Battery State-of-Power Peak Current Calculation and ...

4512 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 65, NO. 6, JUNE 2016

Battery State-of-Power Peak Current Calculationand Verification Using an AsymmetricParameter Equivalent Circuit Model

Pawel Malysz, Member, IEEE, Jin Ye, Member, IEEE, Ran Gu, Student Member, IEEE,Hong Yang, and Ali Emadi, Fellow, IEEE

Abstract—In this paper, a higher fidelity battery equivalentcircuit model incorporating asymmetric parameter values is pre-sented for use with battery state estimation (BSE) algorithmdevelopment; particular focus is given to state-of-power (SOP) orpeak power availability reporting. A practical optimization-basedmethod is presented for model parameterization fitting. Two novelmodel-based SOP algorithms are proposed to improve voltage-limit-based power output accuracy in larger time intervals. Thefirst approach considers first-order extrapolation of resistor valuesand open-circuit voltage (OCV) based on the instantaneous equiv-alent circuit model parameters of the cell. The second proposedapproach, which is referred to as multistep model predictive it-erative (MMPI) method, incorporates the cell model in a modelpredictive fashion. Finally, a SOP verification methodology ispresented that incorporates drive cycle data to realistically excitethe battery model. Simulation results compare the proposed SOPalgorithms to conventional approaches, where it is shown thathigher accuracy can be achieved for larger time horizons.

Index Terms—Battery management systems, battery model-ing, cell parameter variation, model predictive iterative, powerestimation.

I. INTRODUCTION

BATTERIES are a fundamental component to enable clean,electrified, and sustainable transportation [1], [2]. Battery

state estimation (BSE) plays a key role in a battery managementsystem [3]–[5]. Two major functions of BSE are state-of-charge(SOC) [3]–[5] and state-of-power (SOP) estimation [8]–[20];the latter reports peak power capability, e.g., maximum powerfor an arbitrary time interval. To accelerate the developmentof BSE algorithms and to facilitate the verification and valida-tion effort, a model-in-the-loop (MIL) methodology is needed,

Manuscript received October 31, 2014; revised March 29, 2015; acceptedMay 19, 2015. Date of publication June 11, 2015; date of current versionJune 16, 2016. This work was supported in part by Automotive PartnershipCanada, Natural Sciences and Engineering Research Council of Canada; by theCanada Excellence Research Chairs Program; and by Fiat Chrysler Automo-biles (FCA), including FCA USA LLC and FCA Canada Inc. The review ofthis paper was coordinated by Prof. M. Benbouzid.

P. Malysz is with Fiat Chrysler Automobiles USA LLC, Auburn Hills, MI48326 USA (e-mail: [email protected]).

J. Ye, R. Gu, and A. Emadi are with McMaster Institute for AutomotiveResearch and Technology (MacAUTO), McMaster University, Hamilton, ONL8S 4L8, Canada.

H. Yang is with Fiat Chrysler Automobiles USA LLC, Auburn Hills,MI 48326 USA, and also with Department of Electrical and Computer En-gineering, McMaster University, Hamilton, ON L8S 4K1 Canada (e-mail:[email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TVT.2015.2443975

Fig. 1. Outputs for SOC and SOP algorithm benchmarking. LUT denoteslookup table. The input to the battery model is voltage V or current i.

which employs sufficient model fidelity with high-enough exe-cution speed to perform practical simulation and design studies.The battery model should consider nonlinearities and time-varying effects that are typical of electrochemical cells to gener-ate battery state and parameter variables for benchmarking BSEperformance. For SOC verification, this is relatively straightfor-ward since true SOC is an explicit state variable of the modelthat can be outputted to gauge SOC estimation performance.For SOP algorithm testing, this is not the case since calcula-tion of peak power depends not only on the model’s currentstate/parameter values but on its future values and on the timehorizon of interest as well. To properly assess the performanceof a SOP algorithm, one must be able to generate a “true” SOPprofile as a function of time; this can be done by determiningthe maximum sustained current allowed for some chosen timeinterval and multiplying by voltage. Therefore, peak powercalculation can be simplified to peak current calculation. Forverification of both SOC and SOP, a battery model and anSOP calculation method are needed for algorithm testing anddevelopment, e.g., as shown in Fig. 1. Aside from algorithmtesting, the same SOP calculation method, or its simplifiedversion, can be also directly part of SOP calculation within thebattery management system software.

