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锂离子电池电化学降阶模型性能对比

李涛 程夕明 胡晨华

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锂离子电池电化学降阶模型性能对比

李涛, 程夕明, 胡晨华

Comparative study of reduced-order electrochemical models of the lithium-ion battery

Li Tao, Cheng Xi-Ming, Hu Chen-Hua
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  • 可靠而高效的锂离子电池模型是电池管理系统状态估计与故障诊断的基础. 采用偏微分方程描述的准二维(P2D)机理模型的参数多, 虽然模型准确性高, 但计算费时, 需降阶处理才能更好地应用在车载电池管理系统. 为此, 基于相同模型参数, 建立锂离子电池的P2D模型及其降阶模型—单粒子模型(SPM)和集总粒子模型(LPM), 对三种电化学机理模型电池端电压的计算精度和时间进行对比研究. 结合多孔电极模型和浓溶液理论, 基于电池均匀电流密度的假设条件, 按照电极固相和液相体积比重新分配电流密度, 推导了由液相锂离子浓度分布差异所导致的液相浓差过电压和欧姆内阻, 补偿LPM的电压误差. 采用恒流放电、脉冲放电和动态电应力测试(DST)工况对比分析优化的LPM, P2D模型和SPM的电学性能. 结果表明: 优化的LPM不仅能够以更少的参数大幅降低模型计算时间, 而且能够保证模型电压的计算精度.
    A reliable and efficient lithium-ion battery model is the basis of state estimation and fault diagnosis in the battery management system (BMS). The pseudo-two-dimensional (P2D) mechanism model represented by using partial differential equations has many parameters and high accuracy, but its calculation is time-consuming. The model order should be reduced for the on-board BMS. Here in this work, by using the same model parameters, the P2D model and the order-reduced models including the single particle model (SPM) and the lumped particle model (LPM) are built and their voltage accuracies and operation time are compared with each other. Based on the porous electrode model, the concentrated solution theory, and the assumption of uniform current of the battery, the current density is redistributed by the volume ratio of the solid phase to the liquid phase of the electrode. Furthermore, the overpotential and ohmic resistance caused by the lithium-ion concentration distribution in the liquid phase is deduced to offset the battery voltage bias under a large discharge rate. An unconstrained convex optimization method is established to optimize the concentration difference overpotential of the liquid phase of the P2D model battery, the optimized results of which are used to compensate for the LPM voltage.Under the conditions of the constant current discharge (CCD) of 0.1C — 4C at the ambient temperature, pulse discharge and dynamic stress test (DST), the electrical performance and the operation time of both the LPM and the SPM are compared to those of the P2D model. Some results are validated and given by the model simulation. Firstly, the accuracy of LPM is over 30% higher than those of the SPM at the CCD rates of 0.1 C, 0.5 C and 1 C, and about 30% worse than those of the SPM at the CCD rates of 2 C, 3 C, and 4 C. Secondly, the optimized LPM can produce a voltage in good approximation to the voltage of the P2D model with an absolute relative error of the model voltage below 1.5% whether it is caused by the CCD or the DST. Thirdly, the optimized LPM can run efficiently and the needed calculation time cuts down by 85% and 65% for the P2D and the SPM, respectively. For the real-time applications of lithium-ion batteries, the proposed compensation method by adding the overpotential in the liquid phase can make the LPM produce reliable voltage with shorter operation time than either the SPM or the P2D model.
      通信作者: 程夕明, cxm2004@bit.edu.cn
    • 基金项目: 国家重点研发计划(批准号: 2018YFB0106104)和国家自然科学基金(批准号: 51677006)资助的课题
      Corresponding author: Cheng Xi-Ming, cxm2004@bit.edu.cn
    • Funds: Project supported by the National Key R&D Program of China (Grant No. 2018YFB0106104) and the National Natural Science Foundation of China (Grant No. 51677006)
    [1]

    Hu X S, Li S B, Peng H 2012 J. Power Sources 198 359Google Scholar

    [2]

    Wang Y J, Tian J Q, Sun Z D, Wang L, Chen Z h 2020 Renewable Sustainable Energy Rev. 131 110015-1

    [3]

    Lai X, Wang S, Ma S, Xie J Y, Zheng Y J 2020 Electrochim. Acta 330 135239Google Scholar

    [4]

    Tang K, Yu X Q, Sun J P, Li H, Huang X J 2011 Electrochim. Acta 56 4869Google Scholar

    [5]

    Shen Z, Cao L, Rahn C, Wang C Y 2013 J. Electrochem. Soc. 160 A1842Google Scholar

    [6]

