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An approach to internal and external temperature estimation for cylindrical battery based on finite difference method
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IFAC PapersOnLine 51-31 (2018) 258–261 An approach to internal and external temperature estimation for cylindrical An approach to internal and external temperature An approach to internal and external temperature estimation for cylindrical cylindrical battery based on finite difference estimation method for An approach to internal and external temperature estimation for cylindrical battery based on finite difference method An approach to internal external temperature batteryand based on finite difference estimation method for cylindrical battery basedLion finite difference Xinggang, Xiong Rui* method battery basedLion finite difference Xinggang, Xiong Rui* method
Li Xinggang, Xiong Rui* Li Xinggang, Xiong Rui* * National Engineering Laboratory for Electric Vehicles, School of Mechanical Engineering, Beijing Institute of Technology, Li Xinggang, Xiong Rui* ** National Engineering Laboratory for Electric Vehicles, School of Mechanical Beijing Beijing, 100081, China;( e-mail: [email protected], [email protected] National Engineering Laboratory for Electric Vehicles, School of Mechanical Engineering, Engineering, Beijing) Institute Institute of of Technology, Technology, 100081, China;( e-mail: [email protected], [email protected] ) * National Engineering Beijing, Laboratory for Electric Vehicles, School of Mechanical Engineering, Beijing Institute of Technology, Beijing, 100081, China;( e-mail: [email protected], [email protected] ) * National Engineering Beijing, Laboratory for Electric School of Mechanical Engineering, Beijing) Institute of Technology, 100081, China;(Vehicles, e-mail: [email protected], [email protected] Beijing, of 100081, China;( [email protected], [email protected] Abstract: A method estimating the e-mail: temperature for cylindrical battery is expounded.) Based on the Abstract: A method of of estimating estimating the the temperature for cylindrical battery isgenerate expounded. Based on the simplified A one-dimensional（1-D） thermal model for andcylindrical the Bernardi heatis model, theon finite Abstract: method temperature battery expounded. Based the simplified one-dimensional（1-D） thermal model and the Bernardi heat generate model, the finite Abstract: A method of estimating the temperature for cylindrical battery is expounded. Based on the difference method is used to calculate the internal and external temperatures of the battery. And different simplified one-dimensional（1-D） thermal model for andcylindrical the Bernardi heatisgenerate model, theon finite Abstract: A method of estimating the temperature battery expounded. Based the simplified one-dimensional（1-D） thermal model the Bernardi heat model, finite difference method is used used to calculate calculate the internal internal and and external temperatures ofgenerate the battery. battery. Andthe different charge andmethod discharge condition experiments are performed. Comparisons between the estimation results difference is to the and external temperatures of the And different simplified one-dimensional（1-D） thermal model and the Bernardi heat generate model, the finite charge andmethod discharge condition experiments are performed. Comparisons between the estimation results and measured results proved theexperiments accuracy ofare theperformed. model. TheComparisons maximum estimation error isAnd about 4.5℃. difference is used to calculate the internal and external temperatures of the battery. different charge and discharge condition between the estimation results difference method is external used to calculate the internal and external temperatures of thecan battery. And different and measured results proved the accuracy of the model. The maximum estimation error is about 4.5℃. Using the measured temperature of the battery, the internal temperature be corrected. And charge and discharge condition experiments are performed. Comparisons between the estimation results and measured results proved theexperiments accuracy ofare theperformed. model. TheComparisons maximum estimation error is about results 4.5℃. charge and discharge condition betweencan the estimation Using the measured external temperature of the battery, the internal internal temperature be corrected. And the error ismeasured within 1℃. and measured results proved the accuracy of of the the battery, model. The maximum estimation error is about 4.5℃. Using the external temperature the temperature can be corrected. And and measured results proved the accuracy of of the the battery, model. The maximum estimationcan error is about 4.5℃. the error is within 1℃. Using the measured external temperature the internal temperature be corrected. And the error ismeasured within 1℃. © 2018, IFAC (International Federation of Automatic Control) Hosting bytemperature Elsevier Ltd. Alldifference rights reserved. Keywords: cylindrical battery, battery management system, temperature, finite method Using the external temperature of the battery, theinternal internal can be corrected. And the error is within 1℃. Keywords: cylindrical battery, battery management system, internal temperature, finite difference method the error is within 1℃. Keywords: cylindrical battery, battery management system, internal temperature, finite difference method Keywords: cylindrical battery, battery management system, internal temperature, finite difference method Keywords:1.cylindrical battery, battery management system, internalcylindrical temperature, finite difference INTRODUCTION A NMC battery produced method by a domestic 1. INTRODUCTION A NMC cylindrical battery produced domestic manufacturer is used in this study. The specific 1. INTRODUCTION A NMC cylindrical battery produced by by aa parameters domestic Lithium-ion batteries are used increasingly in electric manufacturer is used in this study. The specific parameters are shown in Table 1. 1. INTRODUCTION A NMC cylindrical battery produced by a domestic manufacturer is used in this study. The specific parameters Lithium-ion batteries are used in electric vehicles because of1.the advantages of high energy INTRODUCTION NMC incylindrical battery produced by a parameters domestic are shown Table 1. manufacturer is used in this study. The specific Lithium-ion batteries are used increasingly increasingly in density, electric A are shown in Table 1. in this study. The specific parameters vehicles because of the advantages of high energy density, high power density, and so on. Battery temperature is an manufacturer is used Lithium-ion batteries are used increasingly in electric are shownTable in Table 1. vehicles because of the are advantages of high energy of experimental battery 1. Parameters Lithium-ion batteries used increasingly in density, electric high power density, and so on. Battery temperature is an important parameter the battery management system (Liu are shownTable in Table 1. vehicles because of for the advantages of high energy density, Parameters