : Max. discharge current

e3 : Max. charge current

e2 : |D

ischarg

e| ≥

|Charg

e|

45

1

Fig. 6. Region of the constrained optimization problem when the predictionlength is one.

where p = 2N − j + k + 1.Then,

Tk+2N+1 − Tk+1 =[1 0

]︸︷︷︸C

[xth,k+2N+1 − xth,k+1],

= C[(Adth)2N − I]xth,k + C2N∑j=1

(Adth)j−1Bdth

[Rsu

2p

Tamb,p

],

=u′Wu+ const., (9)

where, defining ϑj := C(Adth)2N+k−jBdthC′Rs,

W = diag([ϑ1, . . . , ϑ2N ]). The constant term inEqn. (9) can be expressed as C[(Adth)2N − I]xth,k +

C∑2Nj=1(Adth)j−1BdthCTamb,2N−j+k+1 where C = [0 1]. As

constant terms in the cost are immaterial to minimizationproblems, the above constant is dropped in the followingexpressions.

Since the evolution of SOC is related to the summation ofthe control variables, the original problem in Eqn. (8) can bere-written in the following form

minu− ‖u‖2W + β

∑j

uj

subject to : Ψu ≤ Υ

|ui| ≤ |Id(T )|, ∀i ∈ 1, 3, . . . , 2N − 1|ui| ≤ |Ic(T )|, ∀i ∈ 2, 4, . . . , 2N|ui| ≥ |ui+1|, ∀i ∈ 1, 3, . . . , 2N − 1

(10)

where Ψ and Υ are as defined in Eqns. (11) and (12). Theabove optimization problem belongs to the class of problemswhere a concave function is minimized over a convex set; suchproblems have been studied extensively in literature. Solversof concave optimization problems can be broadly classified asbeing either approximate or global; global methods generallyemploy cutting-plane and or branch and bound techniques [26],[27]. In general global solvers are computationally expensiveand thus their use may be limited to small-scale problems.

To gain better insight into the nature of the optimizationproblem under investigation, consider the simple case whenthe prediction horizon is of length one. Figure 6 presents thecharacteristic shape of the constraint polytope in R2 whereincoordinates of the vertices represent, in sequence, the magnitudeof discharge and charge pulses. While edge e1 enforces thetrivial condition that charging and discharging pulses cannothave the same polarity, edge e2 ensures that the magnitude of




Algorithm 1: Control Algorithm (open-loop)

set flag=0;set [ud, uc]

′ = [−1, 1]′;set number of samples in block;while !flag do

Compute Pcap;if Pcap <= Pdmd then

Solve optimization problem;set [ud, uc]

′ = [−u∗d, u∗c ]′;wait(ts · number of samples in block) seconds

elseset flag=1;

endend† Variables with an ‘*’ superscript are optimal solutions.

the charge current is never greater than that of the dischargecurrent. Edges e3 and e4 complete the polytope and enforceadherence to voltage and current constraints.

The bounded polytope defined by constraints in the prob-lem under consideration is convex. The solution to concaveminimization problems, when restricted to a convex polytopelies, at one of the vertices of the polytope [28]. For the simplecase depicted in Fig. 6, it can be shown that the solution liesat either ν3 or ν4. As this work is a feasibility study in asimulation framework, the concave minimization problem issolved using a vertex enumeration strategy to find the globalminimizer.

B. Control scheme

In the preceding sub-section, the problem of determining themagnitude of input current of the cell was formulated as anoptimization problem in a receding horizon framework. Incor-porating the termination condition based on power capability,the overall process can be cast into the control scheme depictedin Alg. 1.

The time constant of the thermal dynamics of the cell underconsideration is in the order of tens of minutes. Thus, theincrease in temperature as a result of applying one period ofcurrent (at 10Hz) may not be significant. For this reason the

Block 1 Block 2 Block 3

Cu

rre

nt

Time

Prediction Horizon

Fig. 7. Blockwise implementation of the MPC problem

problem of current magnitude determination is solved in blocks.Periods in the prediction horizon are binned into blocks, witheach block consisting of a pre-set number of pulse periods; theprediction horizon is then described by the number of blocks(refer Fig. 7). The optimization problem as formulated earlieris modified to enforce the constraint that every period in eachblock is identical.