SOP estimation has been previously investigated usingequivalent circuit models to calculate maximum current subjectto cell limits such as voltage and SOC [8]–[20]. To improvevoltage-limit-based SOP estimation, dynamic models consider-ing, for example, resistor–capacitor (RC) pairs have been pro-posed [9], [10]. However, the variation of the circuit parameterssuch as open-circuit voltage (OCV), ohmic resistance, resis-tances, and capacitance values of RC pairs is usually neglectedin the time horizon of interest. In [8], a multiparameter batterydynamic model is presented and a power estimation method is

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MALYSZ et al.: BATTERY SOP PEAK CURRENT CALCULATION AND VERIFICATION USING A CIRCUIT MODEL 4513

evaluated that considers a first-order extrapolation of OCV. In[11], SOP estimation based on a dynamic model is employed,and a bisection method is used to find the charging/dischargingcurrent limits that account for the exact future value of thenonlinear OCV. In [12], joint SOC and SOP estimation using anadaptive extended Kalman filter is presented; SOP calculationin [12] also considers variation in the OCV. In [13], a review ofstate monitoring of lithium-ion (Li-ion) batteries is enumerated,including SOC, state of health, SOP, and remaining useful life.For SOP estimation, adaption of the model to the aging statusof the battery, current dependence of the battery impedance,and multicell difference in the battery pack of power estimationmethods are discussed. In [14], current dependence of batteryimpedance at lower temperature and aged batteries is evaluatedand the method to estimate battery impedance is proposed. In[15], a novel approach to estimate power is proposed, whichincorporates the current dependence into the nonlinear batterymodel. The proposed method shows improvement of powerestimation at lower temperature and aged batteries. In [17], SOPestimation and SOF estimation based on a Kalman filter arepresented. In [18], a new algorithm to estimate the maximumcharge and discharge power values is proposed, which incor-porates the diffusion effect. The diffusion effect is modeled asthe nonlinear resistance in the equivalent circuit model of thebattery, and the proposed method is validated by simulationresults in a vehicle simulation environment. In [19], a model-based dynamic peak power method is presented for LiNMC andLiFePO batteries based on a linear-parameter-varying batterymodel. The Levenberg–Marquardt algorithm is applied to findthe peak current and compared with the bisection method in[11] in terms of accuracy and computational complexity. Theproposed method is validated in different operating conditionssuch as varying temperature and aging levels. In [20], a real-time prediction of peak power for Li-ion batteries is presentedat different temperatures and aging conditions. A dual Kalmanfilter is utilized to estimate the present state/parameters ofthe batteries, and therefore, peak power estimation can beapplicable to different temperatures and aging conditions. Themethod demonstrates good accuracy and adaptability throughexperiments.

For best accuracy, variations in all models parameters shouldbe taken into account, particularly for larger time horizonintervals. In most previous publications, despite consideringaging effect or current dependence of the battery, power isestimated at shorter time horizon intervals, and therefore, thevariations of model parameters are not addressed. This paperincorporates the model variation into the power estimation,which is applicable at longer time horizons.

This paper presents a higher fidelity equivalent circuit modelthat is useful for both SOC and SOP algorithm testing anddevelopment. Asymmetric parameter values and an arbitrarynumber of RC pairs are considered. The use of asymmetricvalues for charge and discharge for all resistances and capac-itance values increases the fidelity of the model. The model isimplemented in a MATLAB Simulink environment; in partic-ular, PLECS Blockset is used for its computational efficiency.Methods to parameterize this model are also described withinan optimization-based parameter fitting context.

Fig. 2. Battery cell asymmetric equivalent circuit model.

Methods to calculate SOP given a known model are pre-sented where two novel approaches are proposed. The firstmethod is a voltage-limit-based method where extrapolationof resistor values and OCV are considered. It incorporatesfirst-order extrapolation of multiple equivalent circuit modelparameters. The second method is a multistep model predictiveiterative (MMPI) method. It updates the circuit parameters insmaller time intervals where constant or linear variations of theparameters are considered. Both work directly with the batterymodel presented in this paper, and both can be directly used inan online SOP calculation algorithm.

Lastly, a SOP verification methodology is described thatincorporates drive cycle data to realistically excite the batterymodel. The same methodology can be used for hardware test-ing to experimentally validate SOP estimation and calculationperformance. Simulation results in a MIL context are providedfor SOP calculation and are assessed using the proposed SOPverification methodology.

The remainder of this paper is organized as follows. Thebattery model is presented in Section II. Parameterization andvalidation of this model are discussed in Section III. Section IVreviews conventional SOP peak current calculation methods.Section V presents the proposed SOP peak current calcula-tion methods. The new SOP verification methodology and itssimulation results comparing the proposed SOP peak currentcalculation methods are shown in Section VI, and the paper isconcluded in Section VII.

II. BATTERY MODEL

An asymmetric parameter equivalent circuit model shown inFig. 2 is employed. The parameter definitions are enumeratedin the Appendix; they are functions of SOC and temperature.

Columbic losses can be modeled as charging/discharginginefficiencies, which contribute to the difference between theamount of energy put into a cell and the energy extracted;therefore, the internal SOC state is updated according to thefollowing formula to include the effect of charging/discharginginefficiencies:

SOC(t) = − 1Cmax

∫ [ηci

−(t) + η−1d i+(t)

]dt (1)

where i−(t) = min(i(t), 0), i+(t) = max(i(t), 0), 0 ≤ ηc ≤ 1,and 0 ≤ ηd ≤ 1 are the charging and discharging efficiencies.