battery 1. high power density, and so on. Battery temperature is an vehicles because the advantages of high energy density, of experimental experimentalΦ18mm*65mm battery TableDimension 1. Parameters of important parameter the battery management system (Liu et al, power 2013). Dueoftofor the poor conductivity, there high density, and so on.thermal Battery temperature is an important parameter for the battery management system (Liu Parameters of experimental battery Table 1. high power density, and so on. Battery temperature is an Dimension Φ18mm*65mm et al, 2013). Due to the poor thermal conductivity, there exists a difference between the cell surface temperature and important parameter forthe thepoor battery management systemthere (Liu Nominal capacity of experimentalΦ18mm*65mm 2500mAh battery Table 1. Parameters Dimension et al, 2013). Due to thermal conductivity, important parameter the battery system (Liu Dimension Φ18mm*65mm exists aa2013). difference the cell surface temperature and the internal temperature (Zhang et management al. 2014). But onlythere the et al, Due between toforthe poor thermal conductivity, Nominal capacity 2500mAh Upper cutoff voltage 4.20V exists difference between the cell surface temperature and Nominal capacity 2500mAh Dimension Φ18mm*65mm et al, Due to the(Zhang poor thermal conductivity, there the internal temperature et al. 2014). the battery surface temperature can cell be measured inBut realonly vehicle exists a2013). difference between the surface temperature and Nominal capacity 2500mAh Upper cutoff voltage 4.20V the internal temperature (Zhang et al. 2014). But only the Lower 3.0V Upper cutoff voltage 4.20V exists a surface difference between the cell surface temperature Nominal capacity 2500mAh battery temperature can be measured in real vehicle operation, which cannot reflect state of and the the internal temperature (Zhang etthe al. internal 2014). But only Upper cutoff voltage 4.20V battery surface temperature can be measured in real vehicle Lower 3.0V Maximum charging current 2400mA the internal temperature (Zhang al. internal 2014).inBut the Lower cutoff voltage 3.0V Upper cutoff voltage 4.20V operation, which cannot reflect state of the battery truly. Therefore, obtaining the accurate internal and surface temperature can beetthe measured realonly vehicle operation, which cannot reflect the internal state of the Lower cutoff voltage 3.0V Maximum charging current 2400mA Maximum current 7200mA battery surface temperature can battery bethe measured instate real vehicle Maximum charging current 2400mA battery truly. Therefore, the accurate internal external temperatures ofobtaining the has very important operation, which cannot reflect internal of and the Lowerdischarging cutoff voltage 3.0V battery truly. Therefore, obtaining the accurate internal and Maximum charging current 2400mA Maximum discharging current operation, which cannot reflect the internal state of and the Charging working temperature 07200mA ℃~45℃ external temperatures of the battery has very important significance for safety and durability (Chen et al, 2018). Maximum discharging current 7200mA battery truly. Therefore, obtaining the accurate internal Maximum charging current 2400mA external temperatures ofobtaining the battery has veryinternal important Maximumworking discharging current 7200mA battery truly. Therefore, the accurate and Charging ℃ significance for safety and (Chen al, 2018). external temperatures of durability the battery haset very important Discharging workingtemperature temperature -20 ℃~45 ~60℃ ℃ Charging temperature ℃ ~45 ℃ 007200mA significance for safety and durability (Chen et al, 2018). Maximumworking discharging current The internal temperature of the battery can be measured by external temperatures of the battery has very important Charging working temperature 0 ℃ ~45 ℃ significance for safety and durability (Chen et al, 2018). Discharging working temperature -20 ℃ ~60 ℃ Discharging workingtemperature temperature ℃~45 ~60℃ ℃ -20 Charging working 0℃ The internal temperature of the battery be by the built-in sensor etcan al,et2010). But it can significance for safety and durability (Chen al,measured 2018). The internaltemperature temperature of the(Forgez battery can be measured by 2.2Discharging working temperature -20℃~60℃ Heat generation model the built-in temperature sensor (Forgez et al, 2010). But it can be performed only during experiments, and not applicable in The internal temperature of the battery can be measured by Discharging working temperature -20℃~60℃ the built-in temperature sensor etcan al, 2010). But it can 2.2 The internal temperature of the(Forgez battery befinite measured by 2.2 Heat Heat generation generation model model be performed only during experiments, applicable in real car.temperature In addition, though the traditional element the built-in sensor (Forgez etand al, not 2010). But it can It is difficult to obtain the heat generation rate of the battery be performed only during experiments, and not applicable in 2.2 Heat generation model the built-in temperature sensor (Forgez al, not 2010). But it can the real car. In addition, the traditional finite element method is usually used tothough calculate theetand internal temperature, be performed only during experiments, applicable in 2.2 Heat generation model It is difficult to obtain the heat generation rate the accurately by experimental methods. The D. the real car. In addition, though the traditional finite element ofexisting the battery battery be performed during experiments, and notfinite applicable in It is difficult to obtain the heat generation rate of method is used to calculate the internal temperature, the calculation is too large to bethe applied online (Song et al, real car.usually Inonly addition, though traditional element accurately by experimental methods. The existing D. Bernardi (Bernardi et al, 1985) model is used to calculate the It is difficult to obtain the heat generation rate of the battery method is usually used to calculate the internal temperature, accurately byto obtain experimental methods. The existing D. the real car.usually In addition, the traditional finite element the calculation is too to be applied online (Song et al, 2013). It is generation difficult the1985) heat generation rate ofcalculate the battery method is usedlarge tothough calculate the internal temperature, Bernardi (Bernardi et al, model is used to the heat rate, which can be expressed as: accurately by experimental methods. The existing D. the calculation is too large to be applied online (Song et al, Bernardi (Bernardi et al, 1985) model is used to calculate the method is usually used to calculate the internal temperature, 2013). accurately by rate, experimental methods. The existing the D. the calculation is too large to be applied online (Song et al, heat generation which can be expressed as: Bernardi (Bernardi et al, 1985) model is used to calculate 2013). I dE generation rate,et as: A finite difference solving online a 1-D (Song cylindrical the calculation is toomethod large tofor be applied et al, heat al, 1985) U1 be Texpressed