In the overall scheme, at each control instant, the powercapability, Pcap, is first estimated and compared to the desiredset-point, Pdmd. If the required power cannot be provided,the optimization problem to compute the magnitudes of thepulses is solved and the optimal solution to the first block isapplied. After waiting a duration that is equal to the durationof the block, the process is repeated and the power capabilityis re-computed. Once the desired power can be delivered, thewarm-up operation is terminated.

Remark : Operation of this kind can be interpreted asintentionally allowing the states of the thermal model to grow.The thermal dynamics of a Li-ion cell is inherently stable,unless the temperature is increased to levels that may triggerthermal run-away. It can be argued that given the couplingbetween the thermal and electrical sub-models, as long as themaximum temperature is bounded away from (from above)a critical temperature (≈ 80C), the thermal model remainsstable and controllable. As for the electrical dynamics, SOC isa constrained state and the value of V1 is implicitly boundedas a function of constraints on the terminal voltage and input

Ψ =

−Dd

el 0 0 · · · 0CdelB

del Dd

el 0 · · · 0−CdelAdelBdel −CdelBdel −Dd

el · · · 0...

......

. . ....

Cdel(Adel)

(2N−2)Bdel Cdel(Adel)

(2N−3)Bdel Cdel(Adel)

(2N−4)Bdel · · · Ddel

(11)

Υ =

−VminVmax−Vmin

...Vmax

−−CdelAdelCdel(A

del)

2

−Cdel(Adel)3...

Cdel(Adel)

2N

xk −

−CdelBdelCdAdBdel

−Cdel(Adel)2Bdel...

Cdel(Adel)

2N−1Bdel

uk −−(Vocv(zk)− Cdel(1)zk)Vocv(zk)− Cdel(1)zk−(Vocv(zk)− Cdel(1)zk)

...Vocv(zk)− Cdel(1)zk

(12)

W = diag([ϑ1, . . . , ϑ2N ]), ϑi > 0, ϑi are functions of thermal system matrices.




current.

V. SIMULATION AND DISCUSSION

In this section, the proposed Pulsed Current Method (PCM)is simulated with both the plant and model dynamics dictatedby the equations in Section II.

A. Simulation Setup

The augmented electro-thermal model (Eqn. (13)) is nonlin-ear in input and output; the proposed algorithm is implementedusing discrete local linear models and is simulated in the MAT-LAB/Simulink environment using a custom vertex enumerator.[

xelxth

]=

[AelxelAthxth

]+

[Bel 00 Bth

] uu2

Tamb

Vt =VOCV (xel) + Celxel +Delu, (13)

where u is the current drawn from the cell.In implementing PCM variable values were chosen as

follows – the cell operating voltage bounds were set at [2, 3.6];the frequency of the pulse train was set to 10Hz basedon electrochemical considerations [21] and the model wassimulated at Nyquist frequency. The energy that is removedfrom the cell is assumed to be stored in an external storagesystem such as an ultracapacitor bank.

TABLE IIMANUFACTURERS SPECIFICATIONS FOR A123 26650 CELLS FOR

CONSTANT OPERATION.

Direction Temperature Continuous CurrentCharge 0–20oC 3ACharge 20–50oC 10A

Discharge -30–60oC 60A

The simulated LFP cell is assumed to be a part of a packthat consists of 60 cells in series and four cells in parallel witha rated nominal continuous power at 25C of 45 kW. Limitson the maximum deliverable current were set by factoringin manufacturers specifications (Table II) and the standardsproposed by USABC [29]. Note that the specifications providedin Table II are for continuous discharge. For pulsed currents, amultiplicative factor of 1.5 is used to amplify the current ratingsfor constant operation. The value of charge current limit belowfreezing was not provided explicitly in specification sheets.In practice, this limit may have to be empirically estimatedif it is not provided. The value of the limit can be taken atthe maximum magnitude of current that does not increase theeffective resistance of the cell after a pre-determined numberof energy cycles (using pulsed currents). In this study this limitis set at 1C. In addition, we assume a Arrhenius relation forthe increase in charge current limit above 0C.

