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The pseudo-two-dimensional (P2D) model is the most widely used electrochemical model for lithium-ion batteries. Because of the complexity and the difficulty in using the complete P2D model, many simplified P2D models, such as the single particle model (SP model) and the parabolic profile approximation model (PP model), have been proposed. However, the using of the SP model can cause a large amount of precision to lose in its simplified process, while the PP model has a high complexity. In this paper, we propose a liquid phase simplification P2D (LSP2D) model. The using of the LSP2D model has a small precision loss and a relatively low complexity. The LSP2D model is based on the electrochemical average kinetics of the lithium ion battery. We first simplify the terminal voltage into an equation containing only the solid phase concentration cs and the liquid phase concentration ce. Then we use the partial differential equation to represent the solid phase concentration cs and the liquid phase concentration ce, and then obtain a final model. The simulation environment is based on COMSOL, and the simulation results show that when the discharge rate is 1C, the estimation accuracy and speed from the LSP2D model are similar to those from the SP model. But when the discharge rate is 3C, the estimation time from the LSP2D model is reduced by 99.73% compared with that from the P2D model, and the estimation accuracy is greatly improved compared with the estimation accuracy from the SP model.
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Keywords:
- electrochemical model /
- battery management system /
- liquid phase diffusion equation /
- parabolic approximation
[1] Zou C F, Manzie C, Nešic D 2016 IEEE Trans. Control Syst. Technol. 24 1594
Google Scholar
[2] Seaman A, Dao T S, McPhee J 2014 J. Power Sources 256 410
Google Scholar
[3] Tang S X, Solorio L C, Wang Y, Krstic M 2017 Automatica 83 206
Google Scholar
[4] 庞辉 2018 物理学报 67 058201
Google Scholar
Pang H 2018 Acta Phys. Sin. 67 058201
Google Scholar
[5] Hu L B, Choi J W, Yang Y, Jeong S, Mantia F L, Cui L F, Cui Y 2009 Proc. Natl. Acad. Sci. 106 51
Google Scholar
[6] Lindgren J, Asghar I, Lund P D 2016 Int. J. Energy Res. 40 1576
Google Scholar
[7] Bizeray A M, Kim J H, Duncan S R, Howey D A 2018 IEEE Trans. Control Syst. Technol.
Google Scholar
[8] Doyle M, Fuller T F, Newman J 1993 J. Electrochem. Soc. 140 1526
Google Scholar
[9] Li J, Lotfi N, Landers R G, Park J 2017 J. Electrochem. Soc. 164 A874
Google Scholar
[10] Tanim T R, Rahn C D, Wang C Y 2015 J. Dyn. Syst. Meas. Contr. 137 011005
Google Scholar
[11] Subramanian V R, Diwakar V D, Tapriyal D 2005 J. Electrochem. Soc. 152 A2002
Google Scholar
[12] 程昀, 李劼, 贾明, 汤依伟, 杜双龙, 艾立华, 殷宝华, 艾亮 2015 物理学报 64 210202
Google Scholar
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Google Scholar
[13] Santhanagopalan S, Guo Q Z, Ramadass P, White R E 2006 J. Power Sources 156 620
Google Scholar
[14] Farag M, Fleckenstein M, Habibi S 2017 J. Power Sources 342 351
Google Scholar
[15] Zheng L F, Zhang L, Zhu J G, Wang G X, Jiang J C 2016 Appl. Energy 180 424
Google Scholar
[16] 庞辉 2017 物理学报 66 238801
Google Scholar
Pang H 2017 Acta Phys. Sin. 66 238801
Google Scholar
[17] Prada E, Di-Domenico D, Creff Y, Bernard J, Sauvant-Moynot V, Huet F 2012 J. Electrochem. Soc. 159 A1508
Google Scholar
[18] Han X B, Ouyang M G, Lu L G, Li J Q 2015 J. Power Sources 278 814
Google Scholar
[19] Dawson-Elli N, Lee S B, Pathak M 2018 J. Electrochem. Soc. 165 A1
Google Scholar
[20] Ramadass P, Haran B, Gomadam P M, 2004 J. Electrochem. Soc. 151 A196
Google Scholar
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图 1 锂离子电池示意图
Figure 1. Lithium ion battery schematic.
图 2 LSP2D模型简化示意图
Figure 2. Simplified schematic diagram of the LSP2D model.
图 3 1C和3C放电率下, SP, LSP2D和P2D模型得到的放电末期液相Li+浓度
${c_{\rm{e}}}$ 比较Figure 3. Comparison of liquid phase Li+ concentration
${c_{\rm{e}}}$ obtained by SP, LSP2D and P2D model at 1C and 3C discharge rates.
图 4 三种模型在放电率为1C和3C时的放电曲线
Figure 4. Discharge curves of three models at 1C and 3C discharge rates.
表 1 各参数的值
Table 1. Values of parameters.
${a_1}$ ${b_1}$ ${c_1}$ ${a_2}$ ${b_2}$ ${a_3}$ ${b_3}$ ${c_3}$ $\displaystyle\frac{1}{8} + \alpha \tau $ $\displaystyle\frac{{15}}{4}$ $ - \displaystyle\frac{{15}}{8}$ $1$ 0 1 $\displaystyle\frac{{30}}{7}\beta \tau $ $ - \displaystyle\frac{{15}}{7}\beta \tau $ 
DownLoad: CSV
表 2 锂离子电池参数表
Table 2. Parameters lists of Li-ion battery.
