# A Comparative Study on Different Online State of Charge Estimation Algorithms for Lithium-Ion Batteries

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## Abstract

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## 1. Introduction

#### Key Contributions

- Seven different widely used SOC estimation algorithms, EKF, AEKF, SPKF, UKF, LO, SMO, and BPNN, were considered to develop a comparative study.
- The effect of temperature on two RC battery model parameters was analyzed, and the obtained parameters were utilized for model-based SOC estimation.
- An experimental dataset collected under a dynamic load profile test on commercial 25 Ah lithium-nickel-manganese-cobalt-oxide (LiNiMnCoO
_{2}) prismatic cells was considered for SOC estimation using different algorithms. Further, the estimated SOC accuracy and computational burden were compared. - Based on the obtained results, future recommendations are explained to assist engineers and researchers in their work.

## 2. Description of Considered Online SOC Estimation Algorithms

#### 2.1. Model-Based Method

#### 2.1.1. Extended Kalman Filter (EKF)

#### 2.1.2. Adaptive Extended Kalman Filter (AEKF)

_{k}) based on the innovation sequence (e

_{i}) inside the moving estimation window (M) is added in the estimation steps of the EKF. With the help of H

_{k}, the measurement (P

_{v,k}) and process (P

_{w,k}) covariance matrices are updated iteratively. Divergence is also an essential factor for the accuracy of EKF. In [17,57], a divergence judgmental condition was introduced in the AEKF to avoid filter divergence and improve stability.

#### 2.1.3. Sigma-Point Kalman Filter (SPKF)

#### 2.1.4. Unscented KF (UKF)

#### 2.1.5. Luenberger Observer (LO)

#### 2.1.6. Sliding Mode Observer (SMO)

#### 2.2. Data-Driven Method

#### Backpropagation Neural Network (BPNN)

## 3. Experimental Setup and Test Conduct

#### 3.1. Experimental Setup

_{2}) prismatic cell, abbreviated as Li-NMC; an EA-PSI 9080–510 power supply; an EA-EL 9500–60 B electronic load; a Vötsch Industrietechnik VTL 4006 climate chamber; and a laptop running the MATLAB software for data acquisition. Each sample of the battery variables, including current, voltage, and the temperature of the cell during charging, discharging, and rest phases, was obtained every 0.05 s (measurement frequency = 20 Hz). The upper (4.2 V) and lower (2.5 V) safety limits for battery voltage were also monitored throughout the experiments. The battery cell was placed inside the climate chamber to test under controlled temperature conditions. The climate chamber had a temperature range of −40 to + 180 °C with a heating rate of 2.5 K/min and a cooling rate of 3.5 K/min. The power supply and electronic load had an inbuilt function generator that could apply constant current pulses with subsequent rest phases in between. These constant current pulses were applied to obtain the voltage response of the battery when subjected to constant current pulses followed by a long rest phase. The instruments were connected and controlled using a prototype graphical user interface (GUI) developed on the app designer development environment in MATLAB, which ran on the laptop.

#### 3.2. Stepwise Test Description

#### 3.3. Pulse Charge/Discharge Test

#### 3.4. Dynamic Load Profile Test

_{0}, R

_{1}, R

_{2}, C

_{1,}and C

_{2}, it was necessary to validate the results. For validation, a dynamic current profile was chosen, and the voltage response of the cell was recorded. As shown in Figure 6, the dynamic profile involved a combination of the Worldwide Harmonized Light Vehicle Test Procedure (WLTP) class 3 drive cycle and the constant current discharge pulses. The current profile under the WLTP drive cycle is depicted in Figure 6a. The steps involved in dynamic discharging were:

- The cell voltage was measured before the start of the test. The voltage across the terminals was checked to be 4.10 V as specified by the manufacturer for the cell to be completely charged. If the voltage was 4.10 V, the test was started at 25 °C. If not, the cell was charged with a very low C-rate (C/30 or C/50) to bring the voltage to 4.10 V and then allowed to rest for at least 2 h to reach equilibrium.
- A discharging pulse of 25 A (1 C-rate) for 12 min was applied to discharge the cell by 20%, as shown in Figure 6b.
- A resting phase of 15 min was applied during which there was no current flow and the cell could relax.
- Figure 6b shows that a programmed WLTP class 3 drive cycle was applied to the cell under test. The drive cycle lasted for 1800 s, and during this test, the full cycle was applied.
- A resting phase of 15 min was applied again to allow the cell to relax. The above four steps were repeated multiple times until the voltage reached 3.0 V, which was the lower limit specified by the cell manufacturer.

