DOI: 10.3303/CET24114070 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Paper Received: 1 June 2024; Revised: 30 August 2024; Accepted: 23 November 2024 
Please cite this article as: Xiao Z., Ji C., Ma F., Wang J., Sun W., Zhai C., 2024, A Mechanism-aid Deep Learning Method for Li-ion Battery 
State-of-charge Estimation, Chemical Engineering Transactions, 114, 415-420  DOI:10.3303/CET24114070 
  

 CHEMICAL ENGINEERING TRANSACTIONS  
 

VOL. 114, 2024 

A publication of 

 

The Italian Association 
of Chemical Engineering 
Online at www.cetjournal.it 

Guest Editors: Petar S. Varbanov, Min Zeng, Yee Van Fan, Xuechao Wang 

Copyright © 2024, AIDIC Servizi S.r.l. 

ISBN 979-12-81206-12-0; ISSN 2283-9216 

A Mechanism-aid Deep Learning Method for Li-ion Battery 

State-of-charge Estimation 

Zhanghua Xiaoa, Cheng Jia, Fangyuan Maa , Jingde Wanga, Wei Suna, *, Chi Zhaib 

aCollege of Chemical Engineering, Beijing University of Chemical Technology, North Third Ring Road 15, Chaoyang District, 

 100029, Beijing, China  
bFaculty of Chemical Engineering, Kunming University of Science and Technology, 650500 Kunming, China 

 sunwei@mail.buct.edu.cn 

An accurate estimation on the battery state of charge (SOC) could serve as a foundation for the secure and 

stable operation of battery management systems. The rapid development of data science and artificial 

intelligence provides a new solution for battery SOC estimation. However, existing methods that directly utilize 

measured data to establish the SOC estimation will suffer from low prediction accuracy due to the insufficient 

incorporation of mechanism information. The equivalent circuit model is a reliable battery mechanism model 

that could be adopted to calculate the open-circuit voltage, which has been proved to be directly correlated with 

the battery SOC. The open-circuit voltage has rarely been applied to the SOC estimation because it is hard to 

be measured online. Therefore, the physical information provided by the equivalent circuit model can be 

combined with data-driven prediction model to obtain the real-time approximation of the open-circuit voltage, 

with which more useful features can be extracted to improve prediction accuracy. For this purpose, a 

mechanism-aid deep learning method is proposed, in which the loss function of the LSTM neural network is 

modified by the equivalent circuit model. And the neural network model could converge to the mechanism 

relationship of the open circuit voltage in the training stage and obtain accurate estimation on the SOC. The 

proposed method is applied to the Panasonic battery dataset, which is collected by conducting the tests at 

various real-world driving profiles. Compared with related methods that only consider voltage, current and 

temperature, the root mean square error and mean absolute error of the SOC estimation decreases 72.49 % 

and 72.23 %, respectively. Then the effectiveness of the proposed method is further verified under different test 

conditions, demonstrating the significance to introduce mechanism information in battery SOC estimation. 

1. Introduction 

With the continuous development of policy support on new energy technology, its application scenarios have 

been increasingly expanded. Under this circumstance, lithium-ion batteries are widely used in the electric vehicle 

(Pattaraprakorn et al, 2023) industry due to their long cycle life, high energy density, and stable performance, 

etc (Fang et al, 2019). Battery Management System (BMS) can help guarantee the safe and efficient use of 

batteries in electric vehicles. Among them, Stage of Charge (SOC) is a very important state indicator, which 

provides the basis for monitoring and controlling other indicators of BMS. However, there is a lack of direct 

measurement of SOC to provide an accurate and real-time estimation. 

Recent research on methods for SOC estimation includes four main categories (Peng et al, 2024), open-circuit 

voltage method, Ampere-time Integration method（AhI）, mechanism-based modeling method, and data-driven 

based method. The open-circuit voltage method establishes a look-up table relationship by measuring the 

mapping between the open-circuit voltage and the SOC of the battery. This method requires precise 

experimental measurements to obtain an accurate mapping relationship, and the measurement needs to keep 

the terminal voltage stable and eliminate the overpotential effect, which causes significant time consumption 

and limits its practical applications. The AhI method utilizes the integral of current release to calculate the battery 

capacity change to obtain the battery SOC, which is characterized by easy calculation and high efficiency, but 

its performance as an open-loop method is affected by the sampling noise, i.e., there are the problems of high 

