# Comparative Study on Parameter Identification Methods for Dual-Polarization Lithium-Ion Equivalent Circuit Model

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## Abstract

**:**

## 1. Introduction

- Generic-based or data-based empirical models,
- Electrochemical impedance spectroscopy (EIS)-based models,
- Static and dynamic equivalent electrical circuit models (ECM) and
- Electrochemical and physics-based models.

- An experimental characterization and electrical modeling of the Li-ions with three different parameter identification methods,
- A comprehensive comparison among the parameter identification methods from previous studies,
- Analyses on the impedance behavior of the cells in time and frequency domain and estimation on the power capabilities of the cells.

## 2. Experimental Setup

#### 2.1. Test Bench Description

#### 2.2. Experimental Process

#### 2.2.1. Open Circuit Voltage Test

#### 2.2.2. Voltage Response to Current Test

#### 2.2.3. Impedance Response to Frequency Test

#### 2.2.4. Capacity Test

## 3. Model Development

#### 3.1. In Discrete-Time Domain

#### 3.2. In Frequency Domain

^{®}fitting software based on Equation (5) [55]

^{®}fitting software.

#### 3.2.1. Ohmic Resistance

#### 3.2.2. Charge-Transfer and Total Resistance

#### 3.2.3. Diffusion Resistance

## 4. Review of the Most Frequently Used Identification Methods

- Analytical equations,
- Least-square-based methods,
- Heuristic optimization algorithms,
- Impedance spectroscopy methods and
- Kalman or adaptive filters and observer-based methods.

#### 4.1. Parameter Extraction with Analytical Equations

#### 4.2. Parameter Extraction with Least-Square Methods

#### 4.3. Parameter Extraction with Heuristic Optimization

#### 4.4. Parameter Extraction with Impedance Spectroscopy

#### 4.5. Parameter Extraction with Kalman Filters Based Techniques

## 5. Results, Validations and Discussions

#### 5.1. Resistance and Power Capability

#### 5.1.1. Lower Time-Constant Resistances

#### 5.1.2. Total Resistances

#### 5.2. Current Profiles and Voltage Responses

#### 5.3. Discussion of Methodology Robustness

#### 5.4. Parameter Identification Comprehensive Assessment

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Lithium-ion cells placed on a A4 paper. On the left lays the LTO and right is the NMC. (

**b**) Lithium-ion cells nominal properties (datasheet) comparison at the BoL. no.#2 indicates a twice better behavior in absolute values.

**Figure 2.**Topology of test bench and the battery cells during experiments [13].

**Figure 3.**Division of this work in three main parts, the experimental characterization of the cells, the parameter identification process and the model validation.

**Figure 4.**Open circuit voltage of (

**a**) LTO cell and (

**c**) NMC cell during charge and discharge. Hysteresis effect on both cells in (

**b**) and (

**d**) respectively.

**Figure 6.**Dual-polarization electrical equivalent modeling approach of the Li-ion prismatic cells. Model input is the current, the initial SoC and the temperature. Model output is the voltage response, resistances and energy/power capability of cells.

**Figure 10.**Approximation of the time intervals for the polarization instances based on the frequency domain EIS.

**Figure 11.**Experimental current pulses and generated voltage responses for (

**a**) LTO and (

**b**) NMC battery cells with f

_{sample}= 100 Hz.

**Figure 12.**Total resistance at 50% SoC for (

**a**) LTO and (

**b**) NMC. Total charge resistance at various SoC for (

**c**) LTO and (

**e**) NMC. Generated exponential overvoltage for the R

_{DC}for (

**d**) LTO and (

**f**) NMC. Pulse power capabilities of the cells at $25{\phantom{\rule{4pt}{0ex}}}^{\circ}$C, three different C-rates, SoCs and pulse lengths (

