Development of Improved Li-Ion Battery Model

Development of Improved Li-ion Battery Model Incorporating Thermal and Rate Factor Effects Sachin Bhide and Taehyun Shim Department of Mechanical Engineering University Of Michigan-Dearborn Dearborn, MI-48128, USA [email protected], [email protected] the temperature [1] or totally neglected to concentrate on the circuitry [2, 4, 9]. Also, some models do not consider the temperature effect on internal resistance of the battery [2] and use the temperature function of SOC in the batter model [3, 5, 6].

Abstract— This paper presents development of an improved electrical circuit based Li-ion battery model using AMESim. This model considers charge extraction due to current, battery capacity, and effect of internal resistance and extends the thermal aspects which represent temperature rise in the core and crust. The thermal aspects that characterize the temperature rise in core and crust and its effects on the internal resistance and discharge rate factor affecting the terminal voltage of the battery along with the charge extracted are incorporated. The characteristics of the proposed model was compared with published data and showed good correlation.

The thermal aspects of battery modeling are of great significance as inappropriate battery temperatures result in degradation of the battery performance and life. The 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 the core to the surface [7, 8].

Keywords- Li-ion Battery; Battery Modeling; AMESim

I.

INTRODUCTION

Hybrid Electric vehicle (HEV) technologies are considered to be one of the most promising solutions to cope with environmental and energy problems caused by automotive industry. In particular, plug-in HEV and vehicle-to-grid (V2G) concepts have received special attention in recent years due to their potential impacts on the reduction of greenhouse gases and electricity distribution system. The key element to the success of this system is the battery technology. Knowledge in the battery dynamics is crucial in order to predict the chargedischarge behavior for different vehicle operations such as cruise, acceleration, braking, etc.

This paper considers the electrical aspects of a Li-ion battery such as charge extraction due to current, effect of internal resistance and extends the thermal effects beyond only temperature rise due to power lost. The cooling effects, the consequent effects on electrolyte resistance and the terminal voltage are considered. The thermal model that considers the temperature rise in the core and crust of each individual cell is employed. The charge and discharge rates play an important role in characteristics of Li-ion battery as change in rate shows change in battery voltage and capacity. A rate factor function that corrects the amount of charge extracted ( q ) was developed to improve the accuracy of the battery characteristics by considering the change in discharge rate. The whole modeling and simulation is done in AMESim 8.0 [10] modeling environment that provides realistic model parameters and environment for system modeling.

Various battery models are being introduced and studied in the HEV applications. It can be classified as Electrochemical, Mathematical, Electrical and Polynomial [6]. Electrochemical models use the fundamental battery chemistry, mathematical models use empirical equations and polynomial models express its responses in terms of a polynomial expression. Electrical models employ the electrolyte, electrode, polarization resistances, and capacitances along with a controlled battery source. Among different battery models, electrical models are more realistic, intuitive, useful and easy to handle. Since it is a parametric model, it can be applied to any battery model irrespective of its chemistry, configuration, and rate of discharge by finding suitable combination of parameters [6].

Section II and III explain the original electrical and thermal models, respectively. Section IV elucidates the combination of these models to form an improved battery model. Section V and VI clarify the rate factor implementation and the effects of stacking on battery performance, respectively.

II. ELECTRICAL MODEL DEVELOPMENT

Almost all 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

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In the electrical model of a battery, the battery voltage is a result of battery constant voltage, charge depletion and the exponential charge extraction zone. Fig.1 shows 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

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charge extracted during discharge and a controlled voltage signal is sent to constant voltage source. Details of the model can be found in [2].

III. THERMAL MODEL DEVELOPMENT 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 as well as 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 air flowing. The I2R losses from the circuitry are converted into heat and are supplied to thermal mass resembling the core. Fig.2 shows the thermal model constructed in AMESim.

Figure 2: Thermal Model of the cell

Figure 1: Electrical Model of the cell

Following equations are used to determine the controlled voltage and the power losses of the cell: i=

dq dt

⎛ Q ⎞ ⎟⎟ + Ae − Bq E = Eo − K ⎜⎜ − Q q ⎝ ⎠ Vbattery = E − (R.i)

A. Modeling of conductive heat transfer The conductive heat flux between core and crust is given by:

(1)

dhcond = (2)

(T2 − T1 ) Rc

(4)

Where, Rc = contact thermal resistance between core and crust (K/W)

(3)

T2 = temperature at the core, T1 = temperature at the crust

PowerLosse s = i 2 R Where, Eo = battery constant voltage (V), E = No-load voltage (V) Q = battery capacity (Ah), i = battery current (A) R = battery internal resistance(Ω), K = Polarization voltage(V) A, B = constants for function involved in exponential region

It is assumed that a uniform temperature within each cell is maintained. This assumption is essential in order to hold it true that for each of the volumes being considered the internal conduction resistance is negligible compared to thermal resistances of other heat transfer modes. Thus, the heat transfer area and contact conductance are selected so that the contact thermal resistance is negligible. Due to the different internal arrangement along radial and axial directions, better heat conduction usually occurs along the axial direction rather than in 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.

