# A Review on Battery Modelling Techniques

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## Abstract

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## 1. Introduction

#### 1.1. Significance of Battery Modelling

- (i)
- Development of efficient BMS.
- (ii)
- Key in the improvement of charging/discharging techniques and the enhancement of battery capacity.
- (iii)
- Need to capture the influence of power consumption on the battery.
- (iv)
- To prevent serious damage to batteries from overcharging or over-discharging.
- (v)
- Faster and safer way to study battery behaviour under different operating conditions.
- (vi)
- Identifying the operating limits that achieve best lifetime for specific applications.

#### 1.2. Contributions of the Proposed Study

## 2. Types of Battery Modelling

#### 2.1. Electrochemical Modelling of a Battery

_{1}/${x}_{k}$ from Equation (1) and K

_{2}·${x}_{k}$ from Equation (2) reflect the polarisation resistances of the battery [24]. K

_{3}·ln (${x}_{k}$) and K

_{4}·ln (1 − ${x}_{k}$) from Equation (3) represent the influence of the internal temperature and material activity during the electrochemical reaction of the battery, respectively [24]. Combining these equations, the model developed is as follows:

_{0}was used to describe the Open Circuit Voltage (OCV) voltage of the battery, which was not accounted for in Equations (1)–(3) [24]. This shows that the mere adding up of known models would not be possible in the framing of these models, and experimental data are needed. With the help of the data that are derived, a relevant model is developed. The importance of empirical formulation of these models is thus stressed in this example.

#### 2.2. Mathematical Modelling of a Battery

#### 2.3. Circuit-Oriented Modelling of a Battery

#### 2.4. Combined Modelling of a Battery

## 3. Battery Modelling Using Black Box Modelling Data-Driven Techniques

- The first phase is sample construction, after which the selected data are cleaned, normalised and transformed before being extracted and selected as samples for a machine learning algorithm. The data are then split into training and test sets. The second phase is to construct the model, which includes the “core” algorithms after which machine learning takes place. Support vector machine (SVM), support vector regression, artificial neural network, Bayesian principles, recurrent neural network and various optimisation algorithms such as genetic algorithms and simulated annealing methods are among the most widely used machine learning algorithms. The information gathered from the samples is preserved in a machine-readable manner for the following phase.
- Phase 2 involves using training and validation sets to find the best model.
- The third stage is model evaluation, which applies the knowledge gained in the previous step. SOC, RUL, etc., are estimated and the performance of the estimation models is evaluated using the estimation models determined in the previous phase. Root mean square error, absolute error and other metrics are commonly employed for model evaluation.

## 4. Battery Parameters Extraction Techniques Using Black Box Modelling Data-Driven Approach

#### 4.1. Estimation of SoC Using Black Box Modelling Data-Driven Approach

^{−5}and the maximum of absolute values of absolute error is not more than 4.44%. Hansen et al. (2005) investigated the use of SVM to estimate the SoC of a large-scale lithium-ion-polymer (LiP) battery pack [40]. When the optimal SVM is tested with US06, the driving cycle is provided by US and the RMSE (root mean square error) obtained is 5.76%.

_{4}) battery cell from an experimental dataset using SVM approach [41]. The SVM SoC estimator maintains a high level of accuracy better than 6% of overall ranges of operation. A hybrid radial basis function neural network method to estimate the SoC of lithium ferro phosphate (LFP) battery was proposed [55] in 2013. The orthogonal least squares algorithm determines the optimal structure of the RBF neural network and adaptive genetic algorithm is used to tune the parameters of the RBF neural network. Average absolute percentage error is 0.021 for the RBF neural network-based method.

#### 4.2. Estimation of SoH Using Black Box Modelling Data-Driven Approach

^{−3}% and 0.75%, in the data set B0006, they are 0.76%, 1.5 × 10

^{−2}% and 1.23%, in the data set B0007, they are 0.65%, 9.5 × 10

^{−3}% and 0.97%, and for data set B0034, they are 1.32%, 8.06 × 10

^{−2}% and 2.84%, respectively.