**g**) LTO and (

**h**) NMC.

**Figure 13.**Characterization of LTO prismatic cell with least-square optimization at (

**a**) WLTC and (

**b**) HD current profile. For the same approach, LTO and NMC at WLTC (

**c**,

**e**) and HD (

**d**,

**f**) profile validation with the respective voltage relative errors in percentage.

**Figure 14.**Voltage relative error for WLTC and HD profiles for (

**a**,

**b**) LTO (

**c**,

**d**) NMC. RMSE of the ten-time consecutive PSO for WLTC (

**e**) and HD (

**f**) for both cells.

**Figure 15.**Ohmic resistance R

_{0}behavior for the different modeling approaches for (

**a**) LTO and (

**b**) NMC battery cells. R

_{int}variation over the various PSO iterations for LTO (

**c**) and NMC (

**d**). Set from lowest to highest deviation from EIS. Linear regression between RMSE and PSO approach oriented according to R

_{int}evolution for (

**e**,

**f**) LTO, (

**g**,

**h**) NMC.

**Figure 16.**Statistical comparisons of the results for all the methods. Relative errors for (

**a**) LTO and (

**b**) NMC battery cells. For heuristic methods B

_{RMSE}are taken into account and for subfigures (

**c**–

**f**) the mean RE is considered. In (

**g**) the voltage RMSE of both cells is shown.

Main Characteristics | Value | Unit | |
---|---|---|---|

Chemistry | LTO | NMC | [-] |

Nominal voltage | 2.3 | 3.65 | [V] |

Nominal capacity | 23 | 43 | [Ah] |

End-of-charge maximum voltage | 2.7 | 4.2 | [V] |

End-of-discharge cut-off voltage | 1.5 | 3 | [V] |

Volumetric energy density | 202 | 424 | [Wh/L] |

Specific energy density | 96.1 | 186.8 | [Wh/kg] |

Specific power | >1200 | >1200 | [W/kg] |

AC impedance (1 kHz) | 0.6 | 1 | [mOhms] |

Recommended charge | 4 C | 1 C | [-] |

current rate (continuous) | |||

10 s max charge C-rate | >8 C | 3 C | [-] |

Height | 103 | 148 | [mm] |

Width | 115 | 91 | [mm] |

Thickness | 22 | 27.5 | [mm] |

Weight | 0.550 | 0.840 | [kg] |

**Table 2.**Charge and discharge capacities of the lithium-ion cells at $25{\phantom{\rule{4pt}{0ex}}}^{\circ}$C.

Value | LTO | NMC | Unit | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

C-rate | C/3 | C/2 | 1C | 2C | 4C | C/3 | C/2 | 1C | 1.5C | 2C | [-] |

Charge Capacity | 23.2 | 23.1 | 23 | 23 | 22.9 | 44.9 | 44.9 | 44.8 | 44.7 | 44.7 | [Ah] |

Discharge Capacity | 23.6 | 23.3 | 23.1 | 23 | 22.8 | 45.3 | 45.2 | 45 | 44.9 | 44.8 |

**Table 3.**Frequency domain impedance model parameters for both prismatic battery cells at $25{\phantom{\rule{4pt}{0ex}}}^{\circ}$C and five SoC steps.

Parameter | LTO | NMC | Unit | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

SoC | 100 | 80 | 50 | 20 | 0 | 100 | 80 | 50 | 20 | 0 | [%] |

R0 | 0.709 | 0.714 | 0.782 | 0.883 | 0.953 | 1.25 | 1.26 | 1.306 | 1.31 | 1.33 | [m$\Omega $] |

R1 | 0.175 | 0.136 | 0.122 | 0.123 | 0.179 | 0.359 | 0.35 | 0.358 | 0.434 | 0.963 | [m$\Omega $] |

Q1 | 46.4 | 283 | 169.1 | 43.27 | 302.8 | 19.34 | 18.23 | 15.45 | 21.14 | 176.1 | [${\Omega}^{-1}\times {s}^{\xi}$] |

Q2 | 847.7 | 6987 | 8449 | 7209 | 883.3 | 9198 | 8822 | 10,970 | 10,726 | 3662 | [${\Omega}^{-1}\times {s}^{\xi}$] |

**Table 4.**Comparison of several referenced works based on various electrical models and parameter identification techniques.