In this model, the output characteristics of the battery can be represented by the variables Eo , K , and q . The combination of these variables will define the battery output voltage curve. In this paper, we will study the influence of temperature and rate factor on the battery output by considering their influence on these variables.

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B. Modeling of convective heat transfer

xq ∝

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,

dhconv = hconv cearea(T1 − T f )

1 xR

(7)

Equation (2) is modified to obtain the following Equation (8). The different temperature functions in (6) for the basic variables are used in conjunction to (2) to incorporate the temperature influence. Fig. 3 shows the schematic of the battery model.

(5)

where, hconv = convective heat transfer coefficient depends on the cooling method and conditions of flow(W/m2K), T1 =

⎛ ⎞ Q ⎟ + Ae − B.[ xq .q ] E = x Eo .Eo − x K .K .⎜ (8) ⎜ Q − xq .q ⎟ ⎝ ⎠ Where, xk = temperature function for battery polarization voltage

crust temperature, T f = temperature of the cooling fluid, cearea = convective exchange area taken as the surface area of the battery cell exposed to the surrounding cooling fluid (m2)

xEo = temperature function for open circuit voltage

This rejection of heat from the battery cell crust depends on the method of fluid flow, the area of heat exchange, the flow 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 mainly aims to ascertain the value of hconv . Air is used as the cooling fluid in this paper.

= temperature function for charge extracted and in turn the internal resistance xq

For a single cell analysis, the convective heat transfer coefficient of air is used for forced convection heat transfer. (General Range: 50 to 100 W/m2K). Battery stacking causes differential cooling of cells and differences in the temperatures of each cell thus affecting the output voltage. In this paper, we consider a two dimensional stack of cells and would analyze it for different methods of cooling like parallel flow heat rejection and series flow heat rejection. IV. IMPROVED SINGLE CELL MODEL

Figure 3: Schematic of the Improved Model of the cell

A.

Model concept The improved model is a combination of thermal and electrical models that can be used for the study of temperature effects on battery terminal voltage, polarization voltage and amount of charge extracted. This model will address the temperature effects on the basic variables of battery, namely Eo , K and q . Temperature functions, polynomials of the third order in terms of temperature, for a particulate battery capacity and discharge rate are developed. By using this, each variable has a unique temperature function defined by (6) at a discharge rate. x n = f (T ) = a1 + a2T + a3T 2 + ...

The core temperature is the governing factor for the temperature functions and that is in turn governed by the arrangement of cells in the stack, method of cooling, cooling fluid, etc.

(6)

Where, n = Eo , 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. Figure 4: Required test data and the parameters extracted

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B. Methodology of the operation of the model: The improved model is the physical depiction of the parameter variations caused by temperature. In order to determine the temperature functions needed for the model, it requires the information of at least 4 sets of battery parameters at respective temperatures for a particular reference battery capacity ( Qref ) as shown in Fig. 4. From these, the

E o (T

2 ,Q1 )

and x q ( Qref ) in (6) can be determined for Eo , K , and q respectively. In addition, one set of test data is needed to extract the base electric model parameters Eo , K , and q shown in [2] at a desired battery capacity ( Q1 ), and one temperature ( T1 ) for a certain discharge rate. These parameters can be written as Eo (T ,Q ) , K (T1,Q1 ) and q(T1 ,Q1 ) . From the known Eo (T ,Q ) , a value of Eo at temperature 1

• • • •

T1 and the desired battery capacity Q1 , the value of Eo at different temperature of the same battery, Eo (T ,Q ) , can be 2

1

determined as, Eo (T

2 ,Q1 )

=

Eo (T ,Q ) 1

(11)

x(i1,iref ) .x Eo(Qref ,T =T2 )

D. Simulation In this section, the improved battery model that incorporates temperature effects is simulated and its output responses are compared with the published Panasonic batter data [11] for the validation purpose. The following Panasonic batteries for 1C discharge rate are compared

1

1

Eo(T1 ,Q1 )

Thus, the final output parameter which will define the output curve is expressed in terms of known variables. Similar equations can be formed for parameters like K and q . The parameter variations are observed and then its implementation for different battery capacities is employed. Thus, now, the parameter can be expressed in terms of temperature and the battery capacity. Therefore, it is possible to represent every parameter as Eo(Ti ,Qi ) .

polynomial temperature function of x Eo (Qref ) , x K (Qref ) ,

1

=

1

x Eo (Q1,T =T2 )

– 3.7V, 830mAh – 3.7V, 1250mAh – 3.7V, 1800mAh – 3.7V, 2250mAh

Constant temperature outputs: Figures 5-8 show the comparison of the battery voltage between the proposed model and published data for constant temperature cases over the range of battery capacities. The solid line represents the manufacturer’s test data and the dotted line indicates simulation results of the improved model. During the simulation, the temperature of the battery does not change while it is being operated. The battery output voltage at − 10 o C , 0 o C , 20 o C , and 45 o C were compared to 1C discharge rates of above batteries. As shown in Figs. 5-8, the output voltage of the model is well matched with the published data over the desired range of battery capacities and temperatures.