The definition of cell capacity can impact the meaning of theefficiency parameters. For example, let Ahc be the (charging)capacity via current integration from 0% to 100% SOC andAhd

be the (discharging) capacity via current integration from 100%to 0% SOC. Inefficiencies result in Ahd < Ahc; this ratio canbe extracted from experimental data. Now, consider the follow-ing scenarios for capacity definition. If Ahc is defined as thecapacity, charging efficiency ηc is one. Thus, the dischargingefficiency ηd is defined as

ηd =Ahd

Ahc. (2)

Similarly, if Ahd is defined as the capacity, the inefficienciesare expressed as

ηc =Ahd

Ahc, ηd = 1. (3)

Assuming symmetric losses, charging efficiency and discharg-ing efficiency can be redefined as

ηc = ηd =

√Ahd

Ahc. (4)

Using the preceding equation, the following definition for ca-pacity is used in this paper:

Cmax =√AhcAhd. (5)

III. MODEL PARAMETERIZATION

The model can be parameterized using high pulse powercharacterization (HPPC) tests and OCV–SOC charge/dischargetests. This type of data comes in the form of voltage andcurrent profiles over time, typically over the full SOC rangeand at multiple temperature points. Optimization-based fittingmethods are used to extract parameters.

A. SOC–OCV Curve

The following equation is used for SOC–OCV fitting:

Voi = k0 − k1/si + k2si + k3 log si

− k4 log(1 − si) + k5s2i + k6s

3i + k7s

4i + k8s

5i = sTi k (6)

where Voi is the ith OCV data point, and si is SOC and is anumber between zero and one. Variables s and k consolidatethe si and kj terms into column vectors. The use of the higherorder terms provides extra fidelity in the model. To ensure anondecreasing fitted curve with these extra terms, the followingderivative constraints are used:

d Voi

d zi= k1/s

2i + k2 + k3/si + k4/(1 − si)

+ 2k5si + 3k6s2i + 4k7s3i + 5k8s4i = dTi k ≥ 0. (7)

Fig. 3. Preprocessing of HPPC pulse data for an example charging pulse(negative current).

The fitting can be formulated as the following quadraticoptimization problem:

mink

(Sk− v)T (Sk− v)

Dk ≥ 0, vT = [Vo1 Vo2 · · · Von ]

ST = [s1 s2 · · · sn] , DT = [d1 d2 · · · dn]. (8)

MATLAB “quadprog” can be used to obtain a solution.Fig. 4(a) shows an example curve fitting result of SOC–OCVfitting via quadratic optimization. Hysteresis effects werenegligible.

B. Resistor and Capacitor Parameterization

HPPC data over the full SOC range is used in anoptimization-based fitting routine to obtain resistor and capaci-tor values. The voltage profiles are preprocessed to remove theeffect of OCV in the manner shown in Fig. 3. Since there is alarge rest time prior to the pulse, it is assumed that the initialvoltages of the RC states are zero.

Consider a current pulse of ipulse amperes of duration tpulsethat starts at time t = 0, its ideal OCV subtracted voltageresponse can be derived as

VOS(t)=

⎧⎪⎪⎪⎨⎪⎪⎪⎩−(R0+

n∑j=1

Rj(1−e−t/τj)

)ipulse, t≤ tpulse

−n∑

j=1

Rj(1−e−tpulse/τj)ipulsee−(t−tpulse)

τj , t>tpulse

(9)

where Ri, i = 0..n, and τj = RjCj , j = 1..n, are parametersrepresenting resistances and time constants, respectively. Theyare assumed to be constant over the pulse interval. Givenvoltage/current data and using (9), the following optimizationcan be formulated for parameter fitting:

minr,τ

∑k

(V DataOSk

− VOS(r, τ , tk))2

r = [R0 R1 · · · Rn], τ = [τ1 τ2 · · · τn]. (10)


Fig. 4. ECM parameterization results. (a) OCV. (b) R0, (c) R1, (d) C1,(e) R2, (f) C2.

Fig. 5. Validation of battery model based on estimated circuit parameters.

To solve this optimization problem, MATLAB “fmincon” hasbeen used to perform curve fitting of charging and dischargingpulses. It is noted here that choosing time constants rather thancapacitance values as optimization variables produced morestable fitting results. Sample 2RC equivalent circuit modelfitting results from a Li-ion-based cell are shown in Fig. 4 attwo different temperatures. The asymmetry of parameters withrespect to current direction is evident.

C. Validation Results

The model parameters obtained from the optimization meth-ods described in the previous sections are validated againstUrban Dynamometer Driving Schedule (UDDS) drive cycledata meant to mimic real-world driving conditions. Fig. 5 showsa sample comparison result between voltage response from thebattery model and voltage from drive cycle data. The averagevoltage error over the whole drive cycle was 0.68%. The cellcapacity was 40 Ah.