#### 4.3. Estimation of RUL Using Black Box Modelling Data-Driven Approach

^{3}, respectively, in about (2000) cycles [72]. Yang (2016) proposed an ensemble-based relevance vector machine (RVM) learning algorithm to predict the RUL of degraded lithium-ion batteries. The study also demonstrated that the hybrid prognostic approach using the selective kernel ensemble-based RVM learning algorithm outperformed the hybrid prognostic approaches using the single kernel-based RVM learning algorithm and the ensemble all-based RVM learning algorithm [71].

_{1}regularisation and L

_{2}regularisation are 6.6 × 10

^{−3}, 5.7 × 10

^{−3}and 1.4 × 10

^{−2}, respectively [74].

#### 4.4. Estimation of Capacity Using Black Box Modelling Data-Driven Approach

## 5. Battery Parameters Extraction Techniques Using Grey Box Modelling Data-Driven Approach

#### 5.1. Modelling of Circuit Oriented Model for Grey Box Modelling

#### 5.2. Development of Thevenin’s COM Model

_{0}) is realised as a second-order polynomial function of SoC to accurately replicate the terminal properties while maintaining the model’s simplicity.

_{1}, R

_{2}, C

_{1}and V

_{0}under constant current charging/discharging conditions are represented with polynomial equations as a function of charge rate (C

_{r}) SoC for charging scenario and discharge rate (D

_{r}) DoD for discharging scenario. The behaviour of C

_{r}and D

_{r}features is exponentially growing and reducing with regards to C

_{r}and D

_{r}, as shown in Figure 3. As a result, utilising a polynomial equation with an exponential function is the ideal way to express the battery charge and discharge rate characteristics [9].

#### 5.3. Representation of SoC and DoD Using Polynomial Equations

_{0}is defined as the voltage between the terminals of battery in open circuit condition. SoC

_{cr}, DoD

_{cr}, C

_{r}, D

_{r}, temperature and cycle number are all multi-variable functions in the battery. The parameters V

_{0}and R

_{1}in series with parallel R

_{2}C combination are used to determine the effective capacity, instantaneous voltage drop and self-discharge energy. The model’s R

_{2}C network is comparable to Thevenin’s transient response model. The exponential nature of the battery parameters R

_{1}, R

_{2}, V

_{0}and C is approximated by polynomial functions. The general polynomial equations for the parameters are given in Equations (6)–(9) [9].

_{1}, R

_{2}, C and V

_{0}are represented in terms of polynomial equations and there are 31 coefficients (from a

_{1}to a

_{31}) in total. The following section explains the comprehensive extraction process for finding these polynomial coefficients. The charging process parameters are generated by substituting x and y with C

_{r}and SOC

_{cr}, whereas the discharging process parameters are derived by replacing Dr and (1 − DoD

_{cr}). Here, (1 − DOD

_{cr}) is chosen for discharging voltage as V

_{0}decreases with increase in DOD

_{cr}.

#### 5.4. Battery Terminal Voltage Calculations

_{bc}) and discharging scenario (V

_{bd}) with respect to time is as given [88]. Calculating battery terminal voltage for charging or discharging at various C

_{r}and D

_{r}is performed using equations. The battery’s terminal charge or discharge voltage (Vc

_{j}

^{c}or Vd

_{j}

^{c}) varies depending on the battery’s capacity, SOC

_{cr}/DOD

_{cr}levels and C

_{r}/D

_{r}. The parameters of the non-linear Vc

_{j}

^{c}/Vd

_{j}

^{c}relationship are also expressed in terms of polynomial equations, where i and j signify the ith and jth determined charging and discharging voltage values, respectively. As a result, Equations (10) and (11) yield the battery terminal voltage for charging and discharging scenarios with regard to time when the current is constant (6) [9,97,98].

_{r}is the remaining capacity of the battery and t

_{c}, I

_{c}, t

_{d}and I

_{d}are charge time, charge current, discharge time and discharge current, respectively. Thus, Equations (10) and (11) accurately represent the behaviour of any battery type, if the parameters are well defined. These equations capture the non-linear behaviour of the battery which depends on the actual battery charge/discharge voltage.

