# On the Usage of Battery Equivalent Series Resistance for Shuntless Coulomb Counting and SOC Estimation

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## Abstract

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## 1. Introduction

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- A shuntless SOC estimation method is proposed that exploits the knowledge of the internal resistance of the battery under test;
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- The proposed method is validated by experimental results performed on a lithium polymer (LiPo) rechargeable battery with a realistic current profile emulating an electrical vehicle scenario.

## 2. Proposed Shuntless SOC Measurement Method

Algorithm 1 Estimate SOC. | |

Input: (obtained in the calibration phase) Look-up-tables $LU{T}_{OCV,{I}_{1}},\dots ,LU{T}_{OCV,{I}_{n}}$, $LU{T}_{ESR,{I}_{1}},\dots ,LU{T}_{ESR,{I}_{n}}$, $LU{T}_{SOC}$, battery capacity C | |

Output: SOC estimate, denoted as $\widehat{SOC}$ | |

1: begin | |

2: Start condition: battery fully charged | |

3: Measure ${V}_{load}\left(1\right)$ | ▹ First voltage measurement |

4: $\widehat{I}\left(1\right)\leftarrow \frac{{V}_{load}\left(1\right)-LU{T}_{OCV}\left(1\right)}{LU{T}_{ESR}\left(1\right)}$ | ▹ Initial estimate of current |

5: $\widehat{SOC}\left(1\right)\leftarrow 100\%$ | ▹ Initial estimate of SOC |

6: $k\leftarrow 2$ | |

7: while True do | |

8: $\widehat{q}\leftarrow \widehat{I}(k-1)\times (t\left(k\right)-t(k-1))$ | ▹ Charge increment (coulomb counting) |

9: $\widehat{SOC}\left(k\right)\leftarrow \widehat{SOC}(k-1)+\widehat{q}/C$ | ▹ Estimate SOC at time step k |

10: return $\widehat{SOC}\left(k\right)$ | |

11: Measure ${V}_{load}\left(k\right)$ | ▹ Voltage measurement at time step k |

12: $j\leftarrow $ index of the entry of $LU{T}_{SOC}$ closest to $\widehat{SOC}\left(k\right)$ | |

13: $\widehat{OCV}\leftarrow $ linear fit of $LU{T}_{OCV,{I}_{1}}\left(j\right),\dots ,LU{T}_{OCV,{I}_{n}}\left(j\right)$ eval. at $\widehat{I}(k-1)$ | |

14: $\widehat{ESR}\leftarrow $ linear fit of $LU{T}_{ESR,{I}_{1}}\left(j\right),\dots ,LU{T}_{ESR,{I}_{n}}\left(j\right)$ eval. at $\widehat{I}(k-1)$ | |

15: $\widehat{I}\left(k\right)\leftarrow \frac{{V}_{load}\left(k\right)-\widehat{OCV}}{\widehat{ESR}}$ | ▹ Estimate current at time step k |

16: $k\leftarrow k+1$ | |

17: end while | |

18: end |

## 3. Experimental SOC Estimation Results

#### 3.1. Experimental Setup

#### 3.2. Experimental Procedure

#### 3.2.1. Calibration Phase

#### 3.2.2. Operational Phase

#### 3.3. Results

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

SOC | State of charge |

ESR | Equivalent series resistance |

LUT | Look-up-table |

OCV | Open-circuit voltage |

SMU | Source measurement unit |

WLTP | Worldwide Harmonized Light Vehicles Test Procedure |

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**Figure 1.**Operation of the proposed method, divided into two phases: calibration phase where the LUTs are built, and operational phase, where the LUTs are used together with voltage measurements to estimate the current and SOC.

**Figure 3.**Experimental data: OCV as a function of SOC. The curves were obtained in the calibration phase for different values of the discharging current.

**Figure 4.**Experimental data: ESR as a function of SOC. The curves were obtained in the calibration phase for different values of the discharging current.

**Figure 5.**An example of the best-fit line computed during the operational phase to obtain an estimate of OCV from an estimate of the current. The fit was performed on four data samples from the LUTs acquired at different currents: 250 mA, 500 mA, 750 mA, and 1000 mA.

**Figure 6.**Experimental results in the operational phase. (

**a**): Current estimated by the proposed shuntless method with a WLTP profile; (

**b**): magnification of the curve in (

**a**), showing a single repetition of the WLTP profile; (

**c**): current estimation error, computed as the difference between the estimated and reference value of the current.

**Figure 7.**Experimental results in the operational phase: (

**a**) SOC estimated by the proposed shuntless method; (

**b**) estimation error.

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**MDPI and ACS Style**

De Angelis, A.; Carbone, P.; Santoni, F.; Vitelli, M.; Ruscitti, L.
On the Usage of Battery Equivalent Series Resistance for Shuntless Coulomb Counting and SOC Estimation. *Batteries* **2023**, *9*, 286.
https://doi.org/10.3390/batteries9060286

**AMA Style**

De Angelis A, Carbone P, Santoni F, Vitelli M, Ruscitti L.
On the Usage of Battery Equivalent Series Resistance for Shuntless Coulomb Counting and SOC Estimation. *Batteries*. 2023; 9(6):286.
https://doi.org/10.3390/batteries9060286

**Chicago/Turabian Style**

De Angelis, Alessio, Paolo Carbone, Francesco Santoni, Michele Vitelli, and Luca Ruscitti.
2023. "On the Usage of Battery Equivalent Series Resistance for Shuntless Coulomb Counting and SOC Estimation" *Batteries* 9, no. 6: 286.
https://doi.org/10.3390/batteries9060286