q which E0can (1) Bernardi (Bernardi model is used to calculate the 2013). I can be expressed dE generation rate, which as: dT A difference for solving aa perform 1-D battery modelmethod has been on-line heat 2013). dE VIb EE0can U1 be Texpressed (1) heat generation rate,qq which as: A finite finitethermal difference method forproposed solving to 1-D cylindrical cylindrical (1) I dE 0 U1 T dT V battery thermal model has been proposed to perform on-line b estimation of the internal and external temperatures. In order A finitethermal difference method forproposed solving to a perform 1-D cylindrical dT where Vb is the battery volume; I is the charge q VIb Ecell U T (1)or battery model has been on-line dE 0 1 A verify finitethermal difference method for solving a perform 1-D and cylindrical V dT q E U T estimation of the internal and external temperatures. In order to the validity of this method, the inside outside (1)is battery model has been proposed to on-line b 0battery 1 volume; where V is the battery cell I is the charge is the open circuit voltage; U discharge current; E b 0 estimation of the internal and external temperatures. In order Vb cell volume; dT I is the charge 1 or where Vb is the battery or battery thermal model hasthis been proposed to perform on-line to verify the validity of method, the inside and outside temperatures of the battery at different charge and discharge estimation of the internal and external temperatures. In order battery open circuit voltage; U is discharge the terminal Tthe is cell the volume; thermodynamic temperature; where Vb current; is voltage; the E battery I is the charge 0 is 1 or to verify the validity of this method, the inside and Inoutside is the battery open circuit voltage; U is discharge current; E 0 1 estimation of the internal and external temperatures. order where Vthe isentropy the E battery cell volume; I is the charge or temperatures of the battery different charge discharge rates were maximum error between the the b current; to verify the measured. validity of The thisat method, the insideand and outside terminal voltage; T is the thermodynamic temperature; dE/dT is coefficient. is the battery open circuit voltage; U is discharge 0 1 temperatures of the battery at different charge and discharge the terminal voltage; T is the thermodynamic temperature; to verify the validity of this method, the inside and outside isTthe battery open circuit voltage; U1 is discharge current; E0coefficient. rates were measured. The error between the calculation results experimental isand about 4.5℃. temperatures of theand battery at maximum differentresults charge discharge dE/dT is the entropy the terminal voltage; is the thermodynamic temperature; rates were measured. The maximum error between the dE/dT is the entropy temperatures of theand battery differentresults charge and discharge terminal voltage;coefficient. T is the thermodynamic temperature; calculation results experimental about 4.5℃. Using the measured external of is the battery, the the rates were measured. Theat temperature maximum error between dE/dT is the entropy coefficient. calculation results and experimental results is about 4.5℃. 2.3 Finite difference thermal model rates were measured. The maximum error between the dE/dT is the entropy coefficient. Using the measured external temperature of the battery, the internal temperature can be corrected. And the error is within calculation results and experimental results is about 4.5℃. Using the measured external temperature of is theabout battery, the 2.3 Finite difference thermal model calculation results and experimental results 4.5℃. internal temperature can be corrected. And the error is within 1℃. Using the measured external temperature of the battery, the 2.3 Finite difference thermal model internal temperature can be corrected. And the error is within conduction refers to model the direct contact of different 2.3 Finite difference thermal Using the measured external temperature of the battery, the Heat 1℃. internal temperature can be corrected. And the error is within 2.3 Finite difference thermal model 1℃. Heat conduction refers to contact different objects or the same object which does not occurof to 2. BATTERY THERMAL MODEL internal temperature can be corrected. And the error is within Heat conduction refers to the the direct direct contact ofrelative different 1℃. objects or the same object which does not occur relative to the part of movement due to the existence of temperature Heat conduction refers to the direct contact of different 2. BATTERY THERMAL MODEL 1℃. objects or the same object which does not occurofrelative to 2. BATTERY THERMAL MODEL Heat conduction refers to the direct contact different the part of movement due to the existence of temperature differences relying on material molecules, atoms and free objects or the same object which does not occur relative to 2. BATTERY THERMAL MODEL the part of the movement due to the existence of temperature 2.1 Battery Introduction objects sameinon object which does not diffusion occur relative to 2. BATTERY THERMAL MODEL differences relying molecules, atoms and free electrons occur thematerial heat and process the part or oftothe the movement due totransfer the existence of temperature differences relying on material molecules, atoms and free 2.1 the partetofal, the movement due process totransfer the existence of temperature 2.1 Battery Battery Introduction Introduction electrons to occur in the heat and diffusion process (Yang 2007) The whole can be described by the differences relying on material molecules, atoms and free electrons to relying occur inonthematerial heat transfer and diffusion process 2.1 Battery Introduction differences molecules, atoms and free (Yang et al, whole can described by electrons to 2007) occur The in the heatprocess transfer andbe diffusion process 2.1 Battery Introduction (Yang et al, 2007) The whole process can be described by the the electrons to occur in the heat transfer and diffusion process (Yang et al, 2007) The whole process can be described by the Copyright © 2018 IFAC 281 (Yang et al, 2007) The whole process can be described by the 2405-8963 © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright 2018 responsibility IFAC 281Control. Peer review© of International Federation of Automatic Copyright ©under 2018 IFAC 281 Copyright © 2018 IFAC 281 10.1016/j.ifacol.2018.10.046 Copyright © 2018 IFAC 281
IFAC E-CoSM 2018 Changchun, China, September 20-22, 2018Li Xinggang et al. / IFAC PapersOnLine 51-31 (2018) 258–261
259
1-D cylinder Fourier thermal differential equation as shown in the equation: T 2T T (2) r 2 r q C t r r r where ρ is the cell density; C is the specific heat capacity of the battery; T represents the battery temperature; λr is the thermal conductivity in the radial direction of the cylinder battery. Equations are written in a differential format at the k-th space node and the n-th time node： T T T 2Tn, k Tn, k 1 r Tn, k 1 Tn, k ( ) q (3) C n 1, k n, k r n, k 1 2
this study, the entropy coefficient is measured at 10 different SOCs. The results are shown in the Fig.1.