Refs. | Parameterization/SoC | Covering Aspects | Methodology | Conclusion |
---|---|---|---|---|

A. Analytical equations (TD) | ||||

[75] | CC | Analytical time-domain characterization of the DP ECM. | A 40 Ah NMC cell is tested in a 7s1p module design. | The model is validated on a real-life heavy-duty application profile and found a voltage MAPE ^{1} < 0.2% |

B. Least-square methods (LS) | ||||

[89,90] | CC/KF | RLS with an optimal forgetting factor to characterize several ECMs. | The former compares 7 ECMs, assessed with DST ^{5} and FUDS ^{6} profiles and the latter work compares 2 ECMS for 3 chemistries with DDPT ^{7} and WLTC ^{8}. | The DP models shows the best performance for both works with RMSE ^{2} < 25 mV, and MAE ^{3}< 10 mV, respectively. |

[86] | KF | An NLS technique is used to characterize the DP ECM of the battery cell. | A DST is performed for the 14 Ah LFP/C pouch cell. Discussion on the resistance behavior over high current profiles and SoC estimation with EKF. | An RMSE < 12 mV between the estimated and the measured voltage is observed. |

[81] | CC | The DP EMC is characterized decoupled WRLS method and validated on a motorway-drive data. | A commercial 3 Ah 18650 cylindrical cell is tested in terms of SoC estimation accuracy. | An RMSE < 10 mV with the DWLS with better accuracy on both parameter and SoC identification. |

C. Heuristic algorithms (GA/PSO) | ||||

[94] | CC | MPSO method to parameterize the ECMs. | 12 different ECMS are compared for cylindrical NMC and LFP cells. | A maximum RMSE < 30 mV with a DST and FUDS validation profile is observed- lowest for one ECM with hysteresis (LFP) block. |

[85] | CC/KF | GA method to parameterize the five proposed ECMs for an LMO battery module. | A DST is applied to assess the accuracy of the models with CC. A FUDS validation profile the KF. | A MAE < 50 mV for the DP model is obtained with CC. |

[11] | CC | A comparative study based on various heuristic algorithms for nine different ECMs. | A pouch 32.5 Ah NMC battery cell is validated with NEDC profile. | A relationship on the model-to-algorithm complexity is proposed. PSO with an RMSE < 25 mV |

[96] | KF | A comparative study based on GA of 11 different ECMs and a proposed SoC estimation. | A pouch 32.5 Ah NMC battery cell is validated with NEDC ^{9} profile. | The 2RC is the best among all in terms of accuracy with RMSE < 5 mV. |

[97] | CC | SVM, DP, SPM and combined model are parameterized with GA. | An LFP 10 Ah cell is validated with NEDC to assess the complexity versus accuracy. | Dual-Polarization order ECM is found as a good modeling trade-off. |

D. Impedance spectroscopy (EIS) | ||||

[113] | CC | A hybrid procedure for 2nd order CPE circuit is developed from EIS and HPPC tests. | A pulse test and a frequency regulation validation profile is used on a prismatic LFP/C battery cell. | A max voltage relative error <5% is observed for both profiles. |

[116] | 10% | Seven impedance-based ECMs with RC and CPE elements are characterized by current dependency based on BVE | NMC, LTO and LFP cells are tested in regards to state estimation. | The 3-CPE shows accurate result with an RMSE < 20 mV, but the accuracy of the models is highly dependent on the operating conditions and SoH. |

E. Other methods (KF-based) | ||||

[87] | KF | A three ECMs comparative study (Thevenin, PNGV, and DP) with an EKF-Ah SoC estimation algorithm. | NMC/G pouch 35 Ah cell is modeled and tested with a CCD ^{10} and DST to estimate the influence of each model’s accuracy with the selected SoC estimation. | Best case for RMSE < 20 mV and RMSE < 15 mV for the DP model under the CCD and DST respectively |

[19] | KF | A comparative study on ten ECMs that are parameterized with the dual-EKF technique. | LFP and NMC cells are validated with the NEDC profile. Focus laid on the state estimation (SoC-SoP) with erroneous initial conditions. | The DP model is chosen for having the best performance with less complexity when the hysteresis of the cells can be neglected with RMSE < 10 mV. |

Proposed paper | CC | A DP ECM is proposed and characterized by analytical equations (TD), LS and heuristic optimization techniques (GA and PSO). Impedance behavior is estimated and compared to EIS. | A WLTC urban/suburban and heavy-duty current profiles are performed for LTO and NMC battery cells. Discussion of the Ohmic, total and internal resistance behavior in time and frequency. | A max RE ^{4} < 4% for all techniques is observed. PSO and NLS are optimal identification methods. Trade-off between accuracy, robustness and computational time set PSO as the best approach. |

^{1}MAPE: mean absolute percentage error.