    凌仕刚, 吴妖杨, 张舒, 高健, 王少飞, 李泓 2015 储能科学与技术 4 83Google Scholar

    Ling S G, Wu J Y, Zhang S, Gao J, Wang S F, Li H 2015 Energy Storage Science and Technology 4 83Google Scholar

    [7]

    Subramanian V R, Diwakar V D, Tapriyal D 2005 J. Electrochem. Soc. 152 A2002Google Scholar

    [8]

    Dao T S, Vyasarayani C P, McPhee J 2012 J. Power Sources 198 329Google Scholar

    [9]

    Luo W L, Lyu C, Wang L X, Zhang L Q 2013 Microelectron. Reliab. 53 797Google Scholar

    [10]

    Kemper P, Li S E, Kum D 2015 J. Power Sources 286 510Google Scholar

    [11]

    Deng Z, Yang L, Deng H, Cai Y, Li D 2018 Energy 142 838Google Scholar

    [12]

    Cai L, White R E 2009 J. Electrochem. Soc. 156 A154Google Scholar

    [13]

    Forman J, Bashash S, Stein J, Fathy H 2011 J. Electrochem. Soc. 158 A93Google Scholar

    [14]

    Subramanian V, Boovaragavan V, Ramadesigan V, Arabandi M 2009 J. Electrochem. Soc. 156 A260Google Scholar

    [15]

    Luo W L, Lyu C, Wang L X, Zhang L Q 2013 J. Power Sources 241 295Google Scholar

    [16]

    Romero-Becerril A, Alvarez-Icaza L 2011 J. Power Sources 196 10267Google Scholar

    [17]

    Santhanagopalan S, Guo Q, Ramadass P, White R E 2006 J. Power Sources 156 620Google Scholar

    [18]

    Di Domenico D, Stefanopoulou A, Fiengo G 2010 J. Dyn. Syst. Meas. Contr. 132 061302-1Google Scholar

    [19]

    Prada E, Di Domenico D, Creff Y, Bernard J, Sauvant-Moynot V, Huet F 2012 J. Electrochem. Soc. 159 A1508Google Scholar

    [20]

    Ekström H, Fridholm B, Lindbergh G 2018 J. Power Sources 402 296Google Scholar

    [21]

    Doyle M, Fuller T F, Newman J 1993 J. Power Sources 140 1526Google Scholar

    [22]

    Fuller T F, Doyle M, Newman J 1994 J. Electrochem. Soc. 141 1Google Scholar

    [23]

    Han X B, Ouyang M G, Lu L G, Li J Q 2015 J. Power Sources 278 814Google Scholar

  • 图 1  P2D模型示意图

    Fig. 1.  Schematic diagram of P2D.

    图 2  三种电化学模型电压曲线对比

    Fig. 2.  Comparison of three electrochemical model voltage curves.

    图 3  测试集, 优化LPM电压及其误差 (a) 电压; (b) 电压误差

    Fig. 3.  Voltage curves and voltage error curve of the optimized LPM under test set: (a) Voltage curves; (b) voltage error curves.

    图 4  验证集, 优化LPM电压及其误差 (a) 电压; (b) 电压误差

    Fig. 4.  Voltage curves and voltage error curve of the optimized LPM under verification set: (a) Voltage curves; (b) voltage error curves.

    图 5  脉冲放电条件下, 优化的LPM电压和电压误差曲线 (a) 电压; (b) 电压误差

    Fig. 5.  Voltage and voltage error curve of the optimized LPM under pulse discharge condition: (a) Voltage curves; (b) voltage error curve.

    图 6  DST工况下, 优化的LPM电压和电压误差曲线 (a) 电压; (b) 电压误差

    Fig. 6.  Voltage and voltage error curve of the optimized LPM under DST condition: (a) Voltage curves; (b) voltage error curve.

    表 1  P2D模型参数

    Table 1.  Parameter symbols of P2D.