IV. CONVENTIONAL PEAK CURRENT

CALCULATION METHODS

Here, battery peak power estimation [16]–[20] based on peakcurrent calculation and the circuit model shown in Fig. 2 isdescribed. Conventional methods of battery peak current esti-mation are reviewed and include SOC-limited, voltage-limitedohmic resistance (VLOR)-only, and voltage-limited first-orderextrapolation of OCV (VLEO) with RC dynamics. The VLEOmethod is extended to an arbitrary number of RC pairs.

The maximum voltage when charging and the minimum volt-age when discharging are denoted as Vmax and Vmin, respec-tively. The maximum SOC and the minimum SOC are definedas smax and smin, respectively. Without loss of generality, thesubsequent derivations will consider the single-cell case. More-over, for a cleaner presentation, the following variables are sub-sequently used such that, during charging, ζ = ηc, Rj = Rc

j ,τj=τcj =Rc

jCcj , and ipeak= icmin ≤ 0, and during discharging,

ζ = 1/ηd, Rj = Rdj , τj = τdj = Rd

jCdj , and ipeak = idmax ≥ 0.

A. SOC-Limited Method

The SOC-limited method is used to estimate the batterypeak current based on the minimum and maximum SOCs. Theequation for SOC calculation is

s(t+Δt) = s(t)− ipeakΔt

Cmax(11)

where ζ is the Coulomb efficiency factor, which is equal toηc when charging and is equal to 1/ηd when discharging,and Cmax is the cell capacity. The peak (minimum) chargingand peak (maximum) discharging currents by using the SOC-limited method are

ic,SOCmin =

Cmax(s− smax)

ηcΔt, id,SOC

max =Cmax(s− smin)

η−1d Δt

. (12)

Note that the functional dependence on time has been droppedfor a cleaner presentation.

B. VLOR Method

The VLOR method estimates the peak power based on asimplified circuit model, which only includes an OCV andinternal resistance, i.e.,

V (t+Δt) ≈ Vo (s(t))−R0 (s(t)) ipeak (13)

where V (t) is the terminal voltage of the cell at time t; Vo(s(t))is the OCV at present SOC state s(t), and Ro is the charging ordischarging internal resistance of the cell. The peak (minimum)charging and peak (maximum) discharging currents by usingthe VLOR method are defined as

ic,VLORmin =

Vo − Vmax

Rc0

, id,VLORmax =

Vo − Vmin

Rd0

(14)

where Rc0 and Rd

0 are the ohmic resistances for charging anddischarging, respectively. Although simplistic, this estimate canbe a reasonable initial guess for iterative algorithmic methodsthat perform peak current determination.


C. VLEO Method

The limitation of the VLOR method is that it neglects thetransient dynamics of the battery. A voltage-limited extrap-olation in the OCV method [16] that considers a first-orderextrapolation of OCV and includes RC transients is describedhere; it is called the VLEO method. It considers the dynamics ofthe RC elements and OCV change over a specified time range.The voltage response Vj of the jth RC pair assuming constantparameters during interval Δt is derived as

Vj(t+Δt) = e−Δt/τjVj(t) +Rj(1 − e−Δt/τj)ipeak

j ∈ [1, n]. (15)

A Taylor series expansion of Vo is used to extrapolate OCVand is derived as

Vo (s(t+Δt)) = Vo

(s(t)− ipeakς

Δt

Cmax

)

= Vo (s(t))−ipeakςΔt

Cmax

∂Vo(s)

∂s

∣∣∣∣s=s(t)

+h.o.t.

(16)

where “h.o.t.” denotes higher order terms that are subsequentlyneglected. Using (15) and (16), the voltage response of the cellcan be calculated as

V (t+Δt)

= Vo (s(t+Δt))−R0 (s(t+Δt)) ipeak−n∑

j=1

Vj(t+Δt)

≈ Vo (s(t))−ipeakςΔt

Cmax

∂Vo(s)

∂s

∣∣∣∣s=s(t)

−R0 (s(t)) ipeak

−n∑

j=1

e−Δt/τjVj(t) +Rj(1−e−Δt/τj)ipeak. (17)

Rearranging the preceding equation, the peak charging anddischarging currents can be readily solved as

ic,VLEOmin =

Vo−n∑

j=1

e−Δt/τcj Vj(t)−Vmax

ηcΔtCmax

∂Vo(s)∂s +

n∑j=1

Rcj(1−e−Δt/τc

j )+Rc0

(18)

id,VLEOmax =

Vo−n∑

j=1

e−Δt/τdj Vj(t)−Vmin

ΔtηdCmax

∂Vo(s)∂s +

n∑j=1

Rdj (1−e−Δt/τd

j )+Rd0

. (19)

V. PROPOSED PEAK CURRENT CALCULATION METHODS

The conventional methods do not consider variations of allresistor and capacitor values during the constant current pulseipeak. Since the resistor/capacitor parameters are functions ofSOC, they can potentially vary greatly during the current pulse.Improvements to peak current calculation are described here;they consider these variations in a model predictive fashion.