## 4. Evaluation Method and Battery Modeling

#### 4.1. Evaluation Matrices

#### 4.2. Battery Modeling

_{0}is the internal resistance of the battery, and ${\mathrm{i}}_{k}$ shows the current flowing through the battery. ${\tau}_{1}$ represents the fast time constant for the battery and is equal to the product of ${\mathrm{R}}_{1}$ and ${\mathrm{C}}_{1}$, and ${\tau}_{2}$ represents the slow time constant for the battery and is equal to the product of ${\mathrm{R}}_{2}$ and ${\mathrm{C}}_{2}$. ${\mathrm{V}}_{t}$ and ${\mathrm{V}}_{OCV}$ is the voltage source to represent the battery terminal voltage and open circuit voltage, respectively. Due to the presence of fast and slow polarization effects, the model is sometimes also referred to as the dual-polarization model. This model offers the advantage of ease of implementation in real-time applications and provides sufficiently accurate estimation results.

_{0}is the sum of resistances offered by the electrolyte, the separator, and the electrodes. As shown in Figure 8a,b, the values of R

_{0}are relatively high at lower values of SOC (SOC ≤ 10%). This is because lithium ions are stored at the electrodes at the start of the charging or discharging process. R

_{0}values decrease continuously when the SOC is 0.2 ≤ SOC ≤ 0.9. As the charge continues to be added or removed during charging and discharging due to the constant application of current, the internal resistance keeps on decreasing. It is recommended that a battery be operated in the SOC range (20–90%) when used in an electric vehicle (EV) or hybrid EV (HEV). Furthermore, the variation in R

_{1}, R

_{2}, C

_{1}, C

_{2}, τ

_{1}, and τ

_{2}with SOC and temperature during the constant current charging and the discharging tests is demonstrated in Figure 8c–j.

#### 4.3. OCV vs. SOC Relationship

## 5. Results and Discussion

#### 5.1. Battery Model Parameter Validation

#### 5.2. SOC Estimation Results and Their Comparison

^{®}, and the computation time was considered using the “tic-toc” command. The SOC estimation was performed at 25 °C. The estimated parameters were analyzed for their dependency on the temperature and SOC. The SOC estimation algorithms have been compared based on several criteria, and the optimal method/s are suggested after exhaustive analysis.

## 6. Future Recommendations

- As the battery model parameters change with the battery aging, it is recommended to simultaneously estimate the battery model parameters to improve the SOC estimation accuracy using the model-based method further.
- By considering the adaptive value of sampling time, it would be possible to improve the SOC estimation accuracy and reduce the computational burden.
- Based on the obtained results, it is recommended to develop a hybrid algorithm by combining the algorithms to achieve higher SOC estimation accuracy and lower computational time.
- To achieve high accuracy of online SOC estimation in a real-time application, combining the model-based and data-driven methods in future studies is recommended.
- Because of the advancement of machine learning algorithms, the concept of a digital twin would be a good choice for online battery SOC estimation.
- To improve the efficiency of the BMS, it would be useful to develop a combined state estimation method using a data-driven as well as model-based approach.
- With the advancement of ML algorithms, it is recommended to develop a cloud-based BMS to reduce the overall cost of xEVs.

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Research statistics on battery state of charge estimation in the last decade. (Source: Science Direct, 2021 database).

**Figure 3.**Process of model-based online SOC estimation method [20].

**Figure 6.**Dynamic load profile: (

**a**) WLTP current profile; (

**b**) combined WLTP and pulse discharge current profile.

**Figure 8.**Extracted battery model parameters profiles for SOC under three different temperatures: (

**a**) R

_{0}during charging; (

**b**) R

_{0}during discharging; (

**c**) R

_{1}during charging; (

**d**) R

_{1}during discharging; (

**e**) R

_{2}during charging; (

**f**) R

_{2}during discharging; (

**g**) τ

_{1}during charging; (

**h**) τ

_{1}during discharging; (

**i**) τ

_{2}during charging; (

**j**) τ

_{2}during discharging.

**Figure 9.**Obtained OCV vs. SOC relationship under three different temperatures: (

**a**) for charging (

**b**) for discharging.

**Figure 10.**Comparison of measured and model terminal voltage at 25 °C: (

**a**) terminal voltage during charging; (

**b**) model terminal voltage error during charging; (

**c**) terminal voltage during discharging; (

**d**) model terminal voltage error during discharging.

**Figure 11.**SOC estimation results: (

**a**) SOC estimation using EKF; (

**b**) absolute error using EKF; (

**c**) SOC estimation using AEKF; (

**d**) absolute error using AEKF; (

**e**) SOC estimation using SPKF; (

**f**) absolute error using SPKF; (

**g**) SOC estimation using UKF; (

**h**) absolute error using UKF; (

**i**) SOC estimation using SMO; (

**j**) absolute error using SMO; (

**k**) SOC estimation using LO; (

**l**) absolute error using LO; (

**m**) SOC estimation using BPNN; (

**n**) absolute error using BPNN.

**Table 1.**Different coefficients values of the gaussian expression of the fitted curve of charging and discharging OCV.

