Novel Predictive Electric Li-Ion Battery Model Incorporating Thermal and Rate Factor Effects

Novel Predictive Electric Li-Ion Battery Model Incorporating Thermal and Rate Factor Effects
Sachin Bhide and Taehyun Shim
Abstract—This paper presents the development of the electrical aspects of a Li-ion battery model that includes charge extraction due to current,battery capacity,effect of internal resistance,and thermal effects beyond only temperature rise due to power lost. Thermal models that represent temperature rise in the core and crust of each individual cell and a rate factor function that corrects the amount of charge extracted are developed to improve the accuracy of the battery characteristics.In addition,a predictive feature has been developed for this model so that it can predict the battery output characteristics over the selected operating range of temperatures and batteries with the limited amount of input data.In the end,a nine-cell stacking model is proposed and analyzed for its effect for different cooling methods(series and parallel cooling configurations).The simulation results show that the characteristics of the proposed model compared well with the published data.
Index Terms—AMESim,battery modeling,Li-ion battery, predictive model,thermal effects.
I.I NTRODUCTION
H YBRID electric vehicle(HEV)technologies are consid-
ered to be one of the most promising solutions to cope with environmental and energy problems caused by the auto-motive industry.In particular,plug-in HEV and vehicle-to-grid concepts have received special attention in recent years due to their potential impacts on the reduction of greenhouse gases and electricity distribution systems.The key element to the success of this system is a battery technology.Knowledge in the battery dynamics is crucial to predict the charge-discharge behavior for different vehicle operations such as cruise,acceleration, braking,etc.
Various battery models are being introduced and studied in HEV applications.They can be classified as electrochemical, mathematical,electrical,and polynomial[6].Electrochemical models are mainly used to optimize the physical design as-pects of batteries,characterize the fundamental mechanisms of power generation,and relate battery design parameters with macroscopic(e.g.,battery voltage and current)and microscopic (e.g.,concentration distribution)information.Mathematical
Manuscript received March10,2010;revised August9,2010and November12,2010;accepted November30,2010.Date of publication January10,2011;date of current version March21,2011.This work was supported by the Faculty Research Initiation and Seed Grants at the Uni-versity of Michigan-Dearborn.The review of this paper was coordinated by Mr.D.Diallo.
S.Bhide is with the Chrysler Group LLC Auburn Hills,MI48321-8004USA (e-mail:sbhide@).
T.Shim is with the Department of Mechanical Engineering,University of Michigan-Dearborn,Dearborn,MI48128USA(e-mail:tshim@). Color versions of one or more of thefigures in this paper are available online at .
Digital Object Identifier10.1109/TVT.2010.2103333models use empirical equations or mathematical methods like stochastic approaches to predict system-level behavior such as battery runtime,efficiency,or capacity.Polynomial-based models represent the battery in terms of a simplistic expression containing state of charge(SOC),which is the temperature for a certain range.Battery parameters such as the open-circuit voltage and all the internal impedances are a function of a fixed quantity(mostly state of charge).However,these models have limitations to provide I–V information that is important in circuit simulation and optimization.
However,electrochemical models are complex and time consuming,because they involve a system of coupled time-variant spatial partial differential equations,a solution for which requires long simulation time,complex numerical al-gorithms,and battery-specific information that is difficult to obtain because of the proprietary nature of the technology. Mathematical models cannot provide any I–V information that is important to circuit simulation and optimization.In addition, most mathematical models only work for specific applications and provide inaccurate results on the order of5%–20%error. For example,the maximum error of Peukert’s law predicting runtime can be more than100%for time-variant loads.[6]Elec-trical models employ the electrolyte,electrode,polarization resistances,and capacitances,along with a controlled battery source.Since it is a parametric model,it can be applied to any battery model,irrespective of its chemistry,configuration, and rate of discharge byfinding a suitable combination of parameters[6].Thus,electrical models are more widely used for HEV application among different battery models.However, majority of the electrical aspects of battery models classify the internal resistances in the battery as electrode and electrolyte resistances,and thermal effects on the battery are not modeled in detail for these models[1],[4]–[6].For instance,the change in the battery cell temperature is modeled using a simple polynomial of power lost in the electrolyte resistance that raises the temperature[1]or totally neglected to concentrate on the circuitry[2],[4],[9].In addition,some models do not consider the temperature effect on the internal resistance of the battery [2]and use the temperature function of SOC in the battery model[3],[5],[6].
