1 The Estimation of Temperature Distribution in Cylindrical Battery Cells under Unknown Cooling Conditions Youngki Kim, Student Member, IEEE Shankar Mohan, Student Member, IEEE, Jason B. Siegel, Anna G. Stefanopoulou, Fellow, IEEE, and Yi Ding Abstract—The estimation of temperature inside battery cells requires accurate information about the cooling conditions even when the temperature of the battery surface is measured. This paper presents a novel approach of estimating temperature dis- tribution inside cylindrical batteries under unknown convective cooling conditions. A computationally efficient thermal model is first developed using a polynomial approximation of the temperature profile inside the battery cell. The Dual Extended Kalman Filter (DEKF) is then applied for the identification of the convection coefficient and the estimation of temperature inside the battery. In the proposed modeling approach, the thermal properties are represented by volume averaged lumped values with uniformly distributed heat generation. The model is parameterized and validated using experimental data from a 2.3 Ah 26650 Lithium-Iron-Phosphate (LFP) battery cell with a forced-air convective cooling during hybrid electric vehicle (HEV) drive cycles. Experimental results show that the proposed DEKF- based estimation method can provide an accurate prediction of core temperature under unknown cooling condition by measuring the cell current, voltage, and surface and ambient temperature. The accuracy is such that the scheme cam also be used for fault detection of a cooling system malfunction. Index Terms—Lithium ion batteries, Thermal model, Reduced order model, Dual Extended Kalman Filter, Parameter identifi- cation I. INTRODUCTION O VER the past years, energy storage systems utilizing lithium ion (Li-ion) batteries have become one of the most critical components for realizing efficient and clean transportation systems through electrification of vehicles, e.g., hybrid electric vehicles (HEVs), plug-in hybrid electric ve- hicles (PHEVs), and electric vehicles(EVs). Li-ion batteries have several advantages – no memory effect, wide range of operating temperature, and high energy and power density [1], [2]. However, the Li-ion battery performance, cycle life and capacity are adversely affected by sustained operation at extreme (above 45 o C and below freezing) temperatures [3]– [6], a recurring problem in automotive applications where batteries are exposed to temperature extremes with frequent high current discharge/charge rate that cause internal heating. Y. Kim, J. B. Siegel, and A. Stefanopoulou are with the Department of Me- chanical Engineering, University of Michigan, Ann Arbor, MI, 48109 USA. e-mail: ([email protected], [email protected], [email protected]). S. Mohan is with the Department of Electrical Engineering, University of Michigan, Ann Arbor, MI, 48109 USA. e-mail: ([email protected]). Y. Ding is with U.S. Army Tank Automotive Research, Development, and Engineering Center (TARDEC), Warren, Michigan, 48397 USA. e- mail:([email protected]). Thus, being able to estimate/predict the temperature distri- bution across cells and packs is vital for formulating power management strategies that are mindful of the performance limitations of these versatile power/energy sources. In general, the performance of the cooling system can be degraded due to various reasons such as dust on fan blades, partial blockage in pipes, motor/pump ageing, and even a motor/pump failure. When such a degradation or failure occurs, it is not possible to reject the heat generated from the battery cell. In this condition, the lifespan of the battery exposed to the extreme temperatures will be considerably shortened. Therefore, it is important to identify the convective heat coefficient not only to accurately estimate the temperature distribution inside the battery but also for fault detection to ensure safe and reliable operation of the vehicle system. This paper considers a novel method for estimating temper- ature distribution inside cylindrical batteries with simultaneous estimation of the convective cooling condition. To achieve this goal, a computationally efficient thermal model for a cylindrical cell is utilized to estimate simultaneously the convection coefficient and radial temperature distribution of the cell. Unlike existing reduced order modeling approaches in [7]–[10], and [11], a polynomial approximation to the solution of the heat transfer problem is used; this approach facilitates a systematic estimation of core, surface, volume- averaged temperatures, and volume-averaged temperature gra- dients. Dual Extended Kalman Filter (DEKF) is applied for the identification of the convection coefficient and the temperature distribution inside a cylindrical battery cell. The proposed estimation method provides the capability of detecting the malfunction of cooling system by monitoring the difference between the identified and off-line predetermined convection coefficient. This benefit indicates that a significant rise in temperature can be prevented by augmenting the proposed method with other existing battery management strategies for the safe and robust operation. This paper is organized as follows: Section II presents the convective heat transfer problem for a cylindrical battery cell and the reduced order thermal model. Model reduction is performed using a polynomial approximation of the partial differential equation (PDE) system. The thermal properties of the battery are experimentally identified and sensitivity of parameters is analyzed in Section III. In Section IV, the temperature estimator applying a Dual Extended Kalman filter by using the proposed model for estimating the core
Transcript
1
The Estimation of Temperature Distribution inCylindrical Battery Cells under Unknown Cooling
ConditionsYoungki Kim, Student Member, IEEE Shankar Mohan, Student Member, IEEE, Jason B. Siegel, Anna G.