    符号物理量单位
    $c_{\rm{s}}^i$多孔电极固相锂离子浓度${\rm{mol}} \cdot {{\rm{m}}^{ - 3}}$
    $c_{\rm{e}}^i$多孔电极液相锂离子浓度${\rm{mol}} \cdot {{\rm{m}}^{ - 3}}$
    $c_{\rm{e}}^{{\rm{sep}}}$隔膜液相锂离子浓度${\rm{mol}} \cdot {{\rm{m}}^{ - 3}}$
    $c_{{\rm{s, 0}}}^i$多孔电极固相锂离子初始浓度${\rm{mol}} \cdot {{\rm{m}}^{ - 3}}$
    ${c_{{\rm{e, 0}}}}$液相锂离子初始浓度${\rm{mol}} \cdot {{\rm{m}}^{ - 3}}$
    $c_{{\rm{s, max}}}^i$多孔电极固相粒子最大
    锂离子浓度
    ${\rm{mol}} \cdot {{\rm{m}}^{ - 3}}$
    $c_{{\rm{s, surf}}}^i$多孔电极固相粒子表面
    锂离子浓度
    ${\rm{mol}} \cdot {{\rm{m}}^{ - 3}}$
    $D_{\rm{s}}^i$多孔电极固相锂离子扩散系数$ { {\rm{m} }^{2} } \cdot { {\rm{s} }^{ - 1} } $
    $D_{\rm{e}}^{}$液相锂离子扩散系数$ { {\rm{m} }^{2} } \cdot { {\rm{s} }^{ - 1} } $
    $D_{{\rm{e, eff}}}^i$多孔电极液相锂离子
    有效扩散系数
    $ { {\rm{m} }^{2} } \cdot { {\rm{s} }^{ - 1} } $
    $D_{{\rm{e, eff}}}^{{\rm{sep}}}$隔膜液相锂离子有效扩散系数$ { {\rm{m} }^{2} } \cdot { {\rm{s} }^{ - 1} } $
    $j_{\rm{n}}^i$多孔电极锂离子孔壁通量${\rm{mol}} \cdot {{\rm{s}}^{ - 1}} \cdot {{\rm{m}}^{ - 2}}$
    $j_{}^i$多孔电极电流密度${\rm{A}} \cdot {{\rm{m}}^{ - 3}}$
    ${R^i}$多孔电极球形粒子半径m
    $\varepsilon _{\rm{e}}^i$多孔电极液相体积分数
    $\varepsilon _s^i$多孔电极固相体积分数
    $\varepsilon _{\rm{e}}^{{\rm{sep}}}$隔膜液相体积分数
    ${t_{ + }}$锂离子迁移系数
    ${f_ \pm }$液相平均摩尔活度系数
    $a_{\rm{s}}^i$多孔电极球形粒子的比表面积${{\rm{m}}^{ - 1}}$
    $\sigma _{\rm{s}}^i$多孔电极固相电导率${\rm{S}} \cdot {{\rm{m}}^{ - 1}}$
    $\sigma _{{\rm{s, eff}}}^i$多孔电极固相有效电导率${\rm{S}} \cdot {{\rm{m}}^{ - 1}}$
    $\kappa $液相离子电导率${\rm{S}} \cdot {{\rm{m}}^{ - 1}}$
    $\kappa _{{\rm{eff}}}^i$液相有效离子电导率${\rm{S}} \cdot {{\rm{m}}^{ - 1}}$
    ${\kappa _{{\rm{D}}, i}}$有效液相离子扩散导电系数${\rm{A}} \cdot {{\rm{m}}^2} \cdot {\rm{mo}}{{\rm{l}}^{ - 1}}$
    $\phi _{\rm{s}}^i$多孔电极固相电势V
    $\phi _{\rm{e}}^{}$液相电势V
    ${I_{\rm{b}}}$电池电流A
    $ j_0^i $多孔电极参考交换电流密度${\rm{A}} \cdot {{\rm{m}}^{ - 2}}$
    ${\eta ^i}$电极过电势V
    ${\alpha _{{\rm{a, i}}}}$/${\alpha _{{\rm{c, i}}}}$电极氧化还原反应电荷传递系数, 为0.5
    ${k^i}$电极反应常数${{\rm{m}}^{2.5}} \cdot {\rm{mo}}{{\rm{l}}^{ - 0.5}} \cdot {{\rm{s}}^{ - 1}}$
    $U_{}^i$电极平衡电势V
    $R_{{\rm{SEI}}}^i$SEI膜电阻$\Omega \cdot {{\rm{m}}^{ - 2}}$
    ${V_{{\rm{P2 D}}}}$P2D模型端电压V
    A电池横截面积${{\rm{m}}^2}$
    BrugBruggman系数, 取值1.5
    F法拉第常数${\rm{C}} \cdot {\rm{mo}}{{\rm{l}}^{ - 1}}$
    R普适气体常数${\rm{J}} \cdot {\rm{mo}}{{\rm{l}}^{ - 1}} \cdot {{\rm{K}}^{ - 1}}$
    T温度K
    下载: 导出CSV

    表 2  SPM参数

    Table 2.  Parameter symbols of SPM.