Thermal aspects of battery modeling are of great importance as inappropriate battery temperatures result in degradation of the battery performance and life.Thus,battery-cooling strategy can significantly impact fuel economy and cabin climate control [7].The thermal considerations in a battery include the effects of stacking of cells in rows and column,the cooling medium (mostly air)flowing through the passages,convection occurring from its surface,and conduction that occurs in the cell from
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the core to the surface[7],[8].These topics have been mainly addressed in the research of battery-cooling system design[7], [8],but thermal effects on the performance of the electrical aspect of a battery model has not been thoroughly investigated. This paper presents a development of electrical aspects of a Li-ion battery model that includes charge extraction due to current,effect of internal resistance,and thermal effects beyond only temperature rise due to power lost.Since the charge and discharge rates play an important role in the characteristics of Li-ion battery on its voltage and capacity,a rate factor(RF) function that corrects the amount of charge extracted q was also developed to improve the accuracy of the battery characteris-tics.Thermal models that represent temperature rise in the core and crust of each individual cell are developed,in which the cooling effects,the consequent effects on electrolyte resistance, and the terminal voltage are also considered.In addition,a predictive feature that can study the performance characteristics of the battery over a range of different temperatures and battery capacities for afixed discharge rate is introduced.This model can predict the battery output characteristics over the selected operating range of temperatures and batteries with the limited amount of input data.At the end,a nine-cell stacking model is proposed and analyzed for its effect for different cooling methods(series and parallel cooling configurations).All of the modeling and simulation is done in an AMESim8.0[11] modeling environment that provides realistic model parameters and an environment for system modeling.
The rest of this paper is structured as follows:Sections II and III describe the development of electrical and thermal models,respectively.Section IV explains the combination of these models to form an improved battery model.The predictive feature of a battery model and RF implementation is presented in Sections V and VI,respectively.Section V discusses the effects of stacking on battery performance.
II.E LECTRICAL M ODEL D EVELOPMENT
In the electrical model of a battery,the battery voltage is a re-sult of battery constant voltage,charge depletion,and the expo-nential charge extraction zone.Fig.1shows the electrical model of a battery.In this model,the charge rating and voltage at fully charged condition are taken as inputs,along with the charge extracted during discharge,and a controlled voltage signal is sent to the constant voltage source.Details of the model can be found in[2].The following equations are used to determine the controlled voltage and the power losses of the cell:
i=dq
dt
(1)
E=E0−K
Q
Q−q
+Ae−Bq(2)
V battery=E−(Ri)
P owerLosses=i2R.(3) In this model,the output characteristics of the battery can be represented by the variables E o,K,and q,and the com-bination of these variables will define the battery output volt-age curve.The effects of temperature and RF on the
battery Fig.1.Schematic and AMESim electrical model of the cell.
output are studied later by considering their influence on these variables.
III.T HERMAL M ODEL D EVELOPMENT
In this paper,thermal models are constructed to differentiate the temperature at the core and the crust of a battery cell.The thermal mass associated with each core and crust is connected with a conductive resistance in between.The parameters are designed to have heat rejection from the thermal mass of the crust to the convective airflowing.The I2R losses from the circuitry are converted into heat and are supplied to thermal mass resembling the core.Fig.2shows the thermal model constructed in AMESim.