Stefanopoulou, Fellow, IEEE, and Yi Ding
Abstract—The estimation of temperature inside battery cellsrequires accurate information about the cooling conditions evenwhen the temperature of the battery surface is measured. Thispaper presents a novel approach of estimating temperature dis-tribution inside cylindrical batteries under unknown convectivecooling conditions. A computationally efficient thermal modelis first developed using a polynomial approximation of thetemperature profile inside the battery cell. The Dual ExtendedKalman Filter (DEKF) is then applied for the identification ofthe convection coefficient and the estimation of temperatureinside the battery. In the proposed modeling approach, thethermal properties are represented by volume averaged lumpedvalues with uniformly distributed heat generation. The modelis parameterized and validated using experimental data from a2.3 Ah 26650 Lithium-Iron-Phosphate (LFP) battery cell with aforced-air convective cooling during hybrid electric vehicle (HEV)drive cycles. Experimental results show that the proposed DEKF-based estimation method can provide an accurate prediction ofcore temperature under unknown cooling condition by measuringthe cell current, voltage, and surface and ambient temperature.The accuracy is such that the scheme cam also be used for faultdetection of a cooling system malfunction.
Index Terms—Lithium ion batteries, Thermal model, Reducedorder model, Dual Extended Kalman Filter, Parameter identifi-cation
I. INTRODUCTION
OVER the past years, energy storage systems utilizinglithium ion (Li-ion) batteries have become one of the
most critical components for realizing efficient and cleantransportation systems through electrification of vehicles, e.g.,hybrid electric vehicles (HEVs), plug-in hybrid electric ve-hicles (PHEVs), and electric vehicles(EVs). Li-ion batterieshave several advantages – no memory effect, wide range ofoperating temperature, and high energy and power density[1], [2]. However, the Li-ion battery performance, cycle lifeand capacity are adversely affected by sustained operation atextreme (above 45oC and below freezing) temperatures [3]–[6], a recurring problem in automotive applications wherebatteries are exposed to temperature extremes with frequenthigh current discharge/charge rate that cause internal heating.
Y. Kim, J. B. Siegel, and A. Stefanopoulou are with the Department of Me-chanical Engineering, University of Michigan, Ann Arbor, MI, 48109 USA.e-mail: ([email protected], [email protected], [email protected]).
S. Mohan is with the Department of Electrical Engineering, University ofMichigan, Ann Arbor, MI, 48109 USA. e-mail: ([email protected]).
Y. Ding is with U.S. Army Tank Automotive Research, Development,and Engineering Center (TARDEC), Warren, Michigan, 48397 USA. e-mail:([email protected]).
Thus, being able to estimate/predict the temperature distri-bution across cells and packs is vital for formulating powermanagement strategies that are mindful of the performancelimitations of these versatile power/energy sources. In general,the performance of the cooling system can be degraded dueto various reasons such as dust on fan blades, partial blockagein pipes, motor/pump ageing, and even a motor/pump failure.When such a degradation or failure occurs, it is not possibleto reject the heat generated from the battery cell. In thiscondition, the lifespan of the battery exposed to the extremetemperatures will be considerably shortened. Therefore, it isimportant to identify the convective heat coefficient not onlyto accurately estimate the temperature distribution inside thebattery but also for fault detection to ensure safe and reliableoperation of the vehicle system.
This paper considers a novel method for estimating temper-ature distribution inside cylindrical batteries with simultaneousestimation of the convective cooling condition. To achievethis goal, a computationally efficient thermal model for acylindrical cell is utilized to estimate simultaneously theconvection coefficient and radial temperature distribution ofthe cell. Unlike existing reduced order modeling approachesin [7]–[10], and [11], a polynomial approximation to thesolution of the heat transfer problem is used; this approachfacilitates a systematic estimation of core, surface, volume-averaged temperatures, and volume-averaged temperature gra-dients. Dual Extended Kalman Filter (DEKF) is applied for theidentification of the convection coefficient and the temperaturedistribution inside a cylindrical battery cell. The proposedestimation method provides the capability of detecting themalfunction of cooling system by monitoring the differencebetween the identified and off-line predetermined convectioncoefficient. This benefit indicates that a significant rise intemperature can be prevented by augmenting the proposedmethod with other existing battery management strategies forthe safe and robust operation.
This paper is organized as follows: Section II presents theconvective heat transfer problem for a cylindrical battery celland the reduced order thermal model. Model reduction isperformed using a polynomial approximation of the partialdifferential equation (PDE) system. The thermal propertiesof the battery are experimentally identified and sensitivityof parameters is analyzed in Section III. In Section IV,the temperature estimator applying a Dual Extended Kalmanfilter by using the proposed model for estimating the core
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1. REPORT DATE 06 MAR 2013
2. REPORT TYPE Journal Article
3. DATES COVERED 06-09-2012 to 02-03-2013
4. TITLE AND SUBTITLE The Estimation of Temperature Distribution in Cylindrical BatteryCells under Unknown Cooling Conditions
5a. CONTRACT NUMBER W56HZV-04-2-0001
5b. GRANT NUMBER
5c. PROGRAM ELEMENT NUMBER
6. AUTHOR(S) Youngki Kim; Shankar Mohan; Jason Siegel; Anna Stefanopoulou; Yi Ding
5d. PROJECT NUMBER
5e. TASK NUMBER
5f. WORK UNIT NUMBER
7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) UNIVERSITY OF MICHIGAN,4260 Plymouth Road,Ann Arbor,Mi,48109
8. PERFORMING ORGANIZATION REPORT NUMBER ; #23698
9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) U.S. Army TARDEC, 6501 East Eleven Mile Rd, Warren, Mi, 48397-5000
10. SPONSOR/MONITOR’S ACRONYM(S) TARDEC
11. SPONSOR/MONITOR’S REPORT NUMBER(S) #23698
12. DISTRIBUTION/AVAILABILITY STATEMENT Approved for public release; distribution unlimited
13. SUPPLEMENTARY NOTES IEEE Transactions on Vehicles Journal
14. ABSTRACT The estimation of temperature inside battery cells requires accurate information about the coolingconditions even when the temperature of the battery surface is measured. This paper presents a novelapproach of estimating temperature distribution inside cylindrical batteries under unknown convectivecooling conditions. A computationally efficient thermal model is first developed using a polynomialapproximation of the temperature profile inside the battery cell. The Dual Extended Kalman Filter(DEKF) is then applied for the identification of the convection coefficient and the estimation oftemperature inside the battery. In the proposed modeling approach, the thermal properties arerepresented by volume averaged lumped values with uniformly distributed heat generation. The model isparameterized and validated using experimental data from a 2.3 Ah 26650 Lithium-Iron-Phosphate (LFP)battery cell with a forced-air convective cooling during hybrid electric vehicle (HEV) drive cycles.Experimental results show that the proposed DEKFbased estimation method can provide an accurateprediction of core temperature under unknown cooling condition by measuring the cell current, voltage,and surface and ambient temperature. The accuracy is such that the scheme cam also be used for faultdetection of a cooling system malfunction.