A. VLERO Method

An analytical voltage-limited extrapolation method is pre-sented that considers RC elements, first-order extrapolationof all resistors, extrapolation of OCV, and constant RC time

constants, i.e., voltage limited with extrapolation of resistancesand OCV (VLERO) method. A Taylor series expansion of thevariation of resistances is expressed as

Rj (s(t+Δt)) = Rj

(s(t)− ipeakς

Δt

Cmax

), j ∈ [0, n]

=Rj(s(t))−ipeakςΔt

Cmax

∂Rj(s)

∂s

∣∣∣∣s=s(t)

+h.o.t.

(20)

Using the preceding equation with the h.o.t. neglected, thevoltage response of the RC elements given in (15) is modified to

Vj(t+Δt)

= e−Δtτj Vj(t) +

t+Δt∫t

e−(Δt−T)

τjRj(T )

τjdT · ipeak, j ∈ [1, n]

= e−Δtτj Vj(t) +Rj (s(t))

(1 − e

−Δtτj

)ipeak

+ ςτj

Cmax

∂Rj

∂s

(1 − e

−Δtτj − Δt

τj

)i2peak (21)

where Rj(T)=Rj(s(t))−ipeakς(T/Cmax)(∂Rj(s)/∂s)|s=s(t).Using (20) and (21), the voltage response in (15) or (17) is

replaced with

V (t+Δt) ≈ Vo − ipeakςΔt

Cmax

∂Vo

∂s

−(R0 − ipeakς

Δt

Cmax

∂Ro

∂s

)ipeak

−n∑

j=1

e−Δt/τjVj +Rj(1 − e−Δt/τj)ipeak

−n∑

j=1

ςτj

Cmax

∂Rj

∂s

(1 − e

−Δtτj − Δt

τj

)i2peak.

(22)

This can be rearranged into a quadratic equation of the formaipeak ∗ ipeak + bipeak + c = 0, where

a = ςΔt

Cmax

∂Ro

∂s−

n∑j=1

ςτj

Cmax

∂Rj

∂s

(1 − e

−Δtτj − Δt

τj

)

b = −ςΔt

Cmax

∂Vo

∂s−R0 −

n∑j=1

Rj

(1 − e

−Δtτj

)

c = Vo − V −n∑

j=1

e−Δt/τjVj . (23)

To obtain peak charging current icmin, V =Vmin is used with(23). To obtain peak discharging current idmax, V =Vmax is usedwith (23). The quadratic root formula can be applied in bothcases to obtain closed-form solutions. To avoid complex rootsrelated to nonrealistic extrapolation scenarios, a correction ofthe resistance derivatives is applied prior to using the quadraticroot formula. Fig. 6 shows an example of resistance as a func-tion of SOC where the actual slope is shown as a red solid line.Using this slope can lead to extrapolation to negative resistancevalues. A correction shown in Fig. 6 modifies the slope usingthe minimum resistance. The green dashed line shows thecorrected slope used to obtain real analytical solutions with theVLERO method.


Fig. 6. Resistance variation over SOC and slope corrections. The solid (black)curve is the cell resistance, the solid (red) line is the first-order extrapolation ofresistance, the dotted (gray) line is the minimum resistance, and the dashed(green) line is a corrected slope extrapolation.

Fig. 7. Flowchart for inner stage multistep voltage calculation.

B. MMPI Methods

For power limit prediction over larger time intervals, theextrapolation methods can poorly approximate the variationsin OCV and ECM parameters. An MMPI method is proposedto better capture the variation of resistor/capacitor values overa limited time horizon. This approach incorporates a batterymodel in the form of parameter lookup tables or functions thatdepend on SOC. The MMPI method works in two stages. Aninner stage accepts a peak current input and then incrementsSOC using (11) and calculates RC voltage states based on timesteps of length Δt/M , where M is the number of time-stepdivisions. This inner stage performs M substeps to calculateend-of-horizon RC voltage states Vj ; it then uses them withend-of-horizon SOC to output end-of-horizon cell voltage.Model parameters in the lookup tables are used throughout thisinner stage. A flowchart of this stage is shown in Fig. 7.