^{2}RMSE: root mean square error.

^{3}MAE: mean absolute error.

^{4}RE: relative error.

^{5}DST: dynamic stress test.

^{6}FUDS: federal urban driving schedule.

^{7}DDPT: dynamic discharge pulse test.

^{8}WLTC: worldwide harmonized light vehicles test cycle.

^{9}NEDC: new European driving cycle.

^{10}CCD: constant current discharge.

Li-ion | Method | Mean RE (%) | Max. RE (%) | Min. RE (%) | Std. Deviation | Comp. Time (s) | Robustness (p.u) | RMSE (mV) | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

HD | WLTC | HD | WLTC | HD | WLTC | HD | WLTC | on HPPC Profile | HD | WLTC | HD | WLTC | ||

LTO | TD | 0.622 | 0.714 | 3.295 | 1.787 | 4.717 | 3.871 | 0.808 | 0.939 | <10 | 1 | 1 | 22.12 | 26.79 |

LS | 0.573 | 0.621 | 3.067 | 2.085 | 3.861 | 3.364 | 0.745 | 0.742 | <60 | 1 | 1 | 20.22 | 18.56 | |

GA | 0.635 | 0.634 | 3.982 | 3.444 | 4.899 | 2.086 | 0.805 | 0.791 | <1200 | 0.90 | 0.87 | 24.27 * | 21.49 * | |

PSO | 0.558 | 0.522 | 2.984 | 3.293 | 3.290 | 2.431 | 0.657 | 0.531 | <100 | 0.92 | 0.90 | 17.05 * | 15.02 * | |

NMC | TD | 0.577 | 0.755 | 2.576 | 1.793 | 3.147 | 2.053 | 0.803 | 0.800 | <10 | 1 | 1 | 31.26 | 31.69 |

LS | 0.275 | 0.621 | 1.188 | 2.425 | 1.341 | 0.895 | 0.358 | 0.777 | <60 | 1 | 1 | 14.61 | 26.2 | |

GA | 0.312 | 0.477 | 2.418 | 2.023 | 1.640 | 1.180 | 0.430 | 0.557 | <900 | 0.94 | 0.94 | 17.02 * | 20.87 * | |

PSO | 0.291 | 0.459 | 1.447 | 1.710 | 1.455 | 1.451 | 0.383 | 0.511 | <100 | 0.95 | 0.95 | 15.01 * | 18.72 * |

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**MDPI and ACS Style**

Kalogiannis, T.; Hosen, M.S.; Sokkeh, M.A.; Goutam, S.; Jaguemont, J.; Jin, L.; Qiao, G.; Berecibar, M.; Van Mierlo, J.
Comparative Study on Parameter Identification Methods for Dual-Polarization Lithium-Ion Equivalent Circuit Model. *Energies* **2019**, *12*, 4031.
https://doi.org/10.3390/en12214031

**AMA Style**

Kalogiannis T, Hosen MS, Sokkeh MA, Goutam S, Jaguemont J, Jin L, Qiao G, Berecibar M, Van Mierlo J.
Comparative Study on Parameter Identification Methods for Dual-Polarization Lithium-Ion Equivalent Circuit Model. *Energies*. 2019; 12(21):4031.
https://doi.org/10.3390/en12214031

**Chicago/Turabian Style**

Kalogiannis, Theodoros, Md Sazzad Hosen, Mohsen Akbarzadeh Sokkeh, Shovon Goutam, Joris Jaguemont, Lu Jin, Geng Qiao, Maitane Berecibar, and Joeri Van Mierlo.
2019. "Comparative Study on Parameter Identification Methods for Dual-Polarization Lithium-Ion Equivalent Circuit Model" *Energies* 12, no. 21: 4031.
https://doi.org/10.3390/en12214031