# Evaluation of Lithium-Ion Battery Equivalent Circuit Models for State of Charge Estimation by an Experimental Approach

^{*}

## Abstract

**:**

_{2}O

_{4}battery module. Evaluations on the five models are carried out from the point of view of the dynamic performance and the state of charge (SoC) estimation. The dynamic performances of the five models are obtained by conducting the Dynamic Stress Test (DST) and the accuracy of SoC estimation with the Robust Extended Kalman Filter (REKF) approach is determined by performing a Federal Urban Driving Schedules (FUDS) experiment. By comparison, the DP model has the best dynamic performance and provides the most accurate SoC estimation. Finally, sensitivity of the different SoC initial values is investigated based on the accuracy of SoC estimation with the REKF approach based on the DP model. It is clear that the errors resulting from the SoC initial value are significantly reduced and the true SoC is convergent within an acceptable error.

## 1. Introduction

_{2}O

_{4}battery module with a nominal voltage of 57.6 V and a nominal capacity of 100 Ah is researched. An improved model is proposed based on the investigations of the traditional models from the point of view of the aspects of dynamic performance and SoC estimation. The model parameters are identified by the genetic algorithm along with the experimental data. The dynamic performances of the battery models are compared and the accuracy of the model-based SoC estimations with a robust extended Kalman filter (REKF) are evaluated. Furthermore, the sensitivity of the different SoC initial values on the presented model-based SoC estimation is discussed.

## 2. Equivalent Circuit Models of Lithium-Ion Battery

#### 2.1. The Rint Model

_{oc}to define the battery open-circuit voltage. Both resistance R

_{o}and open-circuit voltage U

_{oc}are functions of SoC, SoH and temperature. I

_{L}is load current with a positive value at discharging and a negative value at charging, U

_{L}is the terminal voltage.

_{L}= U

_{oc}− I

_{L}R

_{o}

#### 2.2. The RC Model

_{c}, C

_{b}) and three resistors (R

_{t}, R

_{e}, R

_{c}). The capacitor C

_{c}, which has a small capacitance and mostly represents the surface effects of a battery, is named surface capacitor. The capacitor C

_{b}, which has a very large capacitance and represents the ample capability of a battery to store charge chemically, is named bulk capacitor. SoC can be determined by the voltage across the bulk capacitor. Resistors R

_{t}, R

_{e}, R

_{c}are named terminal resistor, end resistor and capacitor resistor, respectively. U

_{b}and U

_{c}are the voltages across C

_{b}and C

_{c}, respectively. The electrical behaviour of the circuit can be expressed by Equations (2) and (3).

#### 2.3. The Thevenin Model

_{oc}, internal resistances and equivalent capacitances. The internal resistances include the ohmic resistance R

_{o}and the polarization resistance R

_{Th}. The equivalent capacitance C

_{Th}is used to describe the transient response during charging and discharging. U

_{Th}is the voltages across C

_{Th}. I

_{Th}is the outflow current of C

_{Th}. The electrical behavior of the Thevenin model can be expressed by Equation (4).

#### 2.4. The PNGV Model

_{d}and U

_{PN}are the voltages across $1/{U}_{\text{oc}}^{\prime}$ and C

_{PN}respectively. I

_{PN}is the outflow current of C

_{PN}. The electrical behavior of the PNGV model can be expressed by Equation (5):

#### 2.5. The DP Model

_{oc}; (2) Internal resistances such as the ohmic resistance R

_{o}and the polarization resistances, which include R

_{pa}to represent the effective resistance characterizing electrochemical polarization and R

_{pc}to represent the effective resistance characterizing concentration polarization; (3) the effective capacitances like C

_{pa}and C

_{pc}, which are used to characterize the transient response during transfer of power to/from the battery and to describe the electrochemical polarization and the concentration polarization separately. U

_{pa}and U

_{pc}are the voltages across C

_{pa}and C

_{pc}respectively. I

_{pa}and I

_{pc}are the outflow currents of C

_{pa}and C

_{pc}respectively. The electrical behavior of the circuit can be expressed by Equation (6):

## 3. Model Parameters’ Identification of a Lithium-Ion Power Battery Module

#### 3.1. Battery Test Bench

_{2}O

_{4}battery module are carried out in a thermal chamber with a fixed temperature of 20 °C.