Mode | Coefficients in the Gaussian Expression Fitted Curve | ||||
---|---|---|---|---|---|

Charging OCV | ${a}_{1}$ = 3.761 | ${a}_{2}$ = 1.543 | ${a}_{3}$ = 3.294 | ${a}_{4}$= 1.368 | ${a}_{5}$ = 0.1842 |

${b}_{1}$ = 1.132 | ${b}_{2}$ = 0.7436 | ${b}_{3}$ = 0.3469 | ${b}_{4}$ = 0.00292 | ${b}_{5}$ = 0.04179 | |

${c}_{1}$ = 0.3507 | ${c}_{2}$ = 0.2674 | ${c}_{3}$ = 0.3965 | ${c}_{4}$ = 0.2271 | ${c}_{5}$ = 0.05305 | |

Discharging OCV | ${a}_{1}$ = 4.282 | ${a}_{2}$ = 1.954 | ${a}_{3}$ = 1.599 | ${a}_{4}$ = 0.4318 | $-$ |

${b}_{1}$ = 1.4 | ${b}_{2}$ = 0.4647 | ${b}_{3}$ = 0.0852 | ${b}_{4}$ = 0.0373 | $-$ | |

${c}_{1}$ = 0.8515 | ${c}_{2}$ = 0.5143 | ${c}_{3}$ = 0.3252 | ${c}_{4}$ = 0.0965 | $-$ |

Method | Tunable parameters | RMSE (%) | Max AE (%) | Computational Time (sec) | Limitations |
---|---|---|---|---|---|

EKF | P (3 × 3), Q (3 × 3), R (1 × 1) | 0.5368 | 0.9721 | 145.134 | Linearizes the state and measurement equations using the first-order Taylor expansion; dependence on battery model parameters; prior knowledge of system noise signals is necessary; dependence on experimental conditions |

AEKF | P (3 × 3), Q (3 × 3), R (1 × 1) | 0.3176 | 0.4184 | 199.908 | Complex tuning of the filter performance when simultaneously adapting matrices Q and R; dependence on battery model parameters; dependence on experimental conditions |

SPKF | P (3 × 3), Q (3 × 3), R (1 × 1), h (1 × 1) | 0.5574 | 0.94 | 208.152 | High value of measurement noise causes filter divergence; estimation errors, and slow convergence when the dataset is too large; dependence on battery model parameters; dependence on experimental conditions |

UKF | P (3 × 3), Q (3 × 3), R (1 × 1), α (1 × 1), β (1 × 1), κ (1 × 1) | 0.5585 | 0.91 | 213.308 | High value of measurement noise causes filter divergence; estimation errors, and slow convergence when the dataset is too large; dependence on battery model parameters; dependence on experimental conditions; 3 additional tunable parameters apart from P, Q, and R matrices |

LO | Observer poles (3 × 1) | 1.2709 | 29.39 | 354.972 | Incorrect determination of observer poles can lead to estimation errors, and noise problems can arise when observer poles are placed farther to the left in the complex plane; dependence on battery model parameters; dependence on the experimental condition |

SMO | R (1 × 1), Q (3 × 3), W (1 × 1), ${Q}_{f}$ (3 × 3), ρ (1 × 1) | 0.0439 | 0.0963 | 330.013 | Ricatti Equation constants; Lyapunov dtability constants; observer switching; gain constant |

BPNN | α (1 × 1), network size, lambda (1 × 1) | 0.8172 | 46.99 | 160.142 | Computationally expensive; requires advanced techniques for optimization |

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**MDPI and ACS Style**

Khan, Z.A.; Shrivastava, P.; Amrr, S.M.; Mekhilef, S.; Algethami, A.A.; Seyedmahmoudian, M.; Stojcevski, A.
A Comparative Study on Different Online State of Charge Estimation Algorithms for Lithium-Ion Batteries. *Sustainability* **2022**, *14*, 7412.
https://doi.org/10.3390/su14127412

**AMA Style**

Khan ZA, Shrivastava P, Amrr SM, Mekhilef S, Algethami AA, Seyedmahmoudian M, Stojcevski A.
A Comparative Study on Different Online State of Charge Estimation Algorithms for Lithium-Ion Batteries. *Sustainability*. 2022; 14(12):7412.
https://doi.org/10.3390/su14127412

**Chicago/Turabian Style**

Khan, Zeeshan Ahmad, Prashant Shrivastava, Syed Muhammad Amrr, Saad Mekhilef, Abdullah A. Algethami, Mehdi Seyedmahmoudian, and Alex Stojcevski.
2022. "A Comparative Study on Different Online State of Charge Estimation Algorithms for Lithium-Ion Batteries" *Sustainability* 14, no. 12: 7412.
https://doi.org/10.3390/su14127412