Two methods can be employed to update the RC voltagestates at each step of the inner stage. One method is to assumethat the parameters are constant during the time interval ofthe substep and that it is essentially a zero-order-hold-typeapproach, and it is shown in Fig. 8(a). RC voltages would beupdated using (15) with parameter values corresponding to theSOC at the beginning of each time step in the inner stage. Forsmall time steps, this method can work well; however, using

Fig. 8. Parameter approximations for multistep (M = 4) peak current calcu-lation. (a) Constant parameters. (b) Piecewise linear parameters.

many small time steps comes at the price of greater compu-tational cost. An alternative approach is to linearly interpolateparameters during the time step with endpoints, as shown inFig. 8(b); each endpoint corresponds to a known different valueof SOC during the time horizon of interest. A hybrid approachis also possible that assumes nonvarying RC time constants andfirst-order variations of resistor values similar to (20). A notabledifference is that the derivative used would be calculated byusing the SOC values at the endpoints of the time-step intervals,e.g., the slopes shown in Fig. 8(b). The hybrid approach woulduse (21) to update the RC voltage states in Fig. 7.

The more general approach of independently varying resistorand time-constant values is also possible. Consider (20) and thefollowing Taylor series expansion of time constants:

τj (s(t+Δt)) = τj

(s(t)− ipeakς

Δt

Cmax

), j ∈ [1, n]

= τj (s(t))−ipeakςΔt

Cmax

∂τj(s)

∂s

∣∣∣∣s=s(t)

+ h.o.t.

(24)

Neglecting the higher order terms in (20) and (24), the dynam-ics of the RC voltage can be expressed as

d Vj(t+Δt)

dΔt=

−Vj+(Rj (s(t))−ipeakζ

ΔtCmax

· ∂Rj

∂s

)ipeak

τj (s(t))−ipeakζΔt

Cmax· ∂τj

∂s

.

(25)

The solution to the preceding equation is the following nonlin-ear function that has current as an input; it can be found usingMATLAB “dsolve” and is expressed as follows:

Vj(t+ tΔ) = Vj(t)1

ξ(

Δtτj(s(t))

, δτj

)

+ ipeakRj (s(t))

⎛⎝1 − 1

ξ(

Δtτj(s(t))

, δτj

)⎞⎠

+ ipeakδRjτj (s(t))

Δtτj(s(t))

−1+ 1

ξ(

Δtτj(s(t))

,δτj

)1+δτj

(26)


Fig. 9. Flowchart for the outer stage of the MMPI iterative algorithm forpeak discharge current calculation; for peak charge current calculation, a nearlyidentical flowchart can be used.

where

ξ

(Δt

τj, δτj

)=

(1 + δτj

Δt

τj

) 1δτj

δRj= −ζ

ipeakCmax

· ∂Rj

∂s

δτj = −ζipeakCmax

· δτjδs

.

This expression can then be used in the RC voltage state updatesin Fig. 7. It is worthy to note that, since

limn→0(1 + nx)1n = ex (27)

then

ξ

(Δt

τj, 0

)= e

Δtτj . (28)

It is also noted here that the approach of assuming linear varia-tions in capacitance (instead of time constant) and resistancedoes not lead to a simple analytical expression for voltageanalogous to (26); this arises since integration of nonelementaryfunctions is encountered.

An outer stage in the MMPI method effectively performs aroot-finding task to determine the peak current that matchesvoltage limits of Vmin or Vmax. Well-known root finding algo-rithms such as bisection method [19], [23] and secant method[23] are some examples that can be applied. A simple approachis to iteratively increase/decrease peak current until it is withina tolerance to the voltage limit. A flowchart of this approachis shown in Fig. 9. The VLERO method is used to obtain aninitial peak current value. If the corresponding terminal voltageobtained by the inner stage is within the tolerance δV of thevoltage limit (maximum voltage for charging or minimum volt-age for discharging), the iteration ends. If the terminal voltageis not close enough, the estimated current values are eitherincreased or decreased in steps δi until the voltage is withinthe tolerance.

Fig. 10. Verification methodology for SOP peak current estimation.

VI. VERIFICATION METHODOLOGY

FOR POWER ESTIMATION

A verification methodology for power estimation is devel-oped in this paper to evaluate different approaches for peakcurrent-based SOP algorithms. Verification of power estimationis shown in Fig. 10. The estimated peak current obtainedby power prediction block will be used to excite the batterymodel at predetermined times. Thus, excitation current for thebattery model could be the peak charging (minimum) current,the driving cycle current, or the peak discharging (maximum)current. The minimum voltage is chosen as 2.8 V, and the max-imum voltage is chosen as 4.1 V. The maximum dischargingcurrent is first calculated and then used to excite the batterymodel to observe whether it would discharge to 2.8 V in aspecified time (e.g., 15 or 60 s). Then, UDDS drive cyclecurrent is used to excite the battery, and minimum chargingcurrent is calculated based on the new state of the battery.Minimum charging current is also used to charge the batteryto see whether it will charge to the maximum voltage.

The battery model has been implemented in MATLABSimulink using PLECS add-on toolboxes. The authors haveobserved that PLECS is more computationally efficient thennative Simscape toolbox. An Intel i5 2.5-GHz Apple computerwith 4-GB RAM was used for MIL simulations. Results areshown at two relatively low temperatures of 0 ◦C and −20 ◦Cwhere cell resistances are such that voltage-based limits are arelevant limiting factor for SOP.