1.

(9)

Where, x Eo (Q1,T =T2 ) is the ratio at which the parameter Eo varies with temperature for the battery of capacity Q1 with T2 as the input variable for the polynomial function. The above mentioned temperature function at Q1 is unknown and has to be found out from the known temperature function at Qref . Also, it can be found that the parameters vary with the ratio that is proportional to the battery currents. Thus when we have a ratio at the reference battery capacity, its relation to the base ratio at Q1 is going to be proportional to the respective battery currents. Thus, the ratio at which the parameter Eo varies with temperature for the battery current i1 can be given as:

4.2 45 degC 4 20 degC 3.8

(10)

Battery Voltage (V)

x Eo (Q1 ,T =T2 ) = x(i1,iref ) x Eo (Qref ,T =T2 )

Panasonic CGR17500 Panasonic CGR17670HC Panasonic CGR18650HG Panasonic CGR18650CG

Where, x(i1 ,iref ) is the current ratio that maneuvers the temperature function across different battery capacities. This is an exponential function in terms of i1 and iref .

3.6 3.4 3.2 0 degC 3 -10 degC 2.8

x Eo (Qref , T =T2 ) is the ratio at which the parameter Eo varies

2.6

with temperature for the battery of capacity Qref with T2 as the input to the temperature function.

0

0.1

0.2

0.3 0.4 0.5 Battery Capacity (Ah)

0.6

0.7

Figure 5: Graph for 1C case for Panasonic CGR17500

From Equation (9) and (10),

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0.8

between the model and data is increase as the battery capacity further changes from the reference one. Despite the increased error in the output voltage, it is noticed that the error doesn’t exceed ± 5% in the constant voltage region of the battery. The error is maximum at the end discharge region which is corrected using the rate factor explained in subsequent section.

4.2 45 degC 4 20 degC

Battery Voltage (V)

3.8 3.6 3.4

2.

Voltage response with varying temperature:

3.2

In this section, the output voltage of the improved battery model is simulated when it is operated under varying temperate. While the battery is in operation, it is observed that the temperature of the battery increases. This temperature rise in the battery is due to heat dissipated by the chemical reactions as well as I2R losses. Forced convection is used with free boundary and full area of heat exchange in the model.

3 0 degC

-10 degC

2.8 2.6 0

0.2

0.4

0.6 0.8 Battery Capacity (Ah)

1

1.2

Figure 6: Graph for 1C case for Panasonic CGR17670HC 4.2 3.8

10 degC Manufacturer's curve (solid line)

4 45 degC

20 degC

3.6

Battery Voltage (V)

Battery Voltage

3.8 3.6 3.4 3.2

3.4

3.2

0 degC Manufacturer's curve (solid line)

3

3 0 degC

2.8

-10 degC

2.8 2.6

Manufacturer's data for 0 and 10 degC Model curve for 0 degC Model curve starting from 0 degC to 10 degC

2.6 0

0.2

0.4

0.6 0.8 1 1.2 Battery Capacity (Ah)

1.4

1.6

1.8

0

0.2

0.4


1

1.2

0

0.2

0.4


1

1.2

.

Figure 7: Graph for Panasonic CGR18650HG

12

4.2

10 Battery cell temperature (DegC)

45 degC 4 20 degC

Battery Voltage (Ah)

3.8 3.6 3.4 3.2

0 degC

2.8

4

0

2.6 0

6

2

-10 degC

3

8

0.2

0.4

0.6

0.8 1 1.2 1.4 Battery Capacity (Ah)

1.6

1.8

Figure 9: Graph showing the battery temperature rising to 10 degC with effect on voltage curve

2

Two constant temperature curves of 0 o C and 10 o C [11] were used for the comparison. During the simulation, the temperature of the battery model reach up to 10 o C starting

Figure 8: Graph for 1C case for Panasonic CGR18650CG

CGR17670HC battery was chosen as the reference battery capacity. Since the improved model parameters were based on the reference battery capacity, the output voltage error

from 0 o C . Fig. 9 shows the validation of Panasonic CGR

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An exponential rate factor function is proposed and its optimized values over a range of temperatures from − 10 o C to 45 o C and battery capacities from 0.83 Ah to 2.15 Ah are shown in (13). A linear regression analysis was performed over the entire two dimensional ranges of battery capacities and temperatures and the function was obtained.