Boundary conditions can be written as the following expression： T (4) r h TS Tamb r r R
Fig.1 .Entropy coefficient at different SOC
t
r
r
-4
Entropy Coefficient (V/K)
x 10 Entropy coefficient at different SOC
-3 -4 20%
40% 60% SOC
80%
100%
3.2 Calculation results The experimental verification includes five tests under different charge and discharge rates. They are 1C, 1.5C, 2C and 3C for discharge and 1C for charge. Simultaneously, the internal and external temperatures are measured respectively by two sensors, one of which is inserted into the battery core and the other is attached to the surface. For convenience, Te-in, Te-out, Tm-in and Tm-out are used to indicate the estimate and measured value of T1 and T5. The following are the results. Temperature(℃)
40 35 Te-out Te-in Tm-in Tm-out
30 25 20
0
1000
2000
3000 Time(s) (a)
4000
5000
6000
Error(℃)
1 0.5 0 -0.5 -1
Error-in Error-out
0
1000
2000
3000 Time(s) (b)
4000
5000
6000
Fig.2 Comparison of temperature at 1C discharge rate Temperature(℃)
cr 2 cr r r D hr hr E hr
-2
-5 0
(11)
50 Te-in Te-out Tm-in Tm-out
45 40 35 30 25
(12)
0
500
1000
1500
2000 2500 Time(s) (a)
3000
3500
4000
4500
1.5
(13) Error(℃)
C
r
-1
r
T 0 (5) r r 0 Similarly, it can be written as: T T r n, k max 1 n, k max h(Tn, k max Tamb ) (6) r Tn ,1 Tn，0 =0 (7) r In this study, we divide the battery into five nodes in radial direction. The iteration time is 1s. And the iteration matrix can be written as: Tn 1,0 A B C 0 0 0 0 Tn ,0 q q T T n 1,1 A B C 0 0 0 0 n ,1 T 0 A B C 0 0 0 T q n 1,2 n ,2 1 Tn 1,3 0 0 A B C 0 0 * Tn ,3 q (8) c q Tn 1,4 0 0 0 A B C 0 Tn ,4 T 0 0 0 0 D 0 E T 0 n ,5 n 1,5 0 0 0 0 0 0 0 1 Tamb Tamb Where r A (9) cr 2 2r r (10) B 1 cr 2 cr r
r
0
3. RESULTS 3.1 Entropy coefficient
1 0.5 0 -0.5
The entropy coefficient of the battery is an important parameter affecting the heat generation (Wang et al, 2017). In
Error-in Error-out
0
500
1000
1500
2000 2500 Time(s) (b)
3000
3500
4000
4500
Fig.3 Comparison of temperature at 1.5C discharge rate 282
IFAC E-CoSM 2018 260 Changchun, China, September 20-22, 2018Li Xinggang et al. / IFAC PapersOnLine 51-31 (2018) 258–261
Temperature(℃)
60
high currents. In addition, the battery has a large error in the static cooling stage. This is a flaw of the model. The onedimensional thermal model ignores the heat dissipation of the battery in the axial direction. And the axial thermal conductivity is far greater than the radial thermal conductivity, resulting in the calculation heat dissipation is less than the actual heat.
Te-out Te-in Tm-out Tm-in
50 40 30 20
0
500
1000
1500
2000 2500 Time(s) (a)
3000
3500
4000
At the same time, the results show that there is similarity between the surface temperature estimation error and the internal temperature estimation error. Using this property, we can improve the accuracy of the internal temperature estimation based on the error of the battery surface temperature.
2
Error(℃)
1 0 -1 -2
Error-in Error-out
0
500
1000
1500
2000 2500 Time(s) (b)
3000
3500
4000
The battery external temperature estimation error can be expressed as:
Fig.4 Comparison of temperature at 2C discharge rate
external Te out Tm out
Temperature(℃)
80 Te-out Te-in Tm-in Tm-out
60
The internal temperature of the battery after error compensation can be written as
40 20
Te'in Te in external 0
500
1000
1500 Time(s) (a)
2000
2500
Temperature(℃)
Error(℃)
0 Error-in Error-out
-5
0
500
1000
1500 Time(s) (b)
2000
2500
3000
80 T'e-in Tm-in
60 40 20
Fig.5 Comparison of temperature at 3C discharge rate
0
500
1000
35 Te-out Te-in Tm-in Tm-out
30 25 0
1000
2000 3000 Time(s) (a)
4000
2000
2500
3000
Error-in
5000
1 0 -1
1
0
500
1000
1500 Time(s) (b)
2000
2500
3000
Fig.7 Internal temperature estimation result at 3C discharge rate after error compensation
0.5 0 -0.5 0
1500 Time(s) (a)
2 Error(℃)
Temperature(℃)
(15)
The result with the greatest error is illustrated as an example (3C discharge). Assuming that the external temperature of the battery is measured by the sensor, the internal temperature is estimated using the above method. Fig.7 shows the internal temperature estimation result after the error compensation.
3000
5
Error(℃)
(14)
Error-in Error-out
1000
2000 3000 Time(s) (b)
4000
After error compensation, the battery's internal temperature estimation error is reduced from the previous 4.5°C to 1°C at 3C discharge rate. Its accuracy has greatly improved. The results show that the internal temperature of the battery can be accurately estimated by a combination of the thermal model and surface temperature measurement.
5000
Fig.6 Comparison of temperature at 1C charge rate Fig.2-6 shows the temperature prediction results at different charge and discharge rates. It is found that there exist some errors of the temperature estimation. When the charge/discharge rate is 1C, the temperature error can be controlled within 1°C. However, the error increases as the discharge rate raises. When the discharge rate is 3C, the estimation error is maximum, which can reach about 4.5 °C. It is because that the heat generation model is inaccurate at
4. CONCLUSION In this paper, an approach to estimate the internal and external temperature of the battery is introduced. Cylindrical batteries are considered as a 1-D thermal model. Combined with the Bernardi heat generation model, the temperatures of the battery are solved using the finite difference method. 283
IFAC E-CoSM 2018 Changchun, China, September 20-22, 2018Li Xinggang et al. / IFAC PapersOnLine 51-31 (2018) 258–261
Temperature estimation error is within 4.5°C. If the battery's surface temperature can be measured, the internal temperature estimation error can be reduced to less than 1°C using the similarity of the error. It solves the defect that the battery's internal temperature cannot be obtained in the battery management system. ACKNOWLEDGEMENTS This work was supported in part by the National Natural Science Foundation of China (Grant No. 51507012) and Beijing Municipal Natural Science Foundation of China (Grant No. 3182035). The systemic experiments of the lithium-ion batteries were performed at the Advanced Energy Storage and Application (AESA) Group, Beijing Institute of Technology. REFERENCES Liu, G. M., Ouyang, M. G. and et al. (2013). Online estimation of internal temperature fields of lithium—ion batteries using a transfer function. Journal of Automotive Safety and Engergy, 4 (1), 61-66. Zhang, J. B., Wu, B., and et al. (2014). Simultaneous estimation of thermal parameters for large-format laminated lithium-ion batteries. Journal of Power Sources, 259,106-116. Chen, Z. Y., Xiong, R., Lu, J. H. and Li, X. G. (2018). Temperature rise prediction of lithium-ion battery suffering external short circuit for all-climate electric vehicles application. Applied Energy, volume 213, 375383. Forgez, C., Do, D. V., Friedrich, G., Morcrette, M. and Delacourt, C. (2010). Thermal modeling of a cylindrical lifepo4/graphite lithium-ion battery. Journal of Power Sources, volume 195, 2961-2968. Song, L., Wei, X.Z., Dai, H.F. and Sun, Z.C. (2013). A Review on the Research of Thermal Models for Lithium Ion Battery Cell. Automotive Engineering, 35(3), 285291. Bernardi, D., Pawlikowski, E. and Newman, J. (1985). A general energy balance for battery systems. J. Electrochem. Soc.: Journal of Electrochemical Science and Technology, 132(1), 5-12. Yang S M, Tao W Q. (2007) Heat transfer, 25-28. Higher Education Press, Beijing. Wang, K. K., Gao, F., and et al. (2017) Research of LiFePO4/C Energy Storage Batteries Entropy Coefficient and Discharge Heat Generation Based on the State of Health. High Voltage Engineering, 43(7), 2241-2248.