To obtain a peak power estimate, the peak current estimateis multiplied by cell voltage. A conservative estimate can beobtained by using the minimum cell operating voltage, i.e.,

P = Vminipeak. (29)

The focus here will be on the voltage-limited peak currentsand comparisons between them. This is done here to avoidconvoluting the results with limits such as absolute currentlimits and SOC-limited currents.

Fig. 11 shows simulation results of the peak current ver-ification methodology using a 15-s peak power availabilitytime horizon at a 0 ◦C modeled test condition. The (conven-tional) VLEO method with a 2RC model was used. At thelow SOC range, power is incorrectly overestimated, resultingin undesirable undervoltage conditions. The MMPI methodwas employed for the same test condition, and the results areshown in Fig. 12; they show the elimination of the undervoltageresponses and a correct peak current calculation. Similar testcases for a 60-s peak power availability horizon are shownin Figs. 13 and 14 using the VLEO and MMPI methods,


Fig. 11. Peak current prediction by using the VLEO method with 15-s timehorizon at 0 ◦C.

Fig. 12. Peak current prediction by using the MMPI method with 15-s timehorizon at 0 ◦C.

Fig. 13. Peak current prediction using the VLEO method with 60-s timehorizon at 0 ◦C.

Fig. 14. Peak current prediction using the MMPI method with 60-s timehorizon at 0 ◦C.

Fig. 15. Peak current prediction using the VLEO method with 60-s timehorizon at −20 ◦C.

respectively. The cell model parameter variations during thepulse lead to both under- and overestimates of peak currentusing the VLEO method. The MMPI method has near-perfectperformance. Additional plots are shown at −20 ◦C for the 60-sSOP time horizon in Figs. 15 and 16. At this temperature, thevoltage-limit-based peak currents are nearly the same magni-tude at the drive cycle currents; this highlights the importanceof accurate voltage-limit-based SOP algorithms. The conven-tional VLEO method results in both under/over cell voltageconditions; these are eliminated using the MMPI method.

A comprehensive analysis was performed between theVLEO method, the proposed VLERO method, and variationsof the proposed MMPI approaches. The following percentagevoltage error in reaching the voltage limit was used:

errd =

∣∣V 2t − V 2

min

∣∣V 2min

, errc =

∣∣V 2t − V 2

max

∣∣V 2max

(30)


Fig. 16. Peak current prediction using the MMPI method with 60-s timehorizon at −20 ◦C.

TABLE ICOMPARISON OF 15-s PULSE TERMINAL VOLTAGE ERRORS AT 0 ◦C

TABLE IICOMPARISON OF 60-s PULSE TERMINAL VOLTAGE ERRORS AT 0 ◦C

where errd and errc represent the maximum discharging andcharging errors of the SOP peak current pulse, Vmin and Vmax

are the minimum/maximum cell voltage limits, and Vt is thevoltage at the end of the peak current pulse.

Tables I–IV summarize the results of the comparisons at0 ◦C and −20 ◦C test conditions where mean and maximumvoltage percentage errors according to (28) are reported. Threevariations of MMPI were employed, two of which used threesubsteps (M = 3) to reduce the computational load of the al-

TABLE IIICOMPARISON OF 15-s PULSE TERMINAL VOLTAGE ERRORS AT −20 ◦C

TABLE IVCOMPARISON OF 60-s PULSE TERMINAL VOLTAGE ERRORS AT −20 ◦C

gorithm. The results show that the VLERO method marginallyimproves performance over the VLEO method. The MMPIvariants have similar near-perfect performance in most caseswith maximum error of 1.54%. The usage of fewer subintervals(M = 3) in the time horizon had minimal impact on accuracy.In principle, computational efficiency when using M = 3 isexpected to be 20 times greater compared with M = 60. Theusage of piecewise linear-based updates, e.g., Fig. 4(b), inMMPI generally improved when M = 3 was used; for M =60, there was no difference between piecewise linear and con-stant parameter-based MMPI variants.

VII. CONCLUDING REMARKS

An asymmetric parameter battery model and its parame-terization that is amenable to timely BSE development andverification have been presented. A unique MIL verificationmethodology has been presented and employed to evaluatethe proposed SOP algorithms. Novel SOP calculation meth-ods have been proposed; one method provides relatively sim-ple analytical expressions for peak current considering linearextrapolations of resistor parameter values and OCV. Moreaccurate MMPI methods have been also presented that em-ploy closed-form expressions of voltage response and multiplesubinterval time steps to improve computational efficiency. Themethods have been evaluated at relatively low-temperature sce-narios where voltage-limit-based SOP is relevant. The proposedMMPI methods had the best accuracy, whereas the new VLEROmethod had marginal improvement.