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initial estimation requirements and increased cumulative errors. Mechanism-based modeling approach is to 

combine the battery model and nonlinear state-space algorithm to overcome the drawbacks of the above 

methods, which can be specifically classified into Equivalent Circuit Model (ECM) and Electrochemical Model 

(EM). The ECM simulates the voltage change process by equating the battery operating process to a circuit 

consisting of a voltage source, resistor, capacitor, and other devices. This method is relatively simple in structure 

and efficient in calculation, but its accuracy is limited by several factors, including ambient temperature. Typical 

EM was proposed by (Dolye et al, 1993). This method describes the working behavior of the battery by 

establishing the internal micro-processes of the battery, and the model established by this method contains 

multiple partial differential equations with numerous corresponding parameters, resulting in huge computational 

loads that cannot satisfy the requirement of online application though the relative accuracy is high. 

The data-driven SOC estimation method has received more attention because of the development of computer 

technology. It could achieve reasonable accuracy and practicality by directly extracting the corresponding 

information from the input information without physical modelling. Deep Learning (DL) is widely used for SOC 

estimation especially Recurrent Neural Networks (RNN), which are specialized in processing data sequentially. 

(Chemall et al, 2018) achieved accurate estimation of SOC by using a Long Short-Term Memory Neural Network 

(LSTM-RNN) to map currents, voltages, and temperatures directly to the SOC, thus avoiding the filters and 

inference algorithms used in the mechanism modeling process. (Zou et al, 2023) investigated a CNN to extract 

the feature inputs in the current-voltage into a convolutional informer network and design a loss function based 

on Laplace distribution features for the training process. (Bian et al, 2020) use a Bidirectional LSTM (BiLSTM) 

to improve the model's ability of handle input sequences in both directions. (Chen et al, 2022). enhance the 

LSTM through cope with the fluctuation problem of SOC estimation by expanding the inputs, and at the same 

time, design the AhI-method based state strategy to limit the output of constrained SOC, which obtain a more 

efficient computational speed compared to traditional filter methods. 

Li-ion battery is a very complex nonlinear system, and the existing deep learning SOC estimation methods are 

purely data-driven "black-box" models, ignoring the battery electrochemical principles. This shortcoming makes 

it difficult to further improve the accuracy of deep learning-based SOC estimation methods, and research in 

some other fields (Daw et al, 2021) has also shown that better performance can be achieved by incorporating 

mechanistic information into machine learning. This paper proposes a new method that combines the 

mechanistic information of ECM with neural networks to estimate the SOC of the battery, and evaluates the 

Root Mean Square Error (RMSE), Mean Absolute Error (MAE) and test the results of the batteries under different 

operating conditions. 

2. Methods 

The proposed method is schematically plotted in Figure 1. In a SOC estimation task, BMS sample signal data 

in different conditions. Subsequently, the voltage and current signals are input into the ECM, the voltage signal 

is deconstructed into open-circuit voltage (Uocv), and the current signal is calculated by the AhI method to obtain 

the cumulative current masked as ΔI. Finally, ΔI replaces the original current signal, with the data of Uocv, 

original voltage, and temperature composed of a new input sequence which is inputted into the neural network 

for training, validation, and testing. 

 

Figure 1: Overview of the proposed SOC estimation method 

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2.1 Identification of ECM parameters 

It has been pointed out that the open circuit voltage has a more direct mapping relationship with the SOC 

compared to the terminal voltage. The parameters proposed in this paper to increase the open-circuit voltage 

input are identified by the equivalent circuit model. The reason for choosing the equivalent circuit model instead 

of the electrochemical model is that there is a better balance between the complexity and accuracy of the ECM. 

As shown in Figure. 2, the method proposed in this paper uses a second-order Thevenin model, which consists 

of two series-connected RC loops, resistors, and an ideal voltage source. In the battery system (Hu et al，2012), 

the two series-connected RC loops of the ECM can simulate the electrochemical polarization and concentration 

polarization processes of the battery, respectively; the series-connected resistor R0 can represent the 

resistance to ion transfer within the battery system; and the ideal voltage source represents the amount of power 

that the battery can provide under ideal conditions. 