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IFAC PapersOnLine 51-31 (2018) 258–261 An approach to internal and external temperature estimation for cylindrical An approach to internal and external temperature An approach to internal and external temperature estimation for cylindrical cylindrical battery based on finite difference estimation method for An approach to internal and external temperature estimation for cylindrical battery based on finite difference method An approach to internal external temperature batteryand based on finite difference estimation method for cylindrical battery basedLion finite difference Xinggang, Xiong Rui* method battery basedLion finite difference Xinggang, Xiong Rui* method
Li Xinggang, Xiong Rui* Li Xinggang, Xiong Rui* * National Engineering Laboratory for Electric Vehicles, School of Mechanical Engineering, Beijing Institute of Technology, Li Xinggang, Xiong Rui* ** National Engineering Laboratory for Electric Vehicles, School of Mechanical Beijing Beijing, 100081, China;( e-mail: [email protected], [email protected] National Engineering Laboratory for Electric Vehicles, School of Mechanical Engineering, Engineering, Beijing) Institute Institute of of Technology, Technology, 100081, China;( e-mail: [email protected], [email protected] ) * National Engineering Beijing, Laboratory for Electric Vehicles, School of Mechanical Engineering, Beijing Institute of Technology, Beijing, 100081, China;( e-mail: [email protected], [email protected] ) * National Engineering Beijing, Laboratory for Electric School of Mechanical Engineering, Beijing) Institute of Technology, 100081, China;(Vehicles, e-mail: [email protected], [email protected] Beijing, of 100081, China;( [email protected], [email protected] Abstract: A method estimating the e-mail: temperature for cylindrical battery is expounded.) Based on the Abstract: A method of of estimating estimating the the temperature for cylindrical battery isgenerate expounded. Based on the simplified A one-dimensional（1-D） thermal model for andcylindrical the Bernardi heatis model, theon finite Abstract: method temperature battery expounded. Based the simplified one-dimensional（1-D） thermal model and the Bernardi heat generate model, the finite Abstract: A method of estimating the temperature for cylindrical battery is expounded. Based on the difference method is used to calculate the internal and external temperatures of the battery. And different simplified one-dimensional（1-D） thermal model for andcylindrical the Bernardi heatisgenerate model, theon finite Abstract: A method of estimating the temperature battery expounded. Based the simplified one-dimensional（1-D） thermal model the Bernardi heat model, finite difference method is used used to calculate calculate the internal internal and and external temperatures ofgenerate the battery. battery. Andthe different charge andmethod discharge condition experiments are performed. Comparisons between the estimation results difference is to the and external temperatures of the And different simplified one-dimensional（1-D） thermal model and the Bernardi heat generate model, the finite charge andmethod discharge condition experiments are performed. Comparisons between the estimation results and measured results proved theexperiments accuracy ofare theperformed. model. TheComparisons maximum estimation error isAnd about 4.5℃. difference is used to calculate the internal and external temperatures of the battery. different charge and discharge condition between the estimation results difference method is external used to calculate the internal and external temperatures of thecan battery. And different and measured results proved the accuracy of the model. The maximum estimation error is about 4.5℃. Using the measured temperature of the battery, the internal temperature be corrected. And charge and discharge condition experiments are performed. Comparisons between the estimation results and measured results proved theexperiments accuracy ofare theperformed. model. TheComparisons maximum estimation error is about results 4.5℃. charge and discharge condition betweencan the estimation Using the measured external temperature of the battery, the internal internal temperature be corrected. And the error ismeasured within 1℃. and measured results proved the accuracy of of the the battery, model. The maximum estimation error is about 4.5℃. Using the external temperature the temperature can be corrected. And and measured results proved the accuracy of of the the battery, model. The maximum estimationcan error is about 4.5℃. the error is within 1℃. Using the measured external temperature the internal temperature be corrected. And the error ismeasured within 1℃. © 2018, IFAC (International Federation of Automatic Control) Hosting bytemperature Elsevier Ltd. Alldifference rights reserved. Keywords: cylindrical battery, battery management system, temperature, finite method Using the external temperature of the battery, theinternal internal can be corrected. And the error is within 1℃. Keywords: cylindrical battery, battery management system, internal temperature, finite difference method the error is within 1℃. Keywords: cylindrical battery, battery management system, internal temperature, finite difference method Keywords: cylindrical battery, battery management system, internal temperature, finite difference method Keywords:1.cylindrical battery, battery management system, internalcylindrical temperature, finite difference INTRODUCTION A NMC battery produced method by a domestic 1. INTRODUCTION A NMC cylindrical battery produced domestic manufacturer is used in this study. The specific 1. INTRODUCTION A NMC cylindrical battery produced by by aa parameters domestic Lithium-ion batteries are used increasingly in electric manufacturer is used in this study. The specific parameters are shown in Table 1. 1. INTRODUCTION A NMC cylindrical battery produced by a domestic manufacturer is used in this study. The specific parameters Lithium-ion batteries are used in electric vehicles because of1.the advantages of high energy INTRODUCTION NMC incylindrical battery produced by a parameters domestic are shown Table 1. manufacturer is used in this study. The specific Lithium-ion batteries are used increasingly increasingly in density, electric A are shown in Table 1. in this study. The specific parameters vehicles because of the advantages of high energy density, high power density, and so on. Battery temperature is an manufacturer is used Lithium-ion batteries are used increasingly in electric are shownTable in Table 1. vehicles because of the are advantages of high energy of experimental battery 1. Parameters Lithium-ion batteries used increasingly in density, electric high power density, and so on. Battery temperature is an important parameter the battery management system (Liu are shownTable in Table 1. vehicles because of for the advantages of high energy density, Parameters battery 1. high power density, and so on. Battery temperature is an vehicles because the advantages of high energy density, of experimental experimentalΦ18mm*65mm battery TableDimension 1. Parameters of important parameter the battery management system (Liu et al, power 2013). Dueoftofor the poor conductivity, there high density, and so on.thermal Battery temperature is an important parameter for the battery management system (Liu Parameters of experimental battery Table 1. high power density, and so on. Battery temperature is an Dimension Φ18mm*65mm et al, 2013). Due to the poor thermal conductivity, there exists a difference between the cell surface temperature and important parameter forthe thepoor battery management systemthere (Liu Nominal capacity of experimentalΦ18mm*65mm 2500mAh battery Table 1. Parameters Dimension et al, 2013). Due to thermal conductivity, important parameter the battery system (Liu Dimension Φ18mm*65mm exists aa2013). difference the cell surface temperature and the internal temperature (Zhang et management al. 2014). But onlythere the et al, Due between toforthe poor thermal conductivity, Nominal capacity 2500mAh Upper cutoff voltage 4.20V exists difference between the cell surface temperature and Nominal capacity 2500mAh Dimension Φ18mm*65mm et al, Due to the(Zhang poor thermal conductivity, there the internal temperature et al. 2014). the battery surface temperature can cell be measured inBut realonly vehicle exists a2013). difference between the surface temperature and Nominal capacity 2500mAh Upper cutoff voltage 4.20V the internal temperature (Zhang et al. 2014). But only the Lower 3.0V Upper cutoff voltage 4.20V exists a surface difference between the cell surface temperature Nominal capacity 2500mAh battery temperature can be measured in real vehicle operation, which cannot reflect state of and the the internal temperature (Zhang etthe al. internal 2014). But only Upper cutoff voltage 4.20V battery surface temperature can be measured in real vehicle Lower 3.0V Maximum charging current 2400mA the internal temperature (Zhang al. internal 2014).inBut the Lower cutoff voltage 3.0V Upper cutoff voltage 4.20V operation, which cannot reflect state of the battery truly. Therefore, obtaining the accurate internal and surface temperature can beetthe measured realonly vehicle operation, which cannot reflect the internal state of the Lower cutoff voltage 3.0V Maximum charging current 2400mA Maximum current 7200mA battery surface temperature can battery bethe measured instate real vehicle Maximum charging current 2400mA battery truly. Therefore, the accurate internal external temperatures ofobtaining the has very important operation, which cannot reflect internal of and the Lowerdischarging cutoff voltage 3.0V battery truly. Therefore, obtaining the accurate internal and Maximum charging current 2400mA Maximum discharging current operation, which cannot reflect the internal state of and the Charging working temperature 07200mA ℃~45℃ external temperatures of the battery has very important significance for safety and durability (Chen et al, 2018). Maximum discharging current 7200mA battery truly. Therefore, obtaining the accurate internal Maximum charging current 2400mA external temperatures ofobtaining the battery has veryinternal important Maximumworking discharging current 7200mA battery truly. Therefore, the accurate and Charging ℃ significance for safety and (Chen al, 2018). external temperatures of durability the battery haset very important Discharging workingtemperature temperature -20 ℃~45 ~60℃ ℃ Charging temperature ℃ ~45 ℃ 007200mA significance for safety and durability (Chen et al, 2018). Maximumworking discharging current The internal temperature of the battery can be measured by external temperatures of the battery has very important Charging working temperature 0 ℃ ~45 ℃ significance for safety and durability (Chen et al, 2018). Discharging working temperature -20 ℃ ~60 ℃ Discharging workingtemperature temperature ℃~45 ~60℃ ℃ -20 Charging working 0℃ The internal temperature of the battery be by the built-in sensor etcan al,et2010). But it can significance for safety and durability (Chen al,measured 2018). The internaltemperature temperature of the(Forgez battery can be measured by 2.2Discharging working temperature -20℃~60℃ Heat generation model the built-in temperature sensor (Forgez et al, 2010). But it can be performed only during experiments, and not applicable in The internal temperature of the battery can be measured by Discharging working temperature -20℃~60℃ the built-in temperature sensor etcan al, 2010). But it can 2.2 The internal temperature of the(Forgez battery befinite measured by 2.2 Heat Heat generation generation model model be performed only during experiments, applicable in real car.temperature In addition, though the traditional element the built-in sensor (Forgez etand al, not 2010). But it can It is difficult to obtain the heat generation rate of the battery be performed only during experiments, and not applicable in 2.2 Heat generation model the built-in temperature sensor (Forgez al, not 2010). But it can the real car. In addition, the traditional finite element method is usually used tothough calculate theetand internal temperature, be performed only during experiments, applicable in 2.2 Heat generation model It is difficult to obtain the heat generation rate the accurately by experimental methods. The D. the real car. In addition, though the traditional finite element ofexisting the battery battery be performed during experiments, and notfinite applicable in It is difficult to obtain the heat generation rate of method is used to calculate the internal temperature, the calculation is too large to bethe applied online (Song et al, real car.usually Inonly addition, though traditional element accurately by experimental methods. The existing D. Bernardi (Bernardi et al, 1985) model is used to calculate the It is difficult to obtain the heat generation rate of the battery method is usually used to calculate the internal temperature, accurately byto obtain experimental methods. The existing D. the real car.usually In addition, the traditional finite element the calculation is too to be applied online (Song et al, 2013). It is generation difficult the1985) heat generation rate ofcalculate the battery method is usedlarge tothough calculate the internal temperature, Bernardi (Bernardi et al, model is used to the heat rate, which can be expressed as: accurately by experimental methods. The existing D. the calculation is too large to be applied online (Song et al, Bernardi (Bernardi et al, 1985) model is used to calculate the method is usually used to calculate the internal temperature, 2013). accurately by rate, experimental methods. The existing the D. the calculation is too large to be applied online (Song et al, heat generation which can be expressed as: Bernardi (Bernardi et al, 1985) model is used to calculate 2013). I dE generation rate,et as: A finite difference solving online a 1-D (Song cylindrical the calculation is toomethod large tofor be applied et al, heat al, 1985) U1 be Texpressed q which E0can (1) Bernardi (Bernardi model is used to calculate the 2013). I can be expressed dE generation rate, which as: dT A difference for solving aa perform 1-D battery modelmethod has been on-line heat 2013). dE VIb EE0can U1 be Texpressed (1) heat generation rate,qq which as: A finite finitethermal difference method forproposed solving to 1-D cylindrical cylindrical (1) I dE 0 U1 T dT V battery thermal model has been proposed to perform on-line b estimation of the internal and external temperatures. In order A finitethermal difference method forproposed solving to a perform 1-D cylindrical dT where Vb is the battery volume; I is the charge q VIb Ecell U T (1)or battery model has been on-line dE 0 1 A verify finitethermal difference method for solving a perform 1-D and cylindrical V dT q E U T estimation of the internal and external temperatures. In order to the validity of this method, the inside outside (1)is battery model has been proposed to on-line b 0battery 1 volume; where V is the battery cell I is the charge is the open circuit voltage; U discharge current; E b 0 estimation of the internal and external temperatures. In order Vb cell volume; dT I is the charge 1 or where Vb is the battery or battery thermal model hasthis been proposed to perform on-line to verify the validity of method, the inside and outside temperatures of the battery at different charge and discharge estimation of the internal and external temperatures. In order battery open circuit voltage; U is discharge the terminal Tthe is cell the volume; thermodynamic temperature; where Vb current; is voltage; the E battery I is the charge 0 is 1 or