A.Modeling of Conductive Heat Transfer
The conductive heatflux between core and crust is given by
dh cond=
(T2−T1)
R c
.(4)
The thermal model is constructed in such a way that the region used for the computation in the battery cells is divided into smallerfinite elements that act as the heat sources within the cells.It is assumed that a uniform temperature is maintained within each of these small volumes in the cell.This assumption
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Fig.2.Thermal model of the cell.
is essential because,with each of the volumes being considered, the internal conduction resistance is negligible,compared with the thermal resistances of other heat transfer modes.Thus,the heat transfer area and contact conductance are selected such that the contact thermal resistance is negligible for the small volume of the cell.The conductive resistance of the cell is selected and validated according to the model in[8].Due to the different internal arrangement along radial and axial directions,better heat conduction usually occurs along the axial direction rather than the radial direction[8].Thus,we consider that there is no temperature variation axially,and only the differentiation of core and crust in radial direction is done by conduction.
B.Modeling of Convective Heat Transfer
The convective heat transfer occurring from the battery cell crust to the surrounding is considered in here.The following equation is used for convective heat transfer:
dh conv=h conv cearea(T1−T f).(5) The heat rejection from the battery cell crust depends on the method offluidflow,the area of heat exchange,theflow pattern, etc.The convective heat transfer coefficient is the variable that is affected by these parameters.Hence,the job for modeling a convective heat transfer component in a system mainly aims to ascertain the value of h conv.Air is used as the coolingfluid in this paper.
For a single-cell analysis,the convective heat transfer co-efficient of air(general range:50to100W/m2K)is used for forced convection heat transfer.Since battery stacking causes differential cooling of cells and temperature variations in each cell,it affects the output voltage.In this paper,we have modeled a2-D stack of cells and used this to analyze various cooling methods such as parallel and seriesflow heat rejection.Before further discussing the stacking effects of a battery,an improved single cell model showing the temperature effects on its per-formance is presented next.The stacking of the battery cells and the coolantflow through the intercellular gaps defines the h conv in each cell(explained in subsequent sections).Since the convective area and the temperature at the crust of the cell can be measured,using the conductive resistance(Rc)presented in [8],the temperature at the core of the cell can be determined using(4).
IV.I MPROVED S INGLE-C ELL M ODEL
The improved model is a combination of thermal and electri-cal models that can be used for the study of temperature effects on battery terminal voltage,polarization voltage,and amount of charge extracted.It also has the predictive feature that can be used to accurately determine the performance of the battery at every temperature and battery in the desired range.
bination of Electrical and Thermal Model
This model will address the temperature effects on the basic variables of battery,i.e.,E o,K,and q.Temperature functions, i.e.,polynomials of the third order in terms of core temperature, for a particular battery capacity and discharge rate are devel-oped.By using this,each variable has a unique temperature function defined by the following at a discharge rate:
x n=f(T)=a1+a2T+a3T2+ (6)
where n=E o,K,q,Q,B,....
The effects on the amount of charge extracted are quantified according to the changes in the internal resistance of the battery since the temperature function for q is inversely proportional to that of R.The internal resistance is corrected by creating a resistance temperature function,which is the directional inverse of the temperature function responsible for q.Thus,the cor-rections due to change in temperature that are intended in the amount of extracted charge are cascaded and made observable in the model in terms of corrections in the internal resistance of the battery cell.This is the method used by the proposed model to show the influence of temperature function for the amount of extracted charge on the battery cell voltage,i.e.,
x q∝
1
x R
.(7)
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Fig.3.Schematic of the improved model of the cell.
To adapt temperature effects,(2)is modified to the following:
E =x Eo .Eo −x K .K. Q
Q −x q .q
+Ae −B.[x q .q ].(8)
The different temperature functions in (6)for the basic variables are used in conjunction to (2)to incorporate the temperature influence.As can be seen in (8),the temperature functions are directly multiplied to their respective basic variables E o ,K ,and q .The combined product of the basic variables and tem-perature functions produces the basic variables needed for the battery performance at new temperature.
Fig.3shows the schematic of the combination of thermal and electrical models to form the improved battery model.As shown in Fig.3,the electrical and thermal models are connected together,and the temperature from the battery core is sensed and given as input to the temperature functions x Eo ,x K ,and x q ,where they govern the value of nominal voltage,battery capacity,and battery internal resistance.The battery performance is altered by correcting the parameters associated with E o ,K ,and q .For instance,the x q function influences the internal resistance of the battery,and the amount of charge extracted is altered accordingly.Thus,the improved model operates with the additional functions that increase the accuracy of the output response.