 

Figure 2: Schematic of the second-order Thevenin model 

The mathematical expression for the model is: 

{
 
 

 
 
𝑈 = 𝑈𝑜𝑐𝑣 − 𝐼𝑅0 − 𝑈1 − 𝑈2

𝑑𝑈1
𝑑𝑡

= −
𝑈1
𝐶1𝑅1

+
𝐼

𝐶1
𝑑𝑈2
𝑑𝑡

= −
𝑈2
𝐶2𝑅2

+
𝐼

𝐶2

 (1) 

where U1 and U2 are the polarization voltages corresponding to the R1C1 and R2C2 branches, respectively, and 

U0 is the ohmic voltage corresponding to R1. The Laplace's equation for the equivalent circuit model is obtained 

after discretizing the above equations as follows as: 

𝑈𝑜𝑐𝑣(𝑠) − 𝑈(𝑠) = 𝐼(𝑠)(𝑅0 +
𝑅1

1 + 𝑅1𝐶1𝑠
+

𝑅2
1 + 𝑅2𝐶2𝑠

) (2) 

where s denotes the response of the system in the complex frequency domain, and then a bilinear 

transformation is performed for the above equations to obtain the difference equation between the input and 

output of the system for ECM as: 

𝑦(𝑘) = 𝑈𝑜𝑐𝑣(𝑘) − 𝑈(𝑘) = 𝑎1𝑦(𝑘 − 1) + 𝑎2𝑦(𝑘 − 2) + 𝑎3𝐼(𝑘) + 𝑎4𝐼(𝑘 − 1) + 𝑎5𝐼(𝑘 − 2) (3) 

where I(k) is the current input to the system and y(k) is the output of the system, from which it follows: 

𝜑(𝑘) = [𝑦(𝑘 − 1)  𝑦(𝑘 − 2)  𝐼(𝑘)  𝐼(𝑘 − 1)  𝐼(𝑘 − 2)]𝑇 (4) 

𝜃(𝑘) = [𝑎1  𝑎2  𝑎3  𝑎4  𝑎5] (5) 

This leads to the recursive equation of the ECM, and in this paper, we apply the least squares approach to 

obtain reliable parameter identification results from the ECM, which in turn leads to the value of Uocv. 

2.2 AhI method to obtain the cumulative current value 

The AhI method is a method of calculating SOC by integrating the cumulative battery current, which is expressed 

as follows 

𝑆𝑂𝐶 = ∫ (−
𝜂

𝑄𝑛
) 𝐼(𝑡)

𝑡

0

+ 𝐶 (6) 

417



where η represents the discharge efficiency of the battery and Qn represents the rated capacity of the battery. 

From the above equations, it can be seen that the AhI method is a simple and effective method for calculating 

the SOC. In this paper, the proposed method is to replace the current input with the cumulative current input, 

so as to obtain the integral information of the current, which further helps the neural network to learn the 

characteristics of the input information. However, the AhI method itself includes some defects in practical 

application, as mentioned before there is the effect of cumulative error, and the more to the end of the estimation 

period the greater the error. Therefore, in this work, when the AhI method is used to obtain the integral 

information of the current, the loss function during training will be modified accordingly to penalize the final error 

offset term, so as to improve the accuracy of the estimation. 

The loss function proposed in this work is modified to add a penalty term to the original mean square error (MSE) 

for the part of the SOC estimate that is lower than the true value to satisfy the SOC estimate, and the original 

MSE expression is as follows: 

𝑀𝑆𝐸 =
1

𝑛
∑(𝑆𝑂𝐶𝑖 − 𝑆�̂�𝐶𝑖)

2

𝑛

𝑖=1

 (7) 

where n denotes the overall number of samples, denotes the reference SOC value, and is the estimated value. 

The modified loss function is: 

If ( 𝑆�̂�𝐶𝑖 − 𝑆𝑂𝐶𝑖)>0 

𝐿𝑜𝑠𝑠 = 𝑤1
1

𝑛
∑(𝑆𝑂𝐶𝑖 − 𝑆�̂�𝐶𝑖)

2

𝑛

𝑖=1

 (8) 

Else 

𝐿𝑜𝑠𝑠 = 𝑤2
1

𝑛
∑(𝑆𝑂𝐶𝑖 − 𝑆�̂�𝐶𝑖)

2

𝑛

𝑖=1

 (9) 

where w1 and w2 are both user-defined parameters and w1 < w2. 