to verify the validity of this method, the inside and Inoutside is the battery open circuit voltage; U is discharge current; E 0 1 estimation of the internal and external temperatures. order where Vthe isentropy the E battery cell volume; I is the charge or temperatures of the battery different charge discharge rates were maximum error between the the b current; to verify the measured. validity of The thisat method, the insideand and outside terminal voltage; T is the thermodynamic temperature; dE/dT is coefficient. is the battery open circuit voltage; U is discharge 0 1 temperatures of the battery at different charge and discharge the terminal voltage; T is the thermodynamic temperature; to verify the validity of this method, the inside and outside isTthe battery open circuit voltage; U1 is discharge current; E0coefficient. rates were measured. The error between the calculation results experimental isand about 4.5℃. temperatures of theand battery at maximum differentresults charge discharge dE/dT is the entropy the terminal voltage; is the thermodynamic temperature; rates were measured. The maximum error between the dE/dT is the entropy temperatures of theand battery differentresults charge and discharge terminal voltage;coefficient. T is the thermodynamic temperature; calculation results experimental about 4.5℃. Using the measured external of is the battery, the the rates were measured. Theat temperature maximum error between dE/dT is the entropy coefficient. calculation results and experimental results is about 4.5℃. 2.3 Finite difference thermal model rates were measured. The maximum error between the dE/dT is the entropy coefficient. Using the measured external temperature of the battery, the internal temperature can be corrected. And the error is within calculation results and experimental results is about 4.5℃. Using the measured external temperature of is theabout battery, the 2.3 Finite difference thermal model calculation results and experimental results 4.5℃. internal temperature can be corrected. And the error is within 1℃. Using the measured external temperature of the battery, the 2.3 Finite difference thermal model internal temperature can be corrected. And the error is within conduction refers to model the direct contact of different 2.3 Finite difference thermal Using the measured external temperature of the battery, the Heat 1℃. internal temperature can be corrected. And the error is within 2.3 Finite difference thermal model 1℃. Heat conduction refers to contact different objects or the same object which does not occurof to 2. BATTERY THERMAL MODEL internal temperature can be corrected. And the error is within Heat conduction refers to the the direct direct contact ofrelative different 1℃. objects or the same object which does not occur relative to the part of movement due to the existence of temperature Heat conduction refers to the direct contact of different 2. BATTERY THERMAL MODEL 1℃. objects or the same object which does not occurofrelative to 2. BATTERY THERMAL MODEL Heat conduction refers to the direct contact different the part of movement due to the existence of temperature differences relying on material molecules, atoms and free objects or the same object which does not occur relative to 2. BATTERY THERMAL MODEL the part of the movement due to the existence of temperature 2.1 Battery Introduction objects sameinon object which does not diffusion occur relative to 2. BATTERY THERMAL MODEL differences relying molecules, atoms and free electrons occur thematerial heat and process the part or oftothe the movement due totransfer the existence of temperature differences relying on material molecules, atoms and free 2.1 the partetofal, the movement due process totransfer the existence of temperature 2.1 Battery Battery Introduction Introduction electrons to occur in the heat and diffusion process (Yang 2007) The whole can be described by the differences relying on material molecules, atoms and free electrons to relying occur inonthematerial heat transfer and diffusion process 2.1 Battery Introduction differences molecules, atoms and free (Yang et al, whole can described by electrons to 2007) occur The in the heatprocess transfer andbe diffusion process 2.1 Battery Introduction (Yang et al, 2007) The whole process can be described by the the electrons to occur in the heat transfer and diffusion process (Yang et al, 2007) The whole process can be described by the Copyright © 2018 IFAC 281 (Yang et al, 2007) The whole process can be described by the 2405-8963 © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright 2018 responsibility IFAC 281Control. Peer review© of International Federation of Automatic Copyright ©under 2018 IFAC 281 Copyright © 2018 IFAC 281 10.1016/j.ifacol.2018.10.046 Copyright © 2018 IFAC 281
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259
1-D cylinder Fourier thermal differential equation as shown in the equation: T 2T T (2) r 2 r q C t r r r where ρ is the cell density; C is the specific heat capacity of the battery; T represents the battery temperature; λr is the thermal conductivity in the radial direction of the cylinder battery. Equations are written in a differential format at the k-th space node and the n-th time node： T T T 2Tn, k Tn, k 1 r Tn, k 1 Tn, k ( ) q (3) C n 1, k n, k r n, k 1 2
this study, the entropy coefficient is measured at 10 different SOCs. The results are shown in the Fig.1.
Boundary conditions can be written as the following expression： T (4) r h TS Tamb r r R
Fig.1 .Entropy coefficient at different SOC
t
r
r
-4
Entropy Coefficient (V/K)
x 10 Entropy coefficient at different SOC
-3 -4 20%
40% 60% SOC
80%
100%
3.2 Calculation results The experimental verification includes five tests under different charge and discharge rates. They are 1C, 1.5C, 2C and 3C for discharge and 1C for charge. Simultaneously, the internal and external temperatures are measured respectively by two sensors, one of which is inserted into the battery core and the other is attached to the surface. For convenience, Te-in, Te-out, Tm-in and Tm-out are used to indicate the estimate and measured value of T1 and T5. The following are the results. Temperature(℃)
40 35 Te-out Te-in Tm-in Tm-out
30 25 20
0
1000
2000
3000 Time(s) (a)
4000
5000
6000
Error(℃)
1 0.5 0 -0.5 -1
Error-in Error-out
0
1000
2000
3000 Time(s) (b)
4000
5000
6000
Fig.2 Comparison of temperature at 1C discharge rate Temperature(℃)
cr 2 cr r r D hr hr E hr
-2
-5 0
(11)
50 Te-in Te-out Tm-in Tm-out
45 40 35 30 25
(12)
0
500
1000
1500
2000 2500 Time(s) (a)
3000
3500
4000
4500
1.5
(13) Error(℃)
C
r
-1
r
T 0 (5) r r 0 Similarly, it can be written as: T T r n, k max 1 n, k max h(Tn, k max Tamb ) (6) r Tn ,1 Tn，0 =0 (7) r In this study, we divide the battery into five nodes in radial direction. The iteration time is 1s. And the iteration matrix can be written as: Tn 1,0 A B C 0 0 0 0 Tn ,0 q q T T n 1,1 A B C 0 0 0 0 n ,1 T 0 A B C 0 0 0 T q n 1,2 n ,2 1 Tn 1,3 0 0 A B C 0 0 * Tn ,3 q (8) c q Tn 1,4 0 0 0 A B C 0 Tn ,4 T 0 0 0 0 D 0 E T 0 n ,5 n 1,5 0 0 0 0 0 0 0 1 Tamb Tamb Where r A (9) cr 2 2r r (10) B 1 cr 2 cr r
r
0
3. RESULTS 3.1 Entropy coefficient
1 0.5 0 -0.5
The entropy coefficient of the battery is an important parameter affecting the heat generation (Wang et al, 2017). In
Error-in Error-out
0
500
1000
1500
2000 2500 Time(s) (b)
3000
3500
4000
4500
Fig.3 Comparison of temperature at 1.5C discharge rate 282
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Temperature(℃)
60
high currents. In addition, the battery has a large error in the static cooling stage. This is a flaw of the model. The onedimensional thermal model ignores the heat dissipation of the battery in the axial direction. And the axial thermal conductivity is far greater than the radial thermal conductivity, resulting in the calculation heat dissipation is less than the actual heat.