B.Inclusion of Predictive Feature
It is desirable to have a battery model that can predict the battery characteristics over a range of operating conditions.A predictive feature that can produce the battery performance in various battery capacities and temperatures for a fixed discharge rate is developed and added to the improved battery model,which was discussed in the previous section.This is done through updating the battery characteristic parameters of E o ,K ,and q according to changing operation conditions.This predictive feature is conceived as a result of studying the trends of battery parameter variation with different temperatures and capacities,along with the influence of particular parameters on a specific region of the voltage curve.
The aim of the predictive feature model is to predict the battery performance for the entire range of temperatures and capacities for any battery sets.For this model,four sets of bat-tery parameters,i.e.,E o ,K ,and q determined at four different temperatures,are needed for a particular battery capacity,which is called Q ref .From four voltage output curves either measured
or given,battery characteristic parameters E o ,K ,and q at cor-responding temperature can be extracted as shown in [2].With these four sets of parameters (E o ,K ,and q ),the polynomial temperature function x Eo (Q ref ),x K (Q ref ),and x q (Q ref )in (6)can be determined for each of the variables E o ,K ,and q ,respectively.Thus,it is possible to determine the voltage output at any temperature at the reference battery capacity Q ref by using the temperature functions.The target battery and target temperature are the battery capacity and temperature at which their values are to be determined,respectively.For instance,Eo (T 2,Q 2)is the value of E o determined at target tempera-ture T 2for the target battery of capacity Q 2.The following explains how the parameter E o (voltage at full charge)can be determined at target temperature T 2and target battery capacity Q 2using the proposed method.If the battery performance at target temperature for the reference battery capacity Q ref is to be found out,then the reference battery capacity Q ref is the same as target battery capacity Q 2.The target temperature T 2is an arbitrary temperature other than four temperatures where voltage output curves are either measured or given.By knowing the reference battery capacity,Q ref ,a value of E o at a temperature T 2,i.e.,Eo (T 2,Qref ),can be determined as in (9)by using Eo (T 1,Q ref )and the polynomial temperature function in (6).Since the performance at temperature T 1and capacity Q ref is available,Eo (T 1,Q ref )can be extracted,as discussed in [2]:
Eo (T 2,Q ref )= Eo (T 1,Q ref )
x Eo (Q ref ,T =T 2)
.(9)
The following explains how the parameter E o (voltage at full charge)can be determined in the predictive feature model at an arbitrary target temperature T 2for the application of a battery that has different capacity Q 2other than reference capacity Q ref .For this,a reference temperature (T ref ),which is common for all batteries,is needed.This can be any temperature within the desired operating temperature range.In this paper,20◦C is used as the reference temperature T ref .It is designed that the polynomial temperature functions converge to 1at the reference temperature.Thus,(6)can be represented as
x n =f (T ref )=1
(10)
where n =E o ,K,q....
With that,this can be done in three steps,as shown in Fig.4.First,Eo (T ref ,Qref )can be determined at the reference battery capacity Q ref at a temperature T ref by using Eo (T 1,Q ref )and the polynomial temperature function x Eo (Q ref ,T =T ref ),i.e.,
Eo (T ref ,Q ref )= Eo (T 1,Q ref )
x Eo (Q ref ,T =T ref )
.(11)
In the next step,the parameter Eo (T ref ,Q 2)at the reference temperature of the target battery for a different capacity Q 2is determined.