15. SUBJECT TERMS Lithium ion batteries, Thermal model, Reduced order model, Dual Extended Kalman Filter, Parameter identification
Standard Form 298 (Rev. 8-98) Prescribed by ANSI Std Z39-18
2
L= 65
.15
R 12.93
r
h
T∞
Heat generation
Temperature
r
rR
R
0
0
Tc
Ts
Fig. 1. Schematic for a A123 26650 cylindrical battery cell
temperature and identifying the convection coefficient. SectionV presents and discusses experimental results and conclusionsare drawn in Section VI.
II. HEAT TRANSFER PROBLEM IN CYLINDRICALBATTERIES
This paper considers the radially distributed (1-D) thermalbehavior of a cylindrical battery cell with convective heattransfer boundary condition as illustrated in Fig. 1 [7], [12],[13]. A cylindrical Li-ion battery, so-called a jelly-roll, is fab-ricated by rolling a stack of cathode/separtor/anode layers. Theindividual layered sheets are thin, therefore, it is reasonableto assume uniform heat generation along the radial direction[13], [14]. Lumped parameters are used so that materialproperties such as thermal conductivity, density, and specificheat coefficient are assumed to be constant in a homogeneousand isotropic body. The thermal conductivity is one or twoorders of magnitude higher in the axial direction than inthe radial direction. Therefore, the temperature distributionin the axial direction will be more uniform [15], [16]. Thegoverning equation of the 1-D temperature distribution T (r, t)and boundary conditions are given by
ρcp∂T (r, t)
∂t= kth
∂2T (r, t)
∂r2+kthr
∂T (r, t)
∂r+Q(t)
Vb, (1)
B.C.’s∂T (r, t)
∂r
∣∣∣r=0
= 0, (2)
∂T (r, t)
∂r
∣∣∣r=R
= − h
kth(T (R, t)− T∞), (3)
where t, ρ, cp and kth represent time, volume-averaged den-sity, specific heat coefficient, and thermal conductivity of thecell respectively. The radius of the battery cell is R, Q isthe heat generation inside the cell, and Vb is the volume ofbattery cell. Ambient temperature for convection is denoted byT∞. The boundary condition in (2) represent the symmetricstructure of the battery about the core. The other boundarycondition shown in (3) represents the convective heat transferat the surface of the battery.
A. Model reduction
With evenly distributed heat generation, the temperaturedistribution along r-direction of the battery cell is assumed
to satisfy the following polynomial approximation proposedin [17]
T (r, t) = a(t) + b(t)( rR
)2
+ d(t)( rR
)4
, (4)
where a(t), b(t), and d(t) are time-varying constants. Tosatisfy the symmetric boundary condition at the core ofthe battery cell, (4) contains only even powers of r. Thus,the temperatures at core and surface of the battery can beexpressed as
where subscripts c and s denote core and surface respectively.The volume-averaged temperature T and temperature gra-
dient γ are introduced as follows:
T =2
R2
∫ R
0
rTdr, (7)
γ =2
R2
∫ R
0
r
(∂T
∂r
)dr. (8)
These volume-averaged values are used as the states unlikeexisting approaches in [8], [9], and [11].
By substituting (4) in (7) and (8), T and γ can be expressedin terms of constants as
T = a(t) +b(t)
2+d(t)
4, (9)
γ =4b(t)
3R+
8d(t)
5R. (10)
By rearranging (6), (9), and (10), time-varying constantsa(t), b(t), and d(t) can be written by
a(t) = 4Ts − 3T − 15R
8γ, (11)
b(t) = − 18Ts + 18T +15R
2γ, (12)
d(t) = 15Ts − 15T − 45R
8γ. (13)
By substituting (11), (12), and (13) in (4), the temperaturedistribution can be expressed as a function of Ts, T , and γ
T (r, t) = 4Ts − 3T − 15R
8γ
+
[−18Ts + 18T +
15R
2γ
]( rR
)2
(14)
+
[15Ts − 15T − 45R
8γ
]( rR
)4
.