#### 3.2. Experimental Design

_{2}O

_{4}battery module at 0.1 SoC intervals (constant current C/3 discharge segments) starting from 1.0 to 0.1 and each interval followed by a 2-hour rest to allow the battery to get an electrochemical and thermal equilibrium condition before applying the next.

#### 3.3. Model Parameters’ Identification Method

#### 3.3.1. The Rint Model

_{L}is conducted at each SoC separately. A confirmed coefficient r

^{2}, which is defined as Equation (7), is selected to evaluate the identification accuracy:

_{L}, ${\stackrel{-}{U}}_{\mathrm{L}}$ is the average value of U

_{L}.

#### 3.3.2. The RC Model

_{c}, C

_{b}, R

_{t}, R

_{e}, R

_{c}) at each SoC separately.

#### 3.3.3. The Thevenin Model

_{Th}= R

_{Th}C

_{Th}) needs to be given in advance based on the battery characteristics:

_{Th}and the objective function of the genetic algorithm is built as follows:

**χ**

_{k}at generation g;

**χ**

_{k}is the current individual k of the population

**χ**, where

**χ**= [τ

_{Th}]; ${\widehat{U}}_{\text{L,k}}$ is the estimation value of U

_{L}at the individual k; N is the estimation length, here N = 200.

#### 3.3.4. The PNGV Model

_{PN}. The same genetic algorithm as Equation (9) where

**χ**=[τ

_{PN}] was used to find the optimal value of τ

_{PN}:

#### 3.3.5. The DP Model

_{pc}and τ

_{pa}. The same genetic algorithm as Equation (9), where $\mathbf{\chi}=\left[\begin{array}{ll}{\tau}_{\text{pa}}\hfill & 0\hfill \\ 0\hfill & {\tau}_{\text{pc}}\hfill \end{array}\right]$ is used to find the optimal values of τ

_{pc}and τ

_{pa}:

#### 3.4. Identification Results

SoC | r^{2} | U_{oc} (V) | R_{o} (Ω) |
---|---|---|---|

0.5 | 0.996 | 63.158 | 0.02486 |

0.6 | 0.997 | 63.676 | 0.02465 |

SoC | r^{2} | C_{b} (F) | R_{e} (Ω) | C_{c} (F) | R_{c} (Ω) | R_{t} (Ω) |
---|---|---|---|---|---|---|

0.5 | 0.976 | 58103 | 0.01776 | 24.73 | 0.00651 | 0.01954 |

0.6 | 0.982 | 70266 | 0.01776 | 27.60 | 0.00651 | 0.01954 |

SoC | r^{2} | U_{oc} (V) | C_{Th} (F) | R_{Th} (Ω) | R_{o} (Ω) | τ_{Th} (s) |
---|---|---|---|---|---|---|

0.5 | 0.999 | 63.294 | 4581 | 0.007360 | 0.02423 | 33.7 |

0.6 | 0.999 | 63.808 | 5141 | 0.007046 | 0.024223 | 34.2 |

SoC | r^{2} | U_{oc} (V) | $1/{U}_{\mathrm{oc}}^{\prime}$ (F) | C_{PN} (F) | R_{PN} (Ω) | R_{o} (Ω) | τ_{PN} (s) |
---|---|---|---|---|---|---|---|

0.5 | 0.999 | 63.327 | 8373761 | 4643 | 0.00719 | 0.02425 | 33.4 |

0.6 | 0.999 | 63.845 | 8597339 | 4635 | 0.00678 | 0.02424 | 34.5 |

SoC | r^{2} | U_{oc} | C_{pa} | R_{pa} | C_{pc} | R_{pc} | R_{o} | τ_{pa} | τ_{pc} |
---|---|---|---|---|---|---|---|---|---|

0.5 | 0.999 | 63.302 | 5630 | 0.00064 | 54277 | 0.00824 | 0.02402 | 3.6 | 44.7 |

0.6 | 0.999 | 63.824 | 5700 | 0.00065 | 53817 | 0.00839 | 0.02406 | 3.7 | 45.2 |

_{oc}and R

_{o}are similar, but those parameters identifying the polarization characteristics are totally different due to the different levels of description of the polarization characteristics.