This step is needed to determine the parameters at the target battery from the reference battery at the reference tempera-ture.It is formulated after observing that the ratio of battery output responses between the reference and target batteries at
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Fig.4.Steps involved in the operation of the predictive model. reference temperature is proportional to that of the respective battery capacities.Thus
Eo(T
ref ,Q2)
=
Q ref
Q2
Eo(T
ref
,Q ref)
.(12)
After determining Eo(T
ref ,Q2)
,the battery parameter at target
temperature T2for a capacity Q2is calculated as
Eo(T
2,Q2)
=x Eo(Q
2
,T=T2)
.Eo(T
ref
,Q2)
.(13)
In(13),x Eo(Q
2,T=T2)
is a temperature function at a capacity
Q2and can be determined in
x Eo(Q
2,T=T2)
=
x(i
2
,i ref)
x Eo(Q
ref
,T=T2)
.(14)
It is observed that the ratio of temperature functions
x Eo(Q
2,T=T2)
/x Eo(Q
ref
,T=T2)
is proportional to the current
function shown in
x(i
2,i ref)
=
i2
i ref
e
|i2−i ref|
i ref
(15)
when the target temperature T2is higher than T ref,whereas it is inversely proportional when T2is lower than T ref.This can be represented as an exponential function in terms of battery currents,as shown in(15).With the target battery current i2 and reference battery current i ref,the battery parameter at target temperature T2for a capacity Q2can be decided.Equations (13)and(14)can be implemented by using the equations given here.
1)For(T2<T ref)
E o(T
2,Q2)
=
x(i
2
,i ref)
x Eo(Q
ref
,T=T2)
E o(T
ref
,Q2)
.(16)
2)For(T2=T ref)
Eo(T
2,Q2)
=Eo(T
ref
,Q2)
.(17)
3)For(T2>T ref)
E o(T
2,Q2)
=
1
x(i
2
,i ref)
x Eo(Q
ref
,T=T2)
E o(T
ref
,Q2)
.(18)
Similar relationships can be formed for parameters K and q.The proposed algorithm was implemented in a Matlab/ Simulink environment and used in the simulation by interfacing with the AMESim battery model discussed in the previous section.Fig.5shows a schematic of AMESim battery model/ Matlab interface.
C.Simulation Results for the Improved Model
In this section,the improved battery model that incorporates temperature effects is simulated,and its output responses are compared with the published Panasonic battery data[11]for the validation purpose.The following Panasonic batteries for 1C discharge rate are compared:
1)Panasonic CGR17500−3.7V,830mAh;
2)Panasonic CGR17670HC−3.7V,1250mAh;
3)Panasonic CGR18650HG−3.7V,1800mAh;
4)Panasonic CGR18650CG−3.7V,2250mAh.
Voltage Output Response for Constant Temperature:Fig.6 shows the battery output voltage responses for different batter-ies at the1C case.The blue solid line represents the manufac-turer’s test data[19],and the red dotted line indicates simulation results of the predictive model.A capacity of CGR17670HC battery was chosen as the reference battery capacity.It is observed that the output voltage responses of the model are well matched to the data sheet over the desired range of battery capacities and temperatures,whereas when the model is used for different battery capacities other than the reference capacity, the voltage output error between the model and the data sheet is increased as the battery capacity deviation becomes large from the reference capacity.However,it is noted that the error does not exceed±8%in the constant voltage region of the battery. The maximum error occurs at the end of the discharge region, which is corrected using the RF explained in the succeeding sections.The purpose of this paper was to construct a battery model with intermediate complexity that can represent the en-tire battery operation curve with good accuracy for application in HEVs.It is assumed that the battery operating range for a HEV ranges from40%to70%SOC to improve the life of the battery and prevent it from overcharging or overdischarging in the high and low SOC regions,respectively.With this assump-tion,the focus of this paper was to minimize the error in theflat nominal voltage region(the region between40%and75%)by updating the parameters that were responsible for the operation in this region.
For instance,the term“A”used in the exponential part of (8)accounts for the rise in the battery voltage from the battery constant voltage region.Since“A”represents the difference between the voltage at100%SOC and the constant voltage in theflat region,the term“A”has no effect on the operating region(40%–75%SOC)and,hence,is not addressed in the model and is held constant.
Voltage Response With Varying Temperature:After the val-idation of a voltage output response of the proposed model to the data set in constant temperature conditions,its responses for varying temperature are compared.When the battery is in operation,it is observed that the temperature of the battery increases.The temperature rise in the battery is due to heat dissipated by the chemical reactions and I2R losses.Forced convection is used with free boundary and full area of heat exchange.In the simulation,a battery model having the parame-ters of Panasonic CGR17670HC battery is used.During the
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Fig.5.Brief AMESim/Simulink interface for predictive algorithm.