The PDE (1) can be converted into ODEs by substituting(14) in volume-averaged governing equation and its partialderivative with respect to r as follows:
dT
dt+
48α
R2T − 48α
R2Ts +
15α
Rγ − α
kthVbQ = 0, (15)
dγ
dt+
320α
R3T − 320α
R3Ts +
120α
R2γ = 0, (16)
where α is thermal diffusivity and is defined as follows:
α = kth/ρcp. (17)
3
10-6
10-4
10-2
100
101
-150
-100
-50
0
50
Magnitude (dB)
Analytical sol.
polynomial app.
10-6
10-4
10-2
-150
-100
-50
0
50
Magnitude (dB)
10-6
10-4
10-2
100
101
-150
-100
-50
0
50
frequency (Hz)
Magnitude (dB)
10-6
10-4
10-2
-150
-100
-50
0
50
frequency (Hz)
Magnitude (dB)
H11(s) H12(s)
H21(s) H22(s)
Fig. 2. Comparison of frequency response functions between analyticalsolution and polynomial approximation
Using (3), the surface temperature Ts can be rewritten as
Ts =24kth
24kth +RhT +
15kthR
48kth + 2Rhγ +
Rh
24kth +RhT∞.
(18)
Finally, a two-state thermal model can be given by thefollowing form:
x =Ax+Bu,
y = Cx+Du, (19)
where x = [T γ]T , u = [Q T∞]T and y = [Tc Ts]T are
states, inputs and outputs respectively. System matrices A, B,C, and D are defined as follows:
A =
[−48αh
R(24kth+Rh)−15αh
24kth+Rh−320αh
R2(24kth+Rh)−120α(4kth+Rh)R2(24kth+Rh)
],
B =
[α
kthVb
48αhR(24kth+Rh)
0 320αhR2(24kth+Rh)
],
C =
[24kth−3Rh24kth+Rh − 120Rkth+15R2h
8(24kth+Rh)24kth
24kth+Rh15Rkth
48kth+2Rh
],
D =
[0 4Rh
24kth+Rh
0 Rh24kth+Rh
]. (20)
This state-space representation is used for the development ofcontrol design.
B. Frequency domain analysisThe transfer function of the thermal system H(s) is calcu-
lated by
H(s) = D + C(sI −A)−1B, (21)
where s is Laplace variable.The frequency response of transfer function of the proposed
model is compared to that of the analytical solution in [7]. Pa-
TABLE IPARAMETERS OF THE BATTERY [7]
Parameter Symbol Value UnitDensity ρ 1824 kg/m3
Specific heat coeff. cp 825 J/kgKThermal conductivity kth 0.488 W/mK
Convection coeff. h 5 W/m2-KRadius R 12.93e-3 mHeight L 65.15e-3 mVolume Vb 3.4219e-5 m3
rameters used to generate the plots in Fig. 2 are summarized inTable I. The heat transfer coefficient of h=5W/m2K is chosensince this value is typical of natural convection condition [18].
Figure 2 shows that the effects of heat generation on coreand surface temperature, denoted by H11(s) and H21(s) re-spectively, can be accurately predicted over the whole range offrequency. On the other hand, the responses of core and surfacetemperature excited by the ambient temperature, H12(s) andH22(s), are nearly identical to the analytical solution forfrequencies below 10−2 Hz. In general, the temperature ofcooling media does not change rapidly; thus, the prediction oftemperature distribution using the proposed approach can beconsidered sufficiently accurate.
C. Heat generation calculation
Since heat generation rate Q is the input to the batterythermal system, the input needs to be accurately calculatedfrom measurement data, such as current and voltage duringoperation. In [19], Bernardi et al. proposed the simplified formof heat generation rate with assumptions that heat generationdue to enthalpy-of-mixing, phase-change, and heat capacityare assumed to be negligible expressed by
Q = i(U − V )− i(T∂U
∂T
), (22)
where i, U , and V represent the current, the open-circuitvoltage (OCV), and the terminal voltage respectively. Asshown in Fig 3, the OCV is a function of the battery state-of-charge(SOC). This function is experimentally obtained byaveraging the measured terminal voltages during chargingand discharging a battery with C/20 current rate under aConstant Current Constant Voltage (CCCV) charging protocol.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
2
2.4
2.8
3.2
3.6
SOC
Voltage (V)
Charging
Discharging
Average(OCV)0.4 0.5
Fig. 3. Open-circuit voltage approximately obtained by averaging terminalvoltages during charging and discharging a battery with C/20 current rate
Fig. 4. Data set used for parameter ID: current (top), voltage (middle), andheat generation rate (bottom) during Urban-Assault Cycle
The OCV is then calculated at the estimated SOC value byintegrating measured current with respect to time as
dSOCdt
= − I
3600Qb(23)
where Qb is a battery capacity in Ah. The sign convention issuch that positive current denotes battery discharging.
The last term in (22) is the heat generation from entropychange. In this paper, heat generation due to entropy changeis neglected for simplicity. This simplification is warrantedsince the typical SOC range of HEV operation is between40% and 60% in which ∂U
∂T of the battery cell is insignificantas shown in [9] for this chemistry. In addition, the reversibleentropic heat generation would have zero mean value whenthe battery is operating in charge-sustaining mode, typical ofHEV operation.