Symbol Anode Cathode Separator $\sigma$/S·m–1 100 100 ${\varepsilon _{\rm s}}$ 0.49 0.59 ${\varepsilon _{\rm e}}$ 0.485 0.365 0.724 Brug 4.0 4.0 4.0 ${c_{{\rm{e,0}}}}$/mol·m–3 1000 1000 1000 ${c_{{\rm{s,0}}}}$/mol·m–3 916.65 48977.25 ${c_{{\rm{s,max}}}}$/mol·m–3 30555 51555 A/m2 $6.03 \!\times\! {10^{ - 4}}$ $5.31 \!\times\! {10^{ - 4}}$ ${D_{\rm{e}}}$/m2·s–1 $7.5 \!\times\! {10^{ - 10}}$ $7.5 \!\times\! {10^{ - 10}}$ $7.5 \!\times\! {10^{ - 10}}$ ${D_{\rm{s}}}$/m2·s–1 $3.9 \!\times\! {10^{ - 14}}$ $1.0 \!\times\! {10^{ - 14}}$ k/mol·(mol·m–3)–1.5 $4.854 \!\times\! {10^{ - 6}}$ $2.252 \!\times\! {10^{ - 6}}$ ${R_{\rm{s}}}$/m $2 \!\times\! {10^{ - 6}}$ $2 \!\times\! {10^{ - 6}}$ x/m $8.8 \!\times\! {10^{ - 5}}$ $8.0 \!\times\! {10^{ - 5}}$ $8.0 \!\times\! {10^{ - 5}}$ ${R_{{\rm{SEI}}}}/\Omega$·m–2 0.01 I/A·m–2 20 $\alpha $ 0.5 0.5 F/C·mol 96487 R/J·mol·K–1 8.314 T/K 298.15
DownLoad: CSV
表 3 放电率为1C和3C时三种模型估算电池端电压的平均误差
Table 3. Average error of the battery terminal voltage is estimated by three models when the discharge rate is 1C and 3C.
Model Average error at
discharge rate 1C/VAverage error at
discharge rate 3C/VP2D 0 0 LSP2D 0.0056 0.014 SP 0.0052 0.142
DownLoad: CSV
表 4 比较三种模型估算速度
Table 4. Comparison of the estimated speeds of the three models.
Model Time for 50 cycles/s P2D 7860 LSP2D 21 SP 12
DownLoad: CSV
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[1] Zou C F, Manzie C, Nešic D 2016 IEEE Trans. Control Syst. Technol. 24 1594
Google Scholar
[2] Seaman A, Dao T S, McPhee J 2014 J. Power Sources 256 410
Google Scholar
[3] Tang S X, Solorio L C, Wang Y, Krstic M 2017 Automatica 83 206
Google Scholar
[4] 庞辉 2018 物理学报 67 058201
Google Scholar
Pang H 2018 Acta Phys. Sin. 67 058201
Google Scholar
[5] Hu L B, Choi J W, Yang Y, Jeong S, Mantia F L, Cui L F, Cui Y 2009 Proc. Natl. Acad. Sci. 106 51
Google Scholar
[6] Lindgren J, Asghar I, Lund P D 2016 Int. J. Energy Res. 40 1576
Google Scholar
[7] Bizeray A M, Kim J H, Duncan S R, Howey D A 2018 IEEE Trans. Control Syst. Technol.
Google Scholar
[8] Doyle M, Fuller T F, Newman J 1993 J. Electrochem. Soc. 140 1526
Google Scholar
[9] Li J, Lotfi N, Landers R G, Park J 2017 J. Electrochem. Soc. 164 A874
Google Scholar
[10] Tanim T R, Rahn C D, Wang C Y 2015 J. Dyn. Syst. Meas. Contr. 137 011005
Google Scholar
[11] Subramanian V R, Diwakar V D, Tapriyal D 2005 J. Electrochem. Soc. 152 A2002
Google Scholar
[12] 程昀, 李劼, 贾明, 汤依伟, 杜双龙, 艾立华, 殷宝华, 艾亮 2015 物理学报 64 210202
Google Scholar
Cheng Y, Li J, Jia M, Tang Y W, Du S L, Ai L H, Yin B H, Ai L 2015 Acta Phys. Sin. 64 210202
Google Scholar
[13] Santhanagopalan S, Guo Q Z, Ramadass P, White R E 2006 J. Power Sources 156 620
Google Scholar
[14] Farag M, Fleckenstein M, Habibi S 2017 J. Power Sources 342 351
Google Scholar
[15] Zheng L F, Zhang L, Zhu J G, Wang G X, Jiang J C 2016 Appl. Energy 180 424
Google Scholar
[16] 庞辉 2017 物理学报 66 238801
Google Scholar
Pang H 2017 Acta Phys. Sin. 66 238801
Google Scholar
[17] Prada E, Di-Domenico D, Creff Y, Bernard J, Sauvant-Moynot V, Huet F 2012 J. Electrochem. Soc. 159 A1508
Google Scholar
[18] Han X B, Ouyang M G, Lu L G, Li J Q 2015 J. Power Sources 278 814
Google Scholar
[19] Dawson-Elli N, Lee S B, Pathak M 2018 J. Electrochem. Soc. 165 A1
Google Scholar
[20] Ramadass P, Haran B, Gomadam P M, 2004 J. Electrochem. Soc. 151 A196
Google Scholar
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