APPENDIX

VARIABLE NOMENCLATURE

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Pawel Malysz (S’06–M’12) received the B.Eng.degree in engineering physics and the M.A.Sc.and Ph.D. degrees in electrical engineering fromMcMaster University, Hamilton, ON, Canada, in2005, 2007, and 2011, respectively.

From 2012 to 2015, he was a PrincipalResearch Engineer with the McMaster Insti-tute for Automotive Research and Technology(MacAUTO), which is a Canada Excellence Re-search Centre. He is currently a Battery ManagementSystem Control Engineer with Fiat Chrysler Au-

tomobiles, Auburn Hills, MI, USA. His current research interests includeenergy systems, battery management software design, and advanced controlengineering. Dr. Malysz co-received the Chrysler Innovation Award in 2014.

Jin Ye (S’13–M’14) received the B.S. and M.S.degrees in electrical engineering from Xi’an JiaotongUniversity, Xi’an, China, in 2008 and 2011, respec-tively, and the Ph.D. degree in electrical engineeringfrom McMaster University, Hamilton, ON, Canada.

She is currently a Postdoctoral Research As-sociate with McMaster Institute for AutomotiveResearch and Technology (MacAUTO). Her mainresearch areas include advanced control methods forpower electronics and electric machines, automotivepower electronics, and energy management systems

for electrified vehicles.


Ran Gu (S’15) received the B.S. degree in automa-tion from Beijing Information Science and Tech-nology University, Beijing, China, in 2010 and theM.S. degree in electrical engineering from IllinoisInstitute of Technology (IIT), Chicago, IL, USA, in2012. In September 2012, he joined the McMasterInstitute for Automotive Research and Technology(MacAUTO), McMaster University, Hamilton, ON,Canada, where he is currently working toward thePh.D. degree in electrical engineering.

From June 2011 to May 2012, he served as aResearch Assistant of electrical engineering with IIT. His main areas of researchinterest are battery management systems, battery modeling, and hybrid energystorage systems.

Hong Yang received the B.S. degree in mechanicalengineering and the M.S. degree in robotics fromSoutheast University, Nanjing, China, and the M.S.degree in electrical engineering and the Ph.D. degreein mechanical engineering from the University ofMichigan, Ann Arbor, MI, USA, in 1998 and 2002,respectively.

He is currently with the Electrified PowertrainDepartment, Fiat Chrysler Automobiles, AuburnHills, MI, USA, as the Manager of the BatteryManagement System Group. He is also an Adjunct

Associate Professor with the Department of Electrical and Computer Engineer-ing, McMaster University, Hamilton, ON, Canada. Before joining Chrysler,he was with General Motors Powertrain as a Project Leader for advancedhybrid electric system development. He was responsible for developing GM’sfirst proof-of-concept plug-in hybrid electric vehicle. His research interest iselectrified power and advanced energy storage systems.

Dr. Yang joined the American Society of Mechanical Engineers Vehicle De-sign Committee in 2009 and is an Editorial Board Member of the InternationalJournal of Powertrain.

Ali Emadi (S’98–M’00–SM’03–F’13) received theB.S. and M.S. degrees in electrical engineering (withhighest distinction) from Sharif University of Tech-nology, Tehran, Iran, in 1995 and 1997, respectively,and the Ph.D. degree in electrical engineering fromTexas A&M University, College Station, TX, USA,in 2000.

He is the Canada Excellence Research Chair inHybrid Powertrain and the Director of the McMasterInstitute for Automotive Research and Technology(MacAUTO), McMaster University, Hamilton, ON,

Canada. Before joining McMaster University, he was the Harris PerlsteinEndowed Chair Professor of Engineering and the Director of the Electric Powerand Power Electronics Center and Grainger Laboratories, Illinois Instituteof Technology (IIT), Chicago, IL, USA. In addition, he was the Founder,Chairman, and President of Hybrid Electric Vehicle Technologies, Inc.: auniversity spin-off company of IIT. He is the principal author/coauthor ofmore than 350 journal and conference papers and several books, includingVehicular Electric Power Systems (2003); Energy Efficient Electric Motors(2004); Uninterruptible Power Supplies and Active Filters (2004); ModernElectric, Hybrid Electric, and Fuel Cell Vehicles, Second Edition (2009); andIntegrated Power Electronic Converters and Digital Control (2009). He is alsothe editor of the Handbook of Automotive Power Electronics and Motor Drives(2005) and Advanced Electric Drive Vehicles (2014).

Dr. Emadi was the inaugural General Chair of the 2012 IEEE TransportationElectrification Conference and Expo and has chaired several IEEE and Societyof Automotive Engineering conferences in the areas of vehicle power andpropulsion. He is the Editor-in-Chief of the IEEE TRANSACTIONS ON TRANS-PORTATION ELECTRIFICATION. He was the Advisor of the Formula HybridTeams at IIT and McMaster University, which won the GM Best EngineeredHybrid Systems Award at the 2010 and 2013 competitions, respectively. He hasreceived numerous awards and recognitions.

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