III. PARAMETER IDENTIFICATION
In this section, the value of the lumped parameters in (19)for a 2.3 Ah 26650 LFP battery cell by A123 are identifiedthrough experimentation. Figure 4 shows current, voltage andcalculated heat generation rate profiles over power demanding
TABLE IIIDENTIFIED THERMAL PROPERTIES
Parameter Symbol Value ReferenceDensity ρ 2047* 2118 [20]
Fig. 5. Comparison between measured and simulated temperatures (top)and errors (bottom)
cycle used for military ground vehicle [22] Urban-AssaultCycle (UAC) that are used for the parametrization. The modelis then validated using a different duty cycle. The numericalanalysis on parameter sensitivity is performed to investigatethe use of constant parameters for thermal conductivity andheat capacity and the importance of identifying the convectioncoefficient on-line.
A. Identifying thermal properties
Parameter identification is important for accurately pre-dicting the temperature distribution inside a battery cell asthe parameters kth, cp, ρ, and h determine the dynamicsof thermal model. As the density can be assumed to bea measurable constant, only three parameters such as kth,cp, and h are considered for the parameter identification.Following an experimental set-up in [21], we measured a cellcurrent, voltage, surface and core temperature of the batterycell, and ambient temperature for parameter identification. Asdiscussed in [21], some parameters are assumed to be knownsince all parameters are not identifiable without using the coretemperature.
Let the error between the measured temperatures and modeloutputs at each time step k in vector form be
Fig. 6. (a) Expected temperature distribution along the normalized radius (r/R) using polynomial approximation; (b) Cell temperature (top) and temperaturegradient (bottom)
MATLAB; the parameters in Table I are used as initial guessfor the identification.
Table II presents the identified thermal properties for theA123 26650 battery; these parameters are close to the valuespresented in the literature. The identified specific heat coeffi-cient cp is 4% larger than the maximum value determined in[9] where cp was determined by measuring transient responsesof the battery under different pulses. It was discussed thatthe deviation in identified value of cp might be caused bymeasurement uncertainty in temperature and the temperaturedependency of the heat capacity.
Despite using similar experimental data and setup, the iden-tified convection coefficient is 10% smaller than the coefficientcalculated by using thermal resistance and battery surfacearea in [21]. This difference between our identified valueand the one in [21] may be due to the two different modelstructures. Lin et al. in [21] considered two different materials,namely one for the core and one for the surface, whereaswe assume the battery is a homogeneous and isotropic body.In order to accurately determine the convection coefficient,the temperature measurements of a pure metal during thermalrelaxation can be used. For instance, the specific heat capacityof copper at 25oC is known as 385 J/kgK. For more detaileddescription about the experiment, the interested reader isreferred to [23].
Figure 5 shows the measured and simulated temperaturesat the core and surface of the battery. The error betweenthe measurements and simulated temperature is less than thesensor accuracy of 0.5oC. Thermocouples used for temperaturemeasurements are T-type whose accuracy is the maximumof 0.5oC or 0.4% according to technical information fromthe manufacture, OMEGA. The parameterized thermal modelaccurately predicts the temperature inside the battery, which isdifficult to measure in practical applications. Using (15), thetemperature distribution inside the battery can be predicted aspresented in Fig 6(a). Figure 6(b) shows the volume-averagedtemperature and its gradient of the battery respectively. As
evidenced in Fig. 6(b), there is no significant difference be-tween the volume-averaged temperature and the linear averageof core and surface temperatures, i.e. (Ts + Tc)/2. It shouldbe noted that existing approaches in [8], [9], [11] have thecapability of predicting the core temperature and have shownthe efficacy of their proposed methods on the prediction oftemperature inside the cell under consideration in this work.However, the phenomena may differ in the case of a cellwith larger radius [24]. The volume-averaged temperaturegradient is different from the linear temperature gradient, i.e.(Ts − Tc)/R; in particular, the volume-averaged temperaturegradient is 1.36 times greater than linear temperature gradientunder the UAC test. Since non-uniform temperature distri-bution can lead to accelerated capacity losses of inner core[24], the volume-averaged temperature gradient is an importantmetric to describe severity of temperature inhomogeneityinside the battery.
B. Model validation
In order to validate the performance of the proposed modelwith identified parameters, the battery was tested under adifferent HEV drive cycle, the Escort Convoy Cycle (ECC).The current and voltage profiles for this cycle are illustratedin Fig 7(a). Figure 7(b) shows that there are slight differencesbetween the measured and simulated temperatures; in partic-ular, the root-mean-square errors (RMSE) of core and surfacetemperatures are 0.4 and 0.3oC respectively. These differencesmay be explained with the assumption of radially uniformheat generation and high conductivity in the axial direction.Additionally, the hysteresis effect of the LFP battery is notproperly considered in heat generation formulation (22), whichmight introduce error in the calculation of heat generationrate. Nevertheless, since the comparison of temperatures showsgood agreement and reasonably small RMSEs, it can beconcluded that the proposed model with identified thermalproperties is sufficiently accurate for HEV drive cycles.
6
0 1000 2000 3000 4000 5000 6000 7000-40
-20
0
20
40
Current(A)
0 1000 2000 3000 4000 5000 6000 70002.5
3
3.5
Voltage(V)
Time(sec)
(a)
0 1000 2000 3000 4000 5000 6000 700025
30
35
40
Temperature ( oC)
Tc,mT
c,e
Ts,m
Ts,e
0 1000 2000 3000 4000 5000 6000 7000-2
-1
0
1
2
Time(sec)
Temperature ( oC)
Ts,m-Ts,e
Tc,m-Tc,e
(b)
Fig. 7. (a) Current (top) and voltage (bottom) profiles during Escort Convoy cycle; (b) Validation data set: comparison between measured and simulatedtemperatures (top) and errors (bottom)
C. Parameter sensitivity analysis
In order to investigate the impact of parameter variationson the performance of temperature prediction, each parameteris varied from the identified value while holding the otherparameters constant. Figure 8 shows that parameters suchas thermal conductivity kth and specific heat capacity cphave more influence on the prediction of core temperaturethan surface temperature. This result corresponds to the factthat the heat inside the battery cell is transferred throughthe conduction. On the other hand, the prediction of surfacetemperature is most sensitive to the variation of convectioncoefficient, which can be explained given the fact that theconvection coefficient is directly related to the convectiveboundary condition (3). The convection coefficient has themost significant influence on the overall prediction of the coreand surface temperature.