Te-out Te-in Tm-out Tm-in
50 40 30 20
0
500
1000
1500
2000 2500 Time(s) (a)
3000
3500
4000
At the same time, the results show that there is similarity between the surface temperature estimation error and the internal temperature estimation error. Using this property, we can improve the accuracy of the internal temperature estimation based on the error of the battery surface temperature.
2
Error(℃)
1 0 -1 -2
Error-in Error-out
0
500
1000
1500
2000 2500 Time(s) (b)
3000
3500
4000
The battery external temperature estimation error can be expressed as:
Fig.4 Comparison of temperature at 2C discharge rate
external Te out Tm out
Temperature(℃)
80 Te-out Te-in Tm-in Tm-out
60
The internal temperature of the battery after error compensation can be written as
40 20
Te'in Te in external 0
500
1000
1500 Time(s) (a)
2000
2500
Temperature(℃)
Error(℃)
0 Error-in Error-out
-5
0
500
1000
1500 Time(s) (b)
2000
2500
3000
80 T'e-in Tm-in
60 40 20
Fig.5 Comparison of temperature at 3C discharge rate
0
500
1000
35 Te-out Te-in Tm-in Tm-out
30 25 0
1000
2000 3000 Time(s) (a)
4000
2000
2500
3000
Error-in
5000
1 0 -1
1
0
500
1000
1500 Time(s) (b)
2000
2500
3000
Fig.7 Internal temperature estimation result at 3C discharge rate after error compensation
0.5 0 -0.5 0
1500 Time(s) (a)
2 Error(℃)
Temperature(℃)
(15)
The result with the greatest error is illustrated as an example (3C discharge). Assuming that the external temperature of the battery is measured by the sensor, the internal temperature is estimated using the above method. Fig.7 shows the internal temperature estimation result after the error compensation.
3000
5
Error(℃)
(14)
Error-in Error-out
1000
2000 3000 Time(s) (b)
4000
After error compensation, the battery's internal temperature estimation error is reduced from the previous 4.5°C to 1°C at 3C discharge rate. Its accuracy has greatly improved. The results show that the internal temperature of the battery can be accurately estimated by a combination of the thermal model and surface temperature measurement.
5000
Fig.6 Comparison of temperature at 1C charge rate Fig.2-6 shows the temperature prediction results at different charge and discharge rates. It is found that there exist some errors of the temperature estimation. When the charge/discharge rate is 1C, the temperature error can be controlled within 1°C. However, the error increases as the discharge rate raises. When the discharge rate is 3C, the estimation error is maximum, which can reach about 4.5 °C. It is because that the heat generation model is inaccurate at
4. CONCLUSION In this paper, an approach to estimate the internal and external temperature of the battery is introduced. Cylindrical batteries are considered as a 1-D thermal model. Combined with the Bernardi heat generation model, the temperatures of the battery are solved using the finite difference method. 283
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Temperature estimation error is within 4.5°C. If the battery's surface temperature can be measured, the internal temperature estimation error can be reduced to less than 1°C using the similarity of the error. It solves the defect that the battery's internal temperature cannot be obtained in the battery management system. ACKNOWLEDGEMENTS This work was supported in part by the National Natural Science Foundation of China (Grant No. 51507012) and Beijing Municipal Natural Science Foundation of China (Grant No. 3182035). The systemic experiments of the lithium-ion batteries were performed at the Advanced Energy Storage and Application (AESA) Group, Beijing Institute of Technology. REFERENCES Liu, G. M., Ouyang, M. G. and et al. (2013). Online estimation of internal temperature fields of lithium—ion batteries using a transfer function. Journal of Automotive Safety and Engergy, 4 (1), 61-66. Zhang, J. B., Wu, B., and et al. (2014). Simultaneous estimation of thermal parameters for large-format laminated lithium-ion batteries. Journal of Power Sources, 259,106-116. Chen, Z. Y., Xiong, R., Lu, J. H. and Li, X. G. (2018). Temperature rise prediction of lithium-ion battery suffering external short circuit for all-climate electric vehicles application. Applied Energy, volume 213, 375383. Forgez, C., Do, D. V., Friedrich, G., Morcrette, M. and Delacourt, C. (2010). Thermal modeling of a cylindrical lifepo4/graphite lithium-ion battery. Journal of Power Sources, volume 195, 2961-2968. Song, L., Wei, X.Z., Dai, H.F. and Sun, Z.C. (2013). A Review on the Research of Thermal Models for Lithium Ion Battery Cell. Automotive Engineering, 35(3), 285291. Bernardi, D., Pawlikowski, E. and Newman, J. (1985). A general energy balance for battery systems. J. Electrochem. Soc.: Journal of Electrochemical Science and Technology, 132(1), 5-12. Yang S M, Tao W Q. (2007) Heat transfer, 25-28. Higher Education Press, Beijing. Wang, K. K., Gao, F., and et al. (2017) Research of LiFePO4/C Energy Storage Batteries Entropy Coefficient and Discharge Heat Generation Based on the State of Health. High Voltage Engineering, 43(7), 2241-2248.
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