Fig.6.(Clockwise from top left)Graphs of battery output voltage(in volts)and battery capacity(Ah)for1C case for Panasonic CGR17500,CGR17670, CGR18650HG,and CGR18650CG.
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Fig.7.Battery temperature rising to10◦C with an effect on the voltage curve. simulation,the temperature of the battery model is increased up to10◦C starting from0◦C to compare the voltage outputs of responses of given data set at two constant temperature(0◦C and10◦C)[11].Fig.7compares the voltage output responses between the Panasonic CGR17670HC battery and the proposed model for a transition from0◦C to10◦C.It can be seen that the voltage curve starts at0◦C response,traces the10◦C response as the temperature reaches10◦C and then reverts back to0◦C. As the temperature is initially at0◦C,the green dot-dashed line in the voltage curve shows a point at the0◦C curve.As the temperature increases and approaches10◦C,the green line navigates to the10◦C curve.As the temperature again falls back to0◦C,wefind the line translating back to the voltage curve representing constant0◦C.
The battery temperature variation for Panasonic CGR18650H battery was tested at1C discharge rate.The different rates of cooling(100,10,5W/m2K and adiabatic) are analyzed[12].The natural convection heat transfer,forced convection heat transfer,adiabatic condition,and the threshold between the natural and forced convection are different cases taken for analysis.Convection is used with free boundary and full area of heat exchange.The battery temperature rise and its effects on voltage can be shown in Fig.8.It can be seen that there is certain capacity enhancement for the battery as
the Fig.8.Temperature rise and battery output voltage during battery operation for different rates of cooling with convection heat transfer in a single cell. temperature rises.In addition,the output voltages can be seen to increase.It can thus be concluded that the rise in temperature has an effect on the output voltage response,and hence,it can be said that the online variations in temperatures are accounted.
D.Implementation of RF Function
It is known that change in the discharge rate affects the battery characteristics as it has direct influence on the magni-tude of voltage and battery capacity.To improve the battery characteristics at the end of the discharge region of the battery, an RF function was employed to the amount of charge extracted in the circuit q during simulation.To accommodate the effects of change in discharge rates,(8)is modified to contain the RF multiplied to variable‘q,’as shown in
E=x Eo.Eo−K
x Q.Q
Q q
+A.e−B.[x q.(RF).q].
(19) An exponential RF function shown in
RF=1±0.01.
e{(0.0736.Q2)+(0.00397.Q.T)+0.618}
(20)
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Fig.9.(a)V oltage response for CGR17670with and without RF.(b)V olt-age response for CGR17500with and without RF.(c)V oltage response for CGR18650with and without RF effect.(d)V oltage response for CGR18650 with and without RF
effect.Fig.10.Schematic of parallelflow heat
exchange.
Fig.11.AMESim custom submodel.
can be used over a range of temperatures from−10◦C to45◦C and battery capacities from0.83to2.15Ah.This function is obtained after performing the linear regression analysis over the entire2-D ranges of battery capacities and temperatures.
With the proposed RF function,the model shows better correlation with the published battery characteristics,as shown in Fig.9(a)–(d).The solid line shows the manufacturer’s curve [19],the dashed line indicates the predictive model with tem-perature function without RF function,and the dot-dashed line shows the predictive model with temperature and RF functions. This RF function is also applicable to other battery capacities for the aforementioned range of temperatures.
V.S TACKING E FFECTS OF B ATTERY P ERFORMANCE Battery stacking causes differential cooling of cells and temperature variations among different cells.Because of this, the output voltage of each cell in the stack is different.In here,2-D four-cell and nine-cell stack models are proposed and analyzed to see the effects of different cooling methods(series and parallel cooling configurations)on the performance of a battery stack.
A.Parallel Flow Heat Exchange in the Stack
Fig.10shows a schematic of a four-cell stack model.The battery cells are stacked,in which the space between the four cells makes a form of a four-lobed star,and the coolingfluid flows through this space.。