#### 5.5. Charge/Discharge Rate and SoC Calculations Using Grey Box Modelling

_{r}and SOC

_{cr}of the battery vary depending on the present condition of the battery. The user-defined C

_{r}limit (C

^{lmt}) and initial battery SoC (SOC

_{ini}) have also been taken into consideration. The ${C}_{r}^{crt}$ and ${D}_{r}^{crt}$ of the battery can be expressed as given below [9,97,98].

^{lmt}and ${C}_{r}^{crt}$ is chosen by the algorithm. A similar type of control algorithm is used for discharging scenarios. The SOC

_{cr}and DOD

_{cr}are calculated from Equations (14) and (15) [9,97,98].

_{ini}is the initial SOC of the battery; SOC

_{max}and DOD

_{max}are the maximum user-defined SOC and DOD limits. The non-linear behaviour of the battery is identified using the polynomial equations mentioned above. It has a unique dependence on the charge/discharge voltage. The actual behaviour of the battery is represented using this model and accurate results are obtained from it. The parameters are easily detectable in this model, and are made to compare with different types of battery manufacturers’ data [99,100,101,102].

#### 5.6. Parameter Extraction of the Grey Box Modelling Using Bio-Inspired Algorithms

- (a)
- The purpose of using an evolutionary algorithm for a battery parameter extraction problem lies in the fact that it requires only manufacturers Cr and Dr characteristics and gives consistent polynomial coefficients of the battery model during relatively fewer iterations.
- (b)
- Evolutionary algorithms are more flexible in extracting the battery parameters with any initial values, while other numerical methods are incapable of obtaining satisfactory solutions.
- (c)
- The algorithm is easy to understand and is optimised using the fitness function.
- (d)
- With optimising capability, the algorithm steers the fitness function to be more representative and yields an accurate solution set even if the initial values are far from the solutions.

_{1}− a

_{31}) in the search space. The next requirement after generation of the random solution set is the measure of the quality/fitness of the solution set. This can be achieved by establishing a fitness function value for the entire population.