The specific heat coefficient and thermal conductivity are
-20 -15 -10 -5 0 +5 +10 +15 +200
0.2
0.4
0.6
0.8
1
Temperature (oC)
k
cp
h
-20 -15 -10 -5 0 +5 +10 +15 +200
0.2
0.4
0.6
0.8
1
Variation (%)
Temperature (oC)
RMSE (Core)
RMSE (Surface)
Fig. 8. The effect of parameter variation to the prediction of core and surfacetemperatures
weakly dependent on temperature [9], [25], [26], so theassumption of constant parameters can be justified. On theother hand, the convection coefficient is highly dependenton fan speed or fluid velocity as expressed by empiricalcorrelations provided by Zukauskas [27]. Consequently, theaccurate identification of convection coefficient is importantfor better prediction of temperature inside the battery. Thisimportance justifies the on-line identification of the convectioncoefficient for better estimation of temperature as detailed inSection IV.
IV. ESTIMATION OF TEMPERATURE ANDCONVECTION COEFFICIENT
As discussed in section III-C, the estimation of temperatureinside the battery cell requires accurate knowledge of theconvection coefficient which depends on cooling condition.In order to identify the convection coefficient on-line, theDual Extended Kalman filter (DEKF) [28] is applied for betterestimation of temperature distribution inside the battery cell.The other thermal parameters, such as thermal conductivityand specific heat coefficient, have less impact on temperatureand do not change significantly over time. Therefore, theconstant values identified in section III can be used.
Assuming the input u(t) is constant over each samplinginterval, a parameter varying (PV) discrete-time model at timestep k can be obtained as
xk+1 = Ad(θk)xk +Bd(θk)uk,
yk = C(θk)xk +D(θk)uk, (26)
where x = [T γ]T , y = [Tc Ts]T , θ = h, and u = [Q T∞]T .
System matrices Ad ≈ I+A∆T and Bd = B∆T are obtainedfrom matrices in (20) where the sampling period is ∆T , andI is the identity matrix.
Let the PV thermal system in discrete-time domain be
7
0 1000 1920 3000 4000 5000 6000 7000
50
75
100
PWM (%)
Time(sec)
Stage I Stage II Stage IIIStage I Stage II Stage III
Fan signal
Fig. 9. Fan schedule for forced-air convective cooling
expressed in a general form by:
xk+1 = f(xk, uk, θk) + wk,
yk = g(xk, uk, θk) + vk,
θk+1 = θk + rk, (27)
where wk, vk, and rk, are independent, zero-mean, Gaussiannoise processes of covariance matrices Σw, Σv , and Σr,respectively. The design of the DEKF estimator is given asfollowing update processes.Time update for the parameter filter:
θ−k = θ+k−1, (28)
S−k = S+k−1 + Σr. (29)
Time update for the state filter:
x−k = f(x+k−1, uk−1, θ
−k ), (30)
P−k =Ak−1P+k−1A
Tk−1 + Σw. (31)
Measurement update for the state filter:
Kk = P−k CxkT[CxkP
−k C
xkT + Σv
]−1
, (32)
x+k = x−k +Kk
[yk − g(x−k , uk, θ
−k )], (33)
P+k = [I −KkC
xk ]P−k . (34)
Measurement update for the parameter filter:
Lk = S−k Cθk
T[CθkP
−k C
θk
T+ Σe
]−1
, (35)
θ+k = θ−k + Lk
[yk − g(x−k , uk, θ
−k )], (36)
S+k =
[I − LkCθk
]S−k , (37)
where the matrices are calculated by
Ak−1 =∂f(xk−1, uk−1, θ
−k )
∂xk−1|xk−1=x+
k−1(38)
Cxk =∂g(xk, uk, θ
−k )
∂xk|xk=x−
k, (39)
Cθk =dg(x−k , uk, θ)
dθ|θ=θ−k . (40)
The details of the matrices are provided in Appendix A.Superscripts − and + denote the a priori and a posteriorivalues respectively.
The identified states x and parameter θ, computed from theabove DEKF algorithm, are used to estimate the core temper-ature in the battery cell from (26). The identified parametercan be also used for monitoring the malfunction or degradation
0 200 400 600 800 1000 1200 1400 1600 1800192040
60
80
100
Convection coefficient
Time(sec)
0 200 400 600 800 1000 1200 1400 1600 1800192025
26
27
28
29
30
31
32
Temperature (oC)
Time(sec)
30.8
31
28.8
28.9
29
θ
*θ θ=ɶ
,c mT
,ˆc KFT
,ˆc DEKFT
,s mT
,ˆs DEKFT
,ˆs KFT
Fig. 10. Comparison of performance between KF estimator, DEKF estimatorduring stage I: convection coefficient (top) and temperature (bottom)
of cooling system. Under the assumption that the relationshipbetween the convection coefficient and fan speed or PWMsignal is known, the malfunction of the cooling system canbe detected by comparing the identified parameter with apredetermined range of values θ∗ based on the identificationprocess performed at various cooling conditions. When thedifference between the identified and predetermined values|θ− θ∗| is bounded and small, it can be considered that thereis no fault in the cooling system. On the other hand, where|θ − θ∗| � ε where ε is a pretuned threshold, a cooling faultcan be detected. Furthermore, |(θ−θ∗)/θ∗| can be interpretedas the severity of degradation of cooling system.