2.3 Data sets and evaluation indicators 

The publicly available dataset used in this work was experimentally measured by Dr. Phillip Kollmeyer 

(phillip.kollmeyer@gmail.com) of the University of Wisconsin-Madison. The battery model used for the 

experiments is Panasonic's 18650PF battery with a rated capacity of 2.9Ah. It contains a test temperature range 

of -20~25°C, and the test conditions also contain a variety of conditions such as US06, HWFET, UDDS, LA92, 

and their combinations, which satisfy the experimental needs of this work. 

In order to evaluate the SOC estimation performance of the proposed method, the RMSE and MAE expressions 

used are as follows: 

𝑅𝑀𝑆𝐸 = √
1

𝑛
∑(𝑆𝑂𝐶𝑖 − 𝑆�̂�𝐶𝑖)

2

𝑛

𝑖=1

 (10) 

𝑀𝐴𝐸 =
1

𝑛
∑|(𝑆𝑂𝐶𝑖 − 𝑆�̂�𝐶𝑖|

𝑛

𝑖=1

 (11) 

The RMSE assesses the standard deviation of the estimation error, while the MAE assesses the robustness of 

the model. 

3. Results and discussion 

This section first validates and discusses the results of the method proposed on the dataset collected at 25°C 

in the battery and then discusses the results under different test conditions. 

3.1 SOC estimation results at 25°C 

As shown in Figure 3, the SOC estimation results of the proposed method at 25°C are first validated. As a 

comparison, the LSTM estimation model with only three inputs of current, voltage and temperature are used as 

a benchmark and is denoted as LSTM. the estimation model using the open circuit voltage parameter is labeled 

as LSTM-1, the estimation model using the cumulative current and the loss function is labeled as LSTM-2, and 

the estimation models that are used by all the improved methods are denoted as LSTM-1&2. A total of four 

operating conditions including US06, HWFET, UDDS, LA92 are used for testing, where the training set is the 

418



first three cycles of cycle1~cycle4 (which is a mix of the four batches of data for the four operating conditions), 

the validation set is the remaining cycle4, and the test set is Neural Network (NN) drive cycles. 

Table 1 demonstrates the estimation results of the proposed method. It can be seen that the traditional current-

voltage-temperature based three-input LSTM has an RMSE as high as 2.639 % and it can be observed from 

the graphs that the estimation fluctuates a lot and the results are not so accurate. Whereas the RMSE of LSTM-

1 with extended open circuit voltage input is 1.857 % and the corresponding RMSE of LSTM-2 is 1.521 %, the 

smallest RMSE is only 0.726 % for LSTM-1&2 using both the improved methods. The proposed method can be 

considered valid for the current test. 

 

Figure 3: SOC estimation results and errors at 25℃. The markers a to d represent LSTM, LSTM-1, LSTM-2 and 

LSTM-1&2 models 

Table 1: The RMSE and MAE of different SOC estimation methods 

Item LSTM LSTM-1 LSTM-2 LSTM-1&2 

RMSE/% 2.639 2.121 1.490 0.726 

MAE/% 2.035 1.706 1.018 0.565 

3.2 Comparison of different test conditions 

In the previous section, when dividing the training and testing sets, cycle1~4 are used for training and NN for 

testing. A comparison experiment is carried out in which cycle1~4 are used as the test conditions and the rest 

are used as the training data in order to analyze the effect of the proposed method under different working 

conditions. The specific estimation results are shown in Figure 4. The corresponding evaluation indicators are 

shown in table 2. It can be seen that neither cycle1, cycle2, cycle3 nor cycle4 obtains higher accuracy than the 

three-input LSTM estimation model in the proposed method. 

 

Figure 4: SOC estimation results under different test conditions. The left sides show the estimation results of 

the three-input LSTM, and the right sides show the estimation results of the proposed method. And a to b denote 

the test conditions cycle1 to 4 

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Table 2: The RMSE and MAE of different SOC estimation conditions 

LSTM Cycle1 Cycle2 Cycle3 Cycle4 LSTM-1&2 Cycle1 Cycle2 Cycle3 Cycle4 

RMSE/% 2.273 2.59 2.642 7.403 RMSE/% 2.012 1.668 1.44 1.856 

MAE/% 1.853 2.075 2.244 6.182 MAE/% 1.316 1.338 1.089 1.607 

The comparison test shows that the proposed method has higher estimation accuracy under different test 

conditions, which indicates that the proposed method can accomplish the estimation task under different test 

conditions. 