The control scheme proposed in Section IV relies on areceding horizon controller. In receding horizon controllers,the length of the prediction horizon is a tuning parameter thattakes integer values. However, for large problems and problemwith fast dynamics, shorter control and prediction horizons are

0 20 40 60 80 100 120 140 160−150

−100

−50

0

Cur

rent

/A

0 20 40 60 80 100 120 140 160

2

2.5

3

3.5

Time /s

Vol

tage

/V

10 10.5 11−20

−10

0

10 10.5 112

3

PCM (β=0)CVM

Charge Limit

Discharge Limit

Limit

1

Fig. 8. Down-sampled simulated trajectory of (down-sampled by 19) voltageand current using Pulse Current Method (β = 0)

preferred; in [30], the authors provide necessary conditionsfor when the prediction horizon of length one is near optimal.In this section, unless stated otherwise, it assumed that theprediction and control lengths are of length one; the impact ofthis assumption is studied numerically in Section V-B.

B. Simulation, Results & Discussion

This section documents result of simulating the electro-thermal model of the battery developed in Sections II and IIIusing the algorithm described in Section IV. Simulations arerun with the following parameters – SOC0 = 0.6, ambienttemperature set to −20oC and under natural cooling condition(h = 5W/m2K).

Baseline

To study the performance of the proposed method and toestablish a baseline, we compare the trajectories of batterytemperature, power capability and SOC from the followingtwo cases:

1) the limiting case when β = 02) the case of maximum permissable continuous discharge.

The second case, when the maximum permissable continuousdischarging current is drawn, generates the maximum possibleheat at every sample and hence is an approximate solution tothe minimum warm-up time problem. In this mode of operation,to satisfy constraints, the terminal voltage is held at Vmin (thatis as long as the discharge current constraint is satisfied); thus,this mode is labeled Constant Voltage Method (CVM).

Figures 8 and 9 present trajectories resulting from simulatingthe electro-thermal model using the proposed reference currentgeneration algorithm, PCM, and CVM using power demand,(Pdmd = 100 W) as terminal constraint. Table III tabulatessome of the key indices from having applied CVM and PCM.

The value of penalty on SOC lost in each period, β,influences the duration of the warm-up operation. Largerpenalties will tend to increase the duration of the warm-upphase; this follows by observing that when operating from




0 50 100 150 200 250−20

0

20

Time /s

Avg

. Tem

p. /o C

0 50 100 150 200 250

50

100

Time /s

Pcap/W

0 50 100 150 200 250

0.4

0.6

Time /s

SO

C

0 50 100 150 200 2500

0.5

1

Time /s

Pol

ariz

atio

n

PCM (β=0)CVMPCM (β=0.58)

Fig. 9. Simulated trajectories of average temperature, power capabilityand polarization using Pulse Current Method (PCM) and Constant Voltage(CVM) Method. The simulation was performed with the pack initializedwith SOC0 = 0.6 operating from −20C with a terminal power demand,Pdmd = 100W under natural cooling conditions (h = 5 W/m2K).

sub-zero temperatures, the current limits are not symmetric.That is, the minimum warm-up time that can be achieved usingPCM is when β = 0. From Fig. 8 and Table III, it is notedthat the warm-up time when using CVM is shorter than whenusing PCM with β = 0. Thus, the warm-up time using PCM,for any value of β, will be longer than when using CVM.

Energy storage elements such as ultra-capacitors do not havevery high energy densities, i.e., it is desirable to transfer aslittle energy as possible to the external energy storage element.From Table III, note that the equivalent SOC stored in externalstorage using CVM is almost twice that of PCM.

Lastly, in comparing the effective energy lost using bothmethods — PCM and CVM — it is noted that CVM is morelossy. More specifically, comparing the CVM with PCM (β =0), we observe that the total energy lost increases by nearly35%; this increased loss manifests itself as increased terminaltemperature of the cell.

The above results bear evidence to the fact that terminatingwarm-up based on terminal temperature is not the same aswhen using power as terminal constraint. While CVM enjoysshorter operating times, it is more lossy and requires largerstorage elements as compared to PCM.

Penalizing energy loss

As formulated, the value of penalty β in the cost can be usedto regulate the amount of energy dissipated as heat. Figure

TABLE IIICOMPARISON BETWEEN PCM∗ AND CVM, KEY INDICES

Method Oper. Time SOCstore Tfinal SOCloss

PCM (β = 0) 172s 0.13 17.5oC 0.11PCM (β = 0.58) 278s 0.12 12.25oC 0.10

CVM 143s 0.23 24.3oC 0.15∗1 block with 5 periods

0.52 0.54 0.56 0.58 0.6 0.62 0.640.09

0.1

0.11

0.12

0.13

0.14

β

Equ

ival

ent S

OC

Energy lostExternal Storage

Fig. 10. Results on increasing penalty on energy loss as percent of whenno penalty is applied. The simulation was performed with the pack initializedwith SOC0 = 0.6 operating from −20C with a terminal power demand,Pdmd = 100 W under natural cooling conditions (h = 5 W/m2K).