V. EXPERIMENTAL RESULTS
In this section, the performance of the proposed temperatureestimator using the DEKF is compared with that of thebaseline Kalman Filter (KF) estimator without parameter iden-tification. The battery is tested using the ECC under differentcooling conditions. Three different forced convective coolingconditions (stage I, stage II, and stage III) are achieved byusing different PWM signals driving the fan as shown in Fig.9 which corresponds to an increase, followed by a decreasein the coolant flow rate. In order to investigate the influenceof change in the parameter on the temperature estimation, theparameter is provided to each estimator as following:• In stage I, the off-line predetermined convection coeffi-
cient is provided to the KF and is used for the DEKF asan initial value: θ = θ∗ and θ(0) = θ∗
• In stage II, the off-line predetermined convection coeffi-cient is provided to the KF only: θ = θ∗
• In stage III, two times larger convection coefficient com-pared to the known value is provided to the KF: θ = 2θ∗
where θ and θ denote fixed and identified parameters forthe KF and the DEKF respectively. Other thermal propertiessuch as thermal conductivity and specific heat coefficient areassumed constant with values identified in section III.
8
1920 2500 3000 3500 4000 4500 500040
60
80
100Convection coefficient
Time(sec)
1920 2500 3000 3500 4000 4500 5000
26
28
30
32
34
36
Temperature (oC)
Time(sec)
28.06
28.08
θ
*θ θ=ɶ
,c mT
,ˆc KFT
,ˆc DEKFT
,s mT
,ˆs DEKFT
,ˆs KFT
Fig. 11. Comparison of performance between KF estimator, DEKF estimatorduring stage II: convection coefficient (top) and temperature (bottom)
It is assumed that the initial temperature distribution insidethe battery is uniform at 30oC and convection coefficient is56.2 W/m2K, i.e. x(0) = [30 0]T and θ(0) = 56.2 respec-tively. The covariance matrix for the state Σw = β1
2diag(1, 1)describes the process noise where β > 0 is a parameterfor tuning relaxed to model inaccuracy. The noise covarianceΣv = σ2 is determined from the standard deviation oftemperature signal σ = 0.05oC. The covariance matrix forthe parameter Σr = β2
2 influences the performance of noisefiltering and the rate of parameter convergence. Ultimately, theinitial condition of the error covariance matrix and the tuningparameter are chosen as P (0) = diag(1, 1), β1 = 0.0005,S(0) = 1, and β2 = 0.01 through repeated simulationsrespectively.
The results for the parameter and state estimation areshown in Fig. 10–12 and summarized in Table III. Figure 10shows that the closed loop estimators can accurately predicttemperature inside the battery. Even though the identifiedvalue is used as an initial guess for the parameter, it can benoticed that there is a large deviation in the initial part of thesimulation. This deviation is caused by the error in the initialstates. Nevertheless, the on-line identified parameter is closeto the off-line determined value. Therefore, the performanceof DEKF estimator is comparable to that of KF estimator.In particular, the RMSE for core temperature estimation byDEKF is 0.26, the same RMSE by the KF. Even though thereis a slight error between the measured and estimated temper-ature, the closed loop estimators show a good performance inpredicting core temperature overall. As discussed in SectionIII-C, thermal properties can vary with respect to operatingtemperature. Therefore, it is expected that better performancecan be achieved by using temperature-dependent parametersfor thermal conductivity and specific heat coefficient.
Figure 11 illustrates the performance of temperature estima-tion by the closed estimator in stage II when there are suddenchanges in the cooling condition. The KF can accuratelyestimate the core temperature with information about the
5000
20
40
60
80
Convection coefficient
Time(sec)
5000 5500 6000 6500 7000
30
35
40
45
Temperature (oC)
Time(sec)
34.1
34.15
34.2
36.5
37
θ
*θ
,c mT
,ˆc KFT
,ˆc DEKFT
,s mT
,ˆs DEKFT
,ˆs KFT
*2θ θ=ɶ
Fig. 12. Comparison of performance between KF estimator, DEKF estimatorduring stage III: convection coefficient (top) and temperature (bottom)
change in parameter value. Since the DEKF is capable ofcompensating inaccuracy in the parameter of the system, theDEKF provides reasonably accurate estimates for the coretemperature by comparing the core temperature predicted bythe KF estimator. Even though the RMSE for core temperatureestimation by DEKF is slightly larger than the RMSE by theKF, the error is still reasonably small with considerations ofthe sensor accuracy.