## 4. Evaluation on the Lithium-Ion Battery Models

#### 4.1. Model Verification

#### 4.2. Evaluation on the Accuracy of the Battery Models

Model | Maximum (V) | Mean (V) | Variance (V^{2}) | Max. Error Rate (%) |
---|---|---|---|---|

Rint model | 1.6229 | 0.3945 | 0.0762 | 2.8176 |

RC model | 1.0785 | 0.2336 | 0.0463 | 2.0337 |

Thevenin model | 0.2967 | 0.0455 | 0.0220 | 0.5151 |

PNGV model | 0.5772 | 0.0875 | 0.0243 | 1.0020 |

DP model | 0.2183 | 0.0429 | 0.0021 | 0.3790 |

#### 4.3. Evaluation on the Adaptability of the Battery Models for SoC Estimation

**X**is a n × 1 state matrix;

**Y**is a m × 1 observe matrix;

**A**,

**B**,

**C**,

**D**and

**Γ**are n × n, n × 1, m × n, m × 1 and n × n matrix respectively;

**w**

_{k}is a process noise with mean of

**q**

_{k}and covariance of

**Q**

_{k};

**v**

_{k}is the measurement noise with mean of

**r**

_{k}and covariance of

**R**

_{k}

**X**and SoC as:

**A**,

**B**,

**C**and

**D**can be written as follows:

**D**

_{k}

_{−1}= [R

_{o}]

**Y**

_{k}= [U

_{L,k}]

_{N}is the nominal capacity of the battery.

**X**

_{0}= [0 0 1]

^{T};

**R**= 10;

**Q**= diag (0.00001, 0.00001, 0.00001, 0.00001, 0.00001);

**P**

_{0}= diag (1, 0.1, 0.1, 0.1, 0.1) (herein,

**P**is the covariance matrix of

**X**). The DP model-based SoC estimation results with REKF are shown in Figure 12.

Current (A) | 30 | 50 | 100 | 150 | 200 | 300 |
---|---|---|---|---|---|---|

Coulombic efficiency in discharging process (%) | 100 | 99.3 | 98.5 | 98.1 | 97.4 | 95.2 |

Coulombic efficiency in charging process (%) | 100 | 98.5 | 97.4 | 96.0 | 94.0 | - |

**Figure 12.**The DP model-based SoC estimation results with REKF: (

**a**) Terminal voltage; (

**b**) Voltage estimation error; (

**c**) Polarization voltages; (

**d**) SoC.

#### 4.3.1. SoC Estimation Accuracy

Model | Maximum | Mean | Variance | Terminal |
---|---|---|---|---|

Rint Model | 0.0462 | 0.0186 | 0.0012 | 0.046 |

RC Model | 0.0681 | 0.0167 | 0.0009 | 0.013 |

Thevenin Model | 0.0500 | 0.0101 | 0.0004 | −0.016 |

PNGV Model | 0.0675 | 0.0126 | 0.0005 | −0.017 |

DP Model | 0.0309 | 0.0047 | 0.00004 | −0.005 |

#### 4.3.2. Evaluation on the SoC Estimation Accuracy Influenced by Its Initial Value

**Table 9.**The statistical list of absolute SoC estimation errors with different initial SoC values after 150 s and the terminal error.

SoC_{0} | Maximum | Mean | Variance | Terminal Error |
---|---|---|---|---|

0.90 | 0.0098 | 0.0051 | 2.65 × 10^{−5} | 0.0070 |

0.96 | 0.0181 | 0.0080 | 1.79 × 10^{−5} | 0.0073 |

0.84 | 0.0279 | 0.0105 | 7.40 × 10^{−5} | 0.0103 |

0.50 | 0.0352 | 0.0144 | 3.68 × 10^{−4} | 0.0156 |

## 5. Conclusions

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**MDPI and ACS Style**

He, H.; Xiong, R.; Fan, J.
Evaluation of Lithium-Ion Battery Equivalent Circuit Models for State of Charge Estimation by an Experimental Approach. *Energies* **2011**, *4*, 582-598.
https://doi.org/10.3390/en4040582

**AMA Style**

He H, Xiong R, Fan J.
Evaluation of Lithium-Ion Battery Equivalent Circuit Models for State of Charge Estimation by an Experimental Approach. *Energies*. 2011; 4(4):582-598.
https://doi.org/10.3390/en4040582

**Chicago/Turabian Style**

He, Hongwen, Rui Xiong, and Jinxin Fan.
2011. "Evaluation of Lithium-Ion Battery Equivalent Circuit Models for State of Charge Estimation by an Experimental Approach" *Energies* 4, no. 4: 582-598.
https://doi.org/10.3390/en4040582