17670HC battery for a transition from 0 o C to 10 o C . We can see that the voltage curve starts at 0 o C response, traces the 10 o C response as the temperature reaches 10 o C and then reverts back to 0 o C . Figure 10 shows the effects of battery temperature rise on the voltage for different heat transfer coefficient. Forced convection is used with free boundary and full area of heat exchange in the model with different rates of cooling (100, 10, 5 W/m2K and adiabatic).

2 RF = 1 ± 0.01.[e {(0.0736 Q )+(0.00397 Q T )+0.618}]

(13)

Figure 11 compares the output voltages of the proposed model with and without rate factor function and the published Panasonic CGR17670HC battery data. The solid line shows the manufacturer’s curve [11], dashed line shows the results after application of the improved model with temperature function and the dot-dashed line shows the application of RF with the improved model. As shown in the figure, with inclusion of the rate factor function in the improved model, the output voltage in the discharge region well matched with the manufacture’s data. This RF function is also applicable to other battery capacities for the above range of temperatures.

4 45 degC

Battery Voltage (V)

20 degC

3.5

0 degC

3

-10 degC Manufacturer's data (solid line) Improved model w/o Rate factor (Dashed) Improved model with Rate Factor (Dot-Dashed)

2.5 0

Figure 10: Voltage curve for different rates of cooling during online temperature variations

0.2

0.4


1

1.2

Figure11: Voltage response for CGR17670 with and without RF effect

V. IMPLEMENTATION OF RATE FACTOR FUNCTION In this section, a rate factor function is proposed and employed during runtime to the amount of charge extracted in the circuit ( q ). It is known that the rate change affects battery characteristics since it has direct influence on the magnitude of voltage and battery capacity. To accommodate the effects of change in the discharge rates, (8) can be modified to consider the rate factor (RF) term that is multiplied to variable ‘ q ’ as shown in (12).

⎛ ⎞ Q ⎟ + Ae − B [ xq ( RF ) q ] E = x Eo Eo − x K K ⎜ ⎜ Q − x q ( RF ) q ⎟ ⎝ ⎠

VI. CONCLUSION An improved electrical circuit based Li-ion battery model which can address the effects of temperature on the battery performance over a range of battery capacities has been developed and simulated using AMESim. This model combines electrical and thermal aspects of battery models and the simulation results show a good correlation in the battery voltage output under different temperature ranges when it’s compared with published data. In addition, the battery rate factor function is proposed and incorporated in the model. With the inclusion of the rate factor effects, the accuracy of the proposed model is further improved. This model is applicable only for a particular discharge rate and requires

(12)

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different temperature and rate factor functions for different rates. The model accommodating the different discharge rates can be studied in the future time. ACKNOWLEDGEMENT This work was supported by the Faculty Research Initiation and Seed Grants at the University of Michigan-Dearborn. REFERENCES [1]

Robyn A. Jackey, “Simple, Effective Lead-Acid Battery Modeling process for Electrical System Component selection”, SAE technical papers, 2007-01-0778 [2] Oliver Trembley, Louis-A. Dessaint, Abdel-Illah Dekkiche, “ A Generic Battery Model for the Dynamic Simulation of Hybrid vehicles”, IEEE technical papers 2007 [3] Ryan C. Kroeze, Philip T. Krein, “Electrical Battery Model for use in Dynamic electric vehicle Simulation” IEEE technical papers 2008 [4] Bernhard Scheighofer, Klaus Raab, Georg Brasseur, “Modelling of High Power Automotive Batteries by the Use of an Automated Test System”, IEEE Instrumentation and Measurement Technology Conference, 2002 [5] Antoni Szumanowski, Yuhua Chang, “Battery Management System Based on Battery Non Linear Dynamics Modeling” IEEE Transactions on Vehicular Technology, Vol.57, 2008 [6] Min Chen, Gabriel A. Rincon-Mora, “Accurate Electrical Battery Model Capable of Predicting Runtime and I-V performance” IEEE Transactions on Energy Conversion, 2006 [7] Chan-Woo Park, Arun Jaura, “Transient Heat Transfer of 42V Ni-MH Batteries for an HEV Application” SAE technical papers,2002-01-1964 [8] Chanwoo Park, Arun Jaura, “Dynamic Thermal Model of Li-ion Battery for Predictive Behavior in Hybrid and Fuel Cell Vehicle” SAE technical papers, 2003-01-2286 [9] N. Medora, A. Kusko, “Enhanced Battery Model of lead acetate batteries using manufacturers data”, IEEE-2006 [10] AMESim users Manual [11] Panasonic battery manufacturer’s datasheet.

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