    符号物理量单位
    $c_{{\rm{s, avg}}}^i$多孔电极粒子平均锂离子浓度${\rm{mol}} \cdot {{\rm{m}}^{ - 3}}$
    $c_{{\rm{s, max}}}^i$多孔电极最大粒子锂离子浓度${\rm{mol}} \cdot {{\rm{m}}^{ - 3}}$
    ${c_{\rm{e}}}$液相锂离子浓度${\rm{mol}} \cdot {{\rm{m}}^{ - 3}}$
    ${\overline j ^i}$平均交换电流密度${\rm{A}} \cdot {{\rm{m}}^{ - 2}}$
    $\phi _{{\rm{SP}}}^i$电池电极电势V
    ${V_{{\rm{SP}}}}$电池端电压V
    下载: 导出CSV

    表 3  LPM参数

    Table 3.  Parameter symbols of LPM.

    符号物理量单位
    cLPM粒子内锂离子浓度${\rm{mol}} \cdot {{\rm{m}}^{ - 3}}$
    ${c_{\max }}$LPM最大锂离子浓度${\rm{mol}} \cdot {{\rm{m}}^{ - 3}}$
    ${\eta _{{\rm{ohm}}}}$欧姆过电压V
    ${\eta _{{\rm{act}}}}$活化过电压V
    ${\eta _{{\rm{conc}}}}$浓度过电压V
    ${c_{{\rm{surf}}}}$粒子表面锂离子浓度${\rm{mol}} \cdot {{\rm{m}}^{ - 3}}$
    ${c_{{\rm{avg}}}}$粒子平均锂离子浓度${\rm{mol}} \cdot {{\rm{m}}^{ - 3}}$
    ${\eta _{{\rm{ohm, 1 C}}}}$1 C欧姆过电压V
    ${I_{{\rm{1 C}}}}$1 C电流A
    ${J_0}$无量纲交换电流密度1
    $\tau $等效扩散系数s
    ${Q_{{\rm{cell}}}}$电池容量${\rm{Ah} }$
    X归一化粒子半径
    下载: 导出CSV

    表 4  电化学模型参数值

    Table 4.  Parameter values of Electrochemical model.

    中文参数名负极隔膜正极单位
    厚度1025100μm
    粒子半径1010μm
    锂离子扩散系数$3.9 \times 1{0^{ - 14}}$$1 \times 1{0^{ - 13}}$$ { {\rm{m} }^{2} } \cdot { {\rm{s} }^{ - 1} } $
    初始固相浓度1962420046${\rm{mol}} \cdot {{\rm{m}}^{ - 3}}$
    最大锂离子浓度2498351228${\rm{mol}} \cdot {{\rm{m}}^{ - 3}}$
    电导率100$\kappa $10S·m–1
    初始电解液浓度100010001000$ { {\rm{m} }^{2} } \cdot { {\rm{s} }^{ - 1} } $
    电解液扩散系数$2.7877 \times 1{0^{ - 10}}$$2.7877 \times 1{0^{ - 10}}$$2.7877 \times 1{0^{ - 10}}$$ { {\rm{m} }^{2} } \cdot { {\rm{s} }^{ - 1} } $
    传递数0.40.40.4
    电解液体积分数0.310.3
    SEI膜电阻$0$$0$${\rm{m}}\Omega $
    反应速率常数$9.6487 \times 1{0^{ - 6}}$$2.89461 \times 1{0^{ - 6}}$${\rm{mo}}{{\rm{l}}^{2.5}} \cdot {{\rm{m}}^{ - 0.5}} \cdot {{\rm{s}}^{ - 1}}$
    活性材料体积分数0.60.5
    填料体积分数0.10.2
    电极横截面积0.1${{\rm{m}}^2}$
    1 C电流30A
    放电截止电压3.4V
    充电截止电压4.2V
    容量30Ah
    温度298K
    普适气体常数8.314${\rm{J}} \cdot {\rm{mo}}{{\rm{l}}^{ - 1}} \cdot {{\rm{K}}^{ - 1}}$
    法拉第常数96487${\rm{C}} \cdot {\rm{mo}}{{\rm{l}}^{ - 1}}$
    下载: 导出CSV

    表 5  测试集电压误差对比

    Table 5.  Comparison of test set voltage errors.

    倍率/CSPM绝对
    误差/mV
    SPM相对
    误差/%
    LPM绝对
    误差/mV
    LPM相对
    误差/%
    0.1–44.50–1.2830.030.87
    0.574.202.1242.301.21
    183.702.3960.701.72
    288.002.51103.402.95
    398.502.81129.303.66
    4125.503.58173.204.75
    下载: 导出CSV

    表 6  不同倍率下参数a取值

    Table 6.  Parameter values of a in the different discharge rate.