## 6. Discussion

## 7. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

BMS | Battery Management System |

SoC | State of Charge |

SoH | State of Health |

DoD | Depth of Discharge |

ANN | Artificial Neural Network |

DFN | Doyle-Fuller-Newman |

SPM | Single Particle Model |

CFD | Computational Fluid Dynamics |

P2D | Pseudo-Two-Dimensional |

LiCoO_{2} | Lithium Cobalt Oxide |

x_{k} | Number of State Vectors in the system |

x | State variable depicting the SoC of the Battery |

y_{k} | Output Voltage of the Battery |

E0 | Output Voltage |

R | Internal Resistance (Ohmic Resistance) |

ik | Current Flowing through the resistance |

V_{0} | Open Circuit Voltage of the Battery |

RC | Resistor-Capacitor |

EKF | Extended Kalman filter |

OCV | Open Circuit Voltage |

COM | Circuit Oriented Model |

EIS | Electrochemical Impedance Spectroscopy |

GA | Genetic Algorithm |

ML | Machine Learning |

WNN | Wavelet Neural Network |

GPU | Graphical processing units |

FL | Fuzzy logic |

SBPM | Sparse Bayesian Predictive Modelling |

PNN | Probabilistic neural network |

SVR | Support vector regression |

RVM | Relevance vector machine |

LSTM | Long Short-Term Memory |

MAPE | Mean Absolute Percentage Error |

CNN | Convolutional Neural Network |

DCCN | Deep Convolutional Neural Network |

SOCcr | State of Charge |

DODcr | Depth of Discharge |

Vbc | Battery’s Terminal Voltage for Charging |

Li-Ion | Lithium-ion |

${C}_{r}^{crt}$ | Current charging rate |

SOCini | Initial SoC of the Battery |

Ic | Charging Current |

NSGA | Non-Dominated Sorting GA |

RUL | Remaining useful life |

SVM | Support Vector Machine |

ANFIS | Adaptive Neuro Fuzzy Inference System |

LCA | Linear Correlation Analysis |

EV | Electric Vehicle |

LiFeMnPO_{4} | Lithium Iron Manganese Phosphate |

RBF | Radial Basis Function |

ESS | Energy Storage System |

BM | Battery Modelling |

MLPNN | Multi layered Perception Neural Network |

ERNN | Elman Recurrent Neural Network |

LM | Levenberg-Marquardt |

AUKF | Adaptive Unscented Kalman Filter |

LSSVM | Least-square support vector machines |

ITDNN | input time-delayed neural network |

BPNN | Back propagation Neural Network |

L-MA | Levenberg-Marquardt Algorithm |

LiP | Lithium-ion-polymer |

LFP | Lithium Ferro Phosphate |

PSO | Particle Swarm Optimisation |

NEDC | National European Driving Cycle |

GRU | Gated Recurrent Unit |

MAE | Mean Absolute Error |

RMSE | Root Mean Square Error |

NN | Neural Network |

FNN | Feed Forward Neural Network |

DDRN | Dynamically driven recurrent network |

IndRNN | Independently recurrent neural network |

HI | Health Indicator |

EoL | End of Life |

CVS | Constant Voltage Source |

τ1 | Time constant in the order of minutes |

τ2 | Time constant in the order of seconds |

Cr | Charge Rate |

Dr | Discharge Rate |

Vcjc | Battery’s Terminal Charge Voltage |

Vdjc | Battery’s Terminal Discharge Voltage |

Vbd | Battery’s Terminal Voltage for Discharging |

LMWNN | L-M-based three-layer Wavelet Neural Network |

LMMWNN | L-M-based Multi hidden layer Wavelet Neural Network |

DODini | Initial DOD of the Battery |

Id | Discharging Current |

RNN | Recurrent Neural Network |

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**Figure 1.**Electrical circuit model with (

**a**) one resistance, (

**b**) one resistance and one RC network, and (

**c**) one resistance and two RC networks.

**Figure 2.**First order battery model [Reprinted with permission from [98] CC BY 4.0].

**Figure 3.**(

**a**) Charging characteristics of EIG battery from manufacturer’s catalogue for first order model in Figure 2. (

**b**) Discharging characteristics of EIG battery from manufacturer’s catalogue [Reprinted with permission from [98] CC BY 4.0].

Data Sets Obtained from Standard | |
---|---|

NASA Prognostics Centre of Excellence Data Repository | [52,69,70,73,75,77,80] |

Lithium Iron | [43] |

Lithium Iron Manganese Phosphate LiFeMnPO_{4} | [41] |

High Power Ni-MH rechargeable battery | [38] |

Lithium Iron Phosphate—LiFePO_{4} | [42,47,48,49,55,86] |

Lithium Titanate | [64] |

NiMH battery | [44] |

Li-Ion cells | [42,44,45,53,54,58,59,60,62,68,71,72,74,76,79,81,82,87,88] |

Li (NiCoMn)_{1/3}O_{2} | [66] |

USO6, US Department of Energy’s Hybrid Electrical Vehicle program | [40] |

8 common cycle conditions ARTERIAL, NYCCOM, UDDSHDV, COMMUTER, WVUINTER, 5PEAK, CSHVR, CBD14 in ADVISOR | [46] |

Lithium nickel-manganese-cobalt oxide | [63] |

Simulated data based on equivalent circuit model for Li-on Battery | [67] |

Lead-acid batteries | [39] |

Li-Co batteries | [61] |

Center for Advanced Life Cycle Engineering (CALCE) | [70] |

Lithium Polymer battery | [83] |

New European Driving Cycle—NEDC | [47,50] |

Federal Urban Driving Schedule—FUDS | [84] |

LiCoNiMnO | [57] |

**Table 2.**Batteries with extracted parameters using different machine learning algorithms discussed in literature.

Sl. No | Estimated Battery Parameter | Type of Battery | Implemented Machine Learning Algorithm |
---|---|---|---|

1 | State of Charge | Lithium Iron Manganese Phosphate (LiFeMnPO_{4}) battery | Support Vector Machine |

2 | State of Charge | High Power Ni-MH rechargeable battery | Adaptive Neuro-Fuzzy Inference System (ANFIS) |

3 | State of Charge | Lithium iron phosphate (LiFePO_{4}) | RBF Neural Network, OLS Algorithm and AGA |