As seen from Fig. 12, the KF estimator overestimatesthe core temperature when the incorrect parameter value isused as a convection coefficient. In other words, the reliableestimation of core temperature is only possible when theaccurate parameter is available. Thus, it can be concludedthat the DEKF estimator outperforms the KF estimator due tothe capability of parameter identification. The RMSE for coretemperature estimation in stage III can be substantially reducedfrom 1.18 to 0.31 by the DEKF. It is worthy to note that theDEKF can be augmented with other existing fault detectionmethods and power management strategies to improve thesystem robustness without cost increase. For instance, in orderto detect partial blockage in cooling system, typically, a massflow or pressure sensor is required. The DEKF enables theidentification of convection coefficient by using sensors whichare already equipped with the battery. Thus, by monitoring thedifference between the identified and off-line predeterminedvalues, the malfunction of cooling system can be detected.
TABLE IIIPERFORMANCE OF TEMPERATURE ESTIMATION: RMSES FOR CORE AND
SURFACE
Method DEKF KF
Location Core Surface Core SurfaceStage I 0.26 0.07 0.26 0.07Stage II 0.39 0.08 0.29 0.08Stage III 0.31 0.11 1.18 0.15
9
This performance tells that a significant rise in temperature canbe prevented by limiting maximum discharge/charge currentrate computed from the estimated parameter and states. Thus,the operation of the battery system can be safe and robust.
VI. CONCLUSION
In this study, a method to estimate the temperature distri-bution in cylindrical batteries under unknown cooling con-dition is proposed. First, a radially distributed 1-D thermalmodeling approach for a cylindrical battery cell is consideredand polynomial approximation is applied to obtain a reducedorder model enabling the development of real-time applica-tions. Frequency domain analysis shows that the proposedmodel provides sufficiently accurate prediction of core andsurface temperatures with a reasonable assumption that thetemperature of cooling media does not change rapidly. Theproposed model is used to identify thermal properties andconvective coefficient for a 2.3 Ah 26650 LFP battery cellusing a set of measured data. The numerical analysis onparameter sensitivity supports the use of constant parametersfor thermal conductivity and heat capacity and the impor-tance of identifying the convection coefficient on-line. Then,the Dual Extended Kalman Filter is applied to estimate thetemperature inside the battery and convection coefficient bythe cooling fan. The proposed method requires no knowledgeof the convective cooling conditions. The results shows thatthe proposed DEKF estimator can provide reasonably accurateestimates of core temperature and convection coefficient byusing surface temperature which is relatively easy to measurein practice. In addition, a faulty operation in the cooling systemcan be detected by monitoring the difference between theidentified and off-line predetermined values. Since forced airis used as a cooling media to reject heat from the cell in thispaper, the range of the convection coefficient of which we areinterested is less than 100 W/m2K. Therefore, the reader isurged to investigate whether the polynomial approximation isvalid for their applications.
In the future, the proposed method can be used to developvarious battery management strategies, e.g. the determinationof maximum current with consideration of thermal constraintsor optimal fan scheduling for energy efficiency, leading to thesafe and efficient operation of the battery system.
APPENDIX A
The matrices Adk−1,Cxk , and Cθk are calculated by
Adk−1 =
1− 48αθ−kR(24kth+Rθ−k )
−15αθ−k24kth+Rθ−k
−320αθ−kR2(24kth+Rθ−k )
1− 120α(4kth+Rθ−k )
R2(24kth+Rθ−k )
,Cxk =
24kth−3Rθ−k24kth+Rθ−k
− 120Rkth+15R2θ−k8(24kth+Rθ−k )
24kth24kth+Rθ−k
15Rkth48kth+2Rθ−k
,Cθk =
[Ψ11 Ψ12
]x−k−1 +
[Φ11 Φ12
]uk−1,
where
Ψ11 = − 24kth(24cpρR
2kth + 3552∆Tk2th) + θ−k (cpρR3 − 148∆TkthR)
cpρR(24kth + Rθ−k )3
Ψ12 = − 15kth(24cpρR
2kth + 3072∆Tk2th) + θ−k (cpρR3 − 168∆TkthR)
2cpρ(24kth + Rθ−k )3
Ψ11 = −24R∆Tkth
Vbcpρ(24kth + Rθ−k )2
Ψ12 = 24kth(24cpρR
2kth + 3552∆Tk2th) + θ−k (cpρR3 − 148∆TkthR)
Rcpρ(24kth + Rθ−k )3
ACKNOWLEDGMENT
The authors wish to acknowledge the technical and fi-nancial support of the Automotive Research Center (ARC)in accordance with Cooperative Agreement W56HZV-04-2-0001 U.S. Army Tank Automotive Research, Development andEngineering Center (TARDEC) Warren, MI.
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PLACEPHOTOHERE
Youngki Kim (S’13) is currently a Ph.D. Studentin the Department of Mechanical Engineering atthe University of Michigan, Ann Arbor, MI. Hereceived his BS and MS degrees from the Schoolof Mechanical and Aerospace Engineering at SeoulNational University, Seoul, Republic of, Korea, in2001 and 2003 respectively. He worked as a Re-search Engineer in the Research and DevelopmentDivision at Hyundai Motor Company, Hwaseong,Republic of, Korea, from 2003 to 2008. His researchinterests include modeling and control of hybrid
electric vehicles and lithium ion batteries.
PLACEPHOTOHERE
Shankar Mohan
PLACEPHOTOHERE
Jason B. Siegel
PLACEPHOTOHERE
Anna G. Stefanopoulou (F’09) is a Professor ofmechanical engineering and the Director of the Au-tomotive 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. Her research interests are on estimation andcontrol of internal combustion engines and electro-chemical processes such as fuel cells and batteries.
She holds 12 U.S. patents and is a Fellow of the ASME.