10 documents the total energy lost and the reduction in sizeof external storage elements in equivalent battery SOC, fordifferent values of β. Inspecting Fig. 10, it it evident thatincreasing the value of β can reduce energy expenditure andexternal sizing. By computing the percent change with respectto when β = 0, the energy lost and external storage sizecan be reduced by as much as 20%. This increased efficiencyof operation does however come at the expense of operationtime. Figure 11 presents a comparison between the increasein warm-up efficiency and time taken to be able to deliver thedesired power; increased energy efficiencies result in increasingwarm-up times.

The observations from Figs. 10 and 11 can be explained bystudying the trajectories of terminal and polarization voltages,when β = 0 and β = 0.58; Figs. 9 and 12 depict thesetrajectories. The first observation from comparing these figuresis that unlike the case when β = 0, the trajectory of terminalvoltage when β = 0.58, does not always hit the lower limitof 2V; however, it does on occasion. Further, the trajectory ofpolarization is different after 40 s; these observations can beinterpreted as follows.

From the problem formulation in Eqn. (10), it is possibleto show that the value of polarization and the cell’s operatingtemperature result in the solution migrating between vertices ofthe constraint polytope (as an example cf. Fig. 6 and verticesν3 and ν4). The vertices between which the solution switchesare dictated by the temperature of the cell, penalty β andthe polarization. As the penalty, β, increases, SOC lost overthe control horizon becomes important; therefore, the optimalsolution tends to be ud = uc, i.e. the amplitudes of currentduring charge and discharge are the same, which can be clearlyobserved in Fig. 11 when β = 0.58 compared to the case ofβ = 0 in Fig. 9. The preference of charging and discharging

0 0.1 0.2 0.3 0.4 0.5 0.6

200

400

600

800

β

Ope

ratio

n tim

e /s

Fig. 11. Comparison between increased efficiency and warm-up operationtime




0 50 100 150 200 250−80−60−40−20

02040

Time /s

Cur

rent

/A

0 50 100 150 200 250

2

2.5

3

3.5

Time /s

Vt /V

150 150.5 151 151.5 152−40

−20

0

Time /s

Cur

rent

/A

150 150.5 151 151.5 152

2

2.5

3

3.5

Time /s

Vt /V

1

Fig. 12. Simulated trajectory of voltage, polarization and current using PulseCurrent Method, β = 0.58. The simulation was performed with the packinitialized with SOC0 = 0.6 operating from −20C with a terminal powerdemand, Pdmd = 100W under natural cooling conditions (h = 5 W/m2K).

at the same current rate has to consequences:1) the average current during the control horizon decreases

to zero and hence polarization voltage drops as well.2) the heat generated during each period reduces and corre-

spondingly the increasing rate of temperature diminishes.As seen from Fig. 8, the increasing rate of temperature becomeslower when β = 0.58 than when no penalty on SOC loss isimposed. It is also observed that polarization voltage decreasesfrom 50 second to 230 second. The polarization state isinherently stable; as the average current during each blockin the control horizon tends to zero, the value of polarizationdecreases. The reduced polarization and rate of heat generationmay result in the solution switching back to the vertex thatextracts maximum current from the cell (vertex ν3 in Fig. 6).This results in the switching behavior observed in Fig. 12.

Figure 11 also highlights another important characteristicof the solution – as the value of β is increased, the operationtime reaches an asymptote, i.e. it becomes impossible to reachthe desired terminal power capability. This is an extension ofthe behavior described above wherein the solution migrates;as β increases, the solution migrates and remains at the vertexthat favors charging and discharging currents being of thesame magnitude (vertex ν4 in Fig. 6). In addition, for βssufficiently large, the solution will remain at the vertex thatfavours negligible SOC loss and hence the power demandcan never be achieved. Thus, for the above algorithm to beimplemented, the value of β needs to be chosen appropriatelyto ensure feasibility of the overall problem.