4 | Aging, State of Charge, State of Health | Lithium iron phosphate (LiFePO_{4}) | Input Time-Delayed Neural Networks |

5 | State of Charge, State of Health | Lithium iron phosphate (LiFePO_{4}) | Dynamically Driven Recurrent Networks (DDRNs) |

6 | State of Charge, State of Health | Lithium Titanate (LTO) | Dynamically Driven Recurrent Networks (DDRNs) |

7 | State of Charge | Lithium Iron | Neural Networks and Extended Kalman Filter (NN and EKF) |

8 | State of Charge | Lithium-ion battery U1-12XP | Neural Networks and Extended Kalman Filter (NN and EKF) |

9 | State of Charge | NiMH battery with 1.2 V and 3.4 Ah | Neural Networks and Extended Kalman Filter (NN and EKF) |

10 | Capacity and State of Charge | Lithium iron phosphate battery cell | Ampere Hour Counting with Correction and Dual Adaptive Extended Kalman Filter Algorithm |

11 | State of Health | Two commercial Li-ion batteries with Li (NiCoMn)_{1/3}O_{2} cathode and graphite anode | Support Vector Machine |

12 | State of Charge | Li-ion cells with 3.2 V/50 Ah supplied by Huanyu New Energy Technology Company Ltd. | Support Vector Machine Based on Particle Swarm Optimisation |

13 | State of Health | Lithium Nickel-Manganese-Cobalt Oxide | Advanced Sparse Bayesian Predictive Modelling (SBPM) |

14 | State of Charge, State of Health | Li-ion cells | Feed-Forward Artificial Neural Network |

15 | Capacity and resistance | lithium-ion battery | SVM |

16 | Capacity | nickel-manganese-cobalt (NMC)/graphite pouch cells | Random Forest Regression |

17 | State of Charge | Panasonic 18650PF battery cells | Recurrent Neural Network with Gated Recurrent Unit (GRU-RNN) |

18 | State of Charge | Samsung 18650-20R battery cells | Recurrent Neural Network with Gated Recurrent Unit (GRU-RNN) |

19 | State of Health | Li-Co batteries | Probabilistic Neural Network |

20 | SoC Estimation | A lithium polymer battery manufactured by KOKAM Company | Adaptive Unscented Kalman Filters (AUKF) and Least-Square Support Vector Machines (LSSVM). |

21 | Charging Current | Lithium Iron Phosphate (LiFePO_{4}) | ANN and Backpropagation Algorithm Ensemble Learning |

22 | RUL | Selected IFP1865140 type batteries were developed by HeFei Guo Xuan High-Tech Power Energy Company Limited of China | Feed Forward Neural Network (FFNN) |

23 | RUL | High-energy 18650 lithium-ion batteries manufactured by Panasonic, labelled NCR18650PF | Long Short-Term Memory (LSTM) Recurrent Neural Network (RNN) |

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Tamilselvi, S.; Gunasundari, S.; Karuppiah, N.; Razak RK, A.; Madhusudan, S.; Nagarajan, V.M.; Sathish, T.; Shamim, M.Z.M.; Saleel, C.A.; Afzal, A.
A Review on Battery Modelling Techniques. *Sustainability* **2021**, *13*, 10042.
https://doi.org/10.3390/su131810042

**AMA Style**

Tamilselvi S, Gunasundari S, Karuppiah N, Razak RK A, Madhusudan S, Nagarajan VM, Sathish T, Shamim MZM, Saleel CA, Afzal A.
A Review on Battery Modelling Techniques. *Sustainability*. 2021; 13(18):10042.
https://doi.org/10.3390/su131810042

**Chicago/Turabian Style**

Tamilselvi, S., S. Gunasundari, N. Karuppiah, Abdul Razak RK, S. Madhusudan, Vikas Madhav Nagarajan, T. Sathish, Mohammed Zubair M. Shamim, C. Ahamed Saleel, and Asif Afzal.
2021. "A Review on Battery Modelling Techniques" *Sustainability* 13, no. 18: 10042.
https://doi.org/10.3390/su131810042