4. Conclusion 

SOC estimation is a fundamental component of the battery management system to ensure the safe and reliable 

operation of the battery power supply system, and neural networks have been evidenced in SOC estimation 

research. In the Panasonic 18650PF lithium battery test dataset, the accuracy of the LSTM model with only 

three inputs provides low accuracy. In contrast, the deep learning algorithm combined with the equivalent circuit 

model proposed in this paper is used for SOC estimation by further extracting the mechanistic information of 

the battery and inputting it into the neural network, which results in lower RMSE and MAE in the test results of 

the 25°C battery data, which are reduced by 72.49 % and 72.23 %, respectively, as compared to the normal 

three-input LSTM model. The results of this paper show that applying battery mechanism information to the 

battery state estimation problem in a data-driven approach is a proven method. 

Reference 

Bian C, He H, and Yang S, “Stacked bidirectional long short-term memory networks for state-of-charge 

estimation of lithium-ion batteries,” Energy, vol. 191, p. 116538, Jan. 2020, doi: 

10.1016/j.energy.2019.116538. 

Chemali E, Kollmeyer P. J, Preindl M, R. Ahmed, and A. Emadi, “Long Short-Term Memory Networks for 

Accurate State-of-Charge Estimation of Li-ion Batteries,” IEEE Transactions on Industrial Electronics, vol. 

65, no. 8, pp. 6730–6739, Aug. 2018, doi: 10.1109/TIE.2017.2787586. 

Chen J, Zhang Y, Wu J, Cheng W, and Zhu Q, “SOC estimation for lithium-ion battery using the LSTM-RNN 

with extended input and constrained output,” Energy, vol. 262, p. 125375, Jan. 2023, doi: 

10.1016/j.energy.2022.125375. 

Daw A, Karpatne A, Watkins W, Read J, and Kumar V, “Physics-guided Neural Networks (PGNN): An 

Application in Lake Temperature Modeling.” arXiv, Sep. 28, 2021. doi: 10.48550/arXiv.1710.11431. 

Doyle M, Fuller T. F, and. Newman J, “Modeling of Galvanostatic Charge and Discharge of the 

Lithium/Polymer/Insertion Cell,” J. Electrochem. Soc., vol. 140, no. 6, p. 1526, Jun. 1993, doi: 

10.1149/1.2221597. 

Fang L, Li J, and Peng B, “Online Estimation and Error Analysis of both SOC and SOH of Lithium-ion Battery 

based on DEKF Method,” Energy Procedia, vol. 158, pp. 3008–3013, Feb. 2019, doi: 

10.1016/j.egypro.2019.01.974. 

Hu X, Li S, and Peng H, “A comparative study of equivalent circuit models for Li-ion batteries,” Journal of Power 

Sources, vol. 198, pp. 359–367, Jan. 2012, doi: 10.1016/j.jpowsour.2011.10.013. 

Pattaraprakorn W, Pattaraprakorn, Chairakorn S, Tippayarom D, Grisdanurak N, and T. Areerob, “Li Recovery 

via Physical-Chemical Treatment from Spent EV Lithium-Ion Battery for Circular Economy,” Chemical 

Engineering Transactions, vol. 106, pp. 1183–1188, Dec. 2023, doi: 10.3303/CET23106198. 

Peng J, Takyi-Aninakwa P, Wang S, Masahudu F, Yang X, and Guerrero J. M, “A hybrid-aided approach with 

adaptive state update for estimating the state-of-charge of LiFePO4 batteries considering temperature 

uncertainties,” Journal of Energy Storage, vol. 76, p. 109758, Jan. 2024, doi: 10.1016/j.est.2023.109758. 

Zou R, Duan Y, Wang Y, Pang J, Liu F, and Sheikh S. R, “A novel convolutional informer network for 

deterministic and probabilistic state-of-charge estimation of lithium-ion batteries,” Journal of Energy Storage, 

vol. 57, p. 106298, Jan. 2023, doi: 10.1016/j.est.2022.106298. 

 

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