Effect of longer prediction horizons

In simulating the results presented thus far, the predictionhorizon was set to be a single block consisting of five pulses.In the context of predictive control, longer prediction horizons

1Simulations were performed on a computer powered by an Intel i5-2500 quad-core processor with 16GB of ram and running Windows 7 withparallelization enabled.

TABLE IVIMPACT OF PREDICTION HORIZON BASED ON KEY INDICES∗

aaaaaaaaIndex

PredictionLength 1 2 3

SOCloss 1 0.99 0.98External Storage 1 0.99 0.97Terminal Time 1 1.01 1.03

Computational Time 1 35 107∗ Entries normalized wrt. results when prediction length is one block

are known to produce better approximations of the globaloptimal solution. In this application, owing to the linearizedMPC implementation, the prediction horizon cannot be takento be arbitrarily large without incurring errors resulting frommodel linearization.

To investigate the influence of prediction horizon on the opti-mal solution trajectory, an iterative test was performed1whereinthe length of the prediction horizon was increased incre-mentally; results of which are presented in Table IV. Theother parameters of the simulation were : Pdmd = 50W ,h = 5W/m2K and β = 0.57 (the power demand is set at50 W in the interest of computational time).

The data presented in Table IV, as expected, indicates thatgiven the same penalty on loss in energy, increasing the lengthof the prediction horizon decreases the total energy lost; thishowever does come at the expense of computational time. Infact, there appears to be a quadratic relation between decrease inloss and total operation-time. Comparing the effective increasein savings and the increase in computational and operationtime, a case for the use of prediction horizon of length oneblock can be made.

VI. CONCLUSION

In this work, a Li-ion battery warm-up strategy that increasesthe cell temperature to meet power demand in an energyefficient method is described. The shape of current usedto shuttle energy between the cell and an external energystorage system was set to be bi-directional pulses to minimizepolarization and reduce damage to electrodes. Magnitude of thepulses were determined by solving a constrained optimizationproblem. From simulations based on models of a 26650 LFPcell, it is noted that it is possible to reduce energy lost as heatand the size of external storage, by as much as 20% whencompared against using just constant voltage discharge. There ishowever, a compromise to be made between reduction in size ofstorage, energy lost and time taken to warm-up. A future workwill study the impact of the proposed technique experimentallyand ascertain if the conclusion in [31]—that pulsed currentscan accelerate degradation—is indeed applicable to operationsof the kind considered in this study.

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Hybrid Electric Vehicles, 2nd ed., U.S. Department of Energy VehicleTechnologies Program, December 2010.

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[31] F. Savoye, P. Venet, M. Millet, and J. Groot, “Impact of periodic currentpulses on li-ion battery performance,” IEEE Trans. Ind. Electron., vol. 59,no. 9, pp. 3481–3488, Sept 2012.

Shankar Mohan received his bachelor’s degree fromthe National University of Singapore in 2011 and hismaster’s degree from the University of Michigan in2013, both in Electrical Engineering. He is currentlyworking towards a PhD. in Electrical Engineering atthe University of Michigan. His current interestsinclude numerical optimization methods and theapplication of control techniques to energy systems.

Youngki Kim (M’14) is currently a PostdoctoralResearch Fellow with the Powertrain Control Labo-ratory, University of Michigan. He received his B.S.and M.S. degrees in the School of Mechanical andAerospace Engineering at Seoul National Universityin 2001 and 2003 respectively, and his Ph.D degreein Mechanical Engineering Department from theUniversity of Michigan in 2014. He worked as aResearch Engineer in the R&D Division at HyundaiMotor Company from 2003 to 2008. His researchinterests include modeling, simulation and optimal

control/estimation of dynamic systems such as electrified vehicles andelectrochemical energy storage systems.

Anna G. Stefanopoulou (F’09) is a Professor ofmechanical engineering and the Director of theAutomotive Research Center, University of Michigan,Ann Arbor. From 1998 to 2000, she was an AssistantProfessor with the University of California, SantaBarbara. From 1996 to 1997, she was a TechnicalSpecialist with Ford Motor Company. She has au-thored or co-authored more than 200 papers and abook on estimation and control of internal combustionengines and electrochemical processes such as fuelcells and batteries. She holds nine U.S. patents. Prof.

Stefanopoulou is a Fellow of the ASME. She was a recipient of four BestPaper Awards.

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http://dx.doi.org/10.1007/BF01720068

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