    倍率参数a
    0.1–0.148
    0.5–0.703
    1–1.331
    2–2.431
    3–3.438
    4–4.476
    下载: 导出CSV

    表 7  优化后LPM电压误差

    Table 7.  Optimized LPM voltage errors.

    倍率/C绝对误
    差/mV
    相对误
    差/%
    倍率/C绝对误
    差/mV
    相对误
    差/%
    0.125.70.730.2522.10.63
    0.520.70.590.7514.80.42
    119.70.561.525.50.73
    228.50.812.5–32.7–0.81
    3–39.9–0.993.5–49.2–1.23
    4–57.6–1.45DST–55.4–1.40
    脉冲
    放电
    41.51.05
    下载: 导出CSV

    表 8  计算用时对比

    Table 8.  Comparison of the calculation time.

    测试集
    用时/s
    验证集
    用时/s
    脉冲放
    电用时/s
    DST工况
    用时/s
    LPM2414811
    P2D1699285108
    占比14.2%15.22%9.41%10.19%
    下载: 导出CSV
  • [1]

    Hu X S, Li S B, Peng H 2012 J. Power Sources 198 359Google Scholar

    [2]

    Wang Y J, Tian J Q, Sun Z D, Wang L, Chen Z h 2020 Renewable Sustainable Energy Rev. 131 110015-1

    [3]

    Lai X, Wang S, Ma S, Xie J Y, Zheng Y J 2020 Electrochim. Acta 330 135239Google Scholar

    [4]

    Tang K, Yu X Q, Sun J P, Li H, Huang X J 2011 Electrochim. Acta 56 4869Google Scholar

    [5]

    Shen Z, Cao L, Rahn C, Wang C Y 2013 J. Electrochem. Soc. 160 A1842Google Scholar

    [6]

    凌仕刚, 吴妖杨, 张舒, 高健, 王少飞, 李泓 2015 储能科学与技术 4 83Google Scholar

    Ling S G, Wu J Y, Zhang S, Gao J, Wang S F, Li H 2015 Energy Storage Science and Technology 4 83Google Scholar

    [7]

    Subramanian V R, Diwakar V D, Tapriyal D 2005 J. Electrochem. Soc. 152 A2002Google Scholar

    [8]

    Dao T S, Vyasarayani C P, McPhee J 2012 J. Power Sources 198 329Google Scholar

    [9]

    Luo W L, Lyu C, Wang L X, Zhang L Q 2013 Microelectron. Reliab. 53 797Google Scholar

    [10]

    Kemper P, Li S E, Kum D 2015 J. Power Sources 286 510Google Scholar

    [11]

    Deng Z, Yang L, Deng H, Cai Y, Li D 2018 Energy 142 838Google Scholar

    [12]

    Cai L, White R E 2009 J. Electrochem. Soc. 156 A154Google Scholar

    [13]

    Forman J, Bashash S, Stein J, Fathy H 2011 J. Electrochem. Soc. 158 A93Google Scholar

    [14]

    Subramanian V, Boovaragavan V, Ramadesigan V, Arabandi M 2009 J. Electrochem. Soc. 156 A260Google Scholar

    [15]

    Luo W L, Lyu C, Wang L X, Zhang L Q 2013 J. Power Sources 241 295Google Scholar

    [16]

    Romero-Becerril A, Alvarez-Icaza L 2011 J. Power Sources 196 10267Google Scholar

    [17]

    Santhanagopalan S, Guo Q, Ramadass P, White R E 2006 J. Power Sources 156 620Google Scholar

    [18]

    Di Domenico D, Stefanopoulou A, Fiengo G 2010 J. Dyn. Syst. Meas. Contr. 132 061302-1Google Scholar

    [19]

    Prada E, Di Domenico D, Creff Y, Bernard J, Sauvant-Moynot V, Huet F 2012 J. Electrochem. Soc. 159 A1508Google Scholar

    [20]

    Ekström H, Fridholm B, Lindbergh G 2018 J. Power Sources 402 296Google Scholar

    [21]

    Doyle M, Fuller T F, Newman J 1993 J. Power Sources 140 1526Google Scholar

    [22]

    Fuller T F, Doyle M, Newman J 1994 J. Electrochem. Soc. 141 1Google Scholar

    [23]

    Han X B, Ouyang M G, Lu L G, Li J Q 2015 J. Power Sources 278 814Google Scholar

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出版历程
  • 收稿日期:  2020-11-11
  • 修回日期:  2021-02-16
  • 上网日期:  2021-06-25
  • 刊出日期:  2021-07-05

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