# Prediction of the Heat Generation Rate of Lithium-Ion Batteries Based on Three Machine Learning Algorithms

^{*}

## Abstract

**:**

^{2}values of interpolation were greater than 0.96; (2) after the discharge voltage was added as an input parameter, the prediction of the ANN was barely affected, whereas the performance of the SVM and GPR were improved; and (3) the ANN exhibited the best performance among the three algorithms. Accurate results can be obtained by using a single hidden layer and no more than 15 neurons without the additional input, where the R

^{2}values were in the range of 0.89–1.00. Therefore, the ANN is preferable for predicting the HGR of lithium-ion batteries.

## 1. Introduction

_{4}pouch cells covered the operating range of the discharge rate and ambient temperature specified by the cell manufacturer. The data of the 15 Ah battery cell were used for the independent assessment of the trained model. The results concurred with the measured data, presenting a correlation coefficient of 0.98627. However, some details of the HGR curve have not been predicted accurately, and only one ANN model with a specific architecture was used in this study.

## 2. Data and Methods

#### 2.1. Data Collection and Preprocessing

_{d}is the discharged capacity by a certain moment (Ah), and C

_{rated}is the rated capacity of the battery (Ah).

#### 2.2. Machine Learning Modeling

_{a}were the inputs for the HGR prediction of different ambient temperatures. Additionally, the discharge voltage U was added to the inputs and the results were compared with those obtained without considering discharge voltage as an input. Therefore, four types of modeling were conducted, as expressed by Equations (3)–(6). Table 1 lists the operating conditions to be predicted and the data used; 12 cases were predicted. The cases equivalent to interpolation and extrapolation were analyzed for different discharge currents and different ambient temperatures, respectively. The interpolation in Table 1 indicates that the operating conditions to be predicted were within the range of those of the training data, whereas the opposite was true for extrapolation.

#### 2.3. Evaluation of the Methods

^{2}), and relative error of the average HGR, δ, were used to evaluate the prediction performance of these models. The RMSE is generally used to measure the deviation between the estimated and actual values, as defined in Equation (7). In this study, the actual value was the experimentally measured value. The closer the value of RMSE is to 0, the higher the prediction accuracy of the model.

_{est}and q

_{act}are the estimated and actual values of the battery HGR for sample i, respectively, and n is the number of samples.

^{2}, or the coefficient of determination, was used to evaluate the goodness of fit of a regression model, which is defined in Equation (8). The closer the value is to 1, the better the regression performance.

## 3. Results

#### 3.1. HGR Prediction at Different Discharge Currents

^{2}values of the regressions. Figure 3 presents the training and testing RMSE values and the relative error of the average HGR, δ.

#### 3.2. HGR Prediction at Different Ambient Temperatures

^{2}values of the regressions. Figure 5 depicts the training and testing RMSE values and the relative error of the average HGR, δ.

## 4. Discussion

^{2}values in Table 2 demonstrate that all three methods achieved good predictions for 1 C discharge (cases No. three and four) that are equivalent to the interpolation, where R

^{2}is greater than 0.96, regardless of whether the inputs contained the discharge voltage. However, the predictions of the extrapolated conditions of 0.5 C (cases No. one and two) and 1.5 C (cases No. five and six) are not as good as those of 1 C. The R

^{2}of the GPR on the 1.5 C discharge prediction was only 0.82 after the discharge voltage was added to the input parameters (Table 2). Figure 3 also demonstrates that the predicted RMSE values of the 1 C discharge were typically the lowest, and the δ values were also low, especially when the discharge voltage was selected as one of the input parameters. Similar results were observed in the predictions at different ambient temperatures. Figure 4 and Figure 5, and the R

^{2}values in Table 3 demonstrate that the three methods achieved good results in the 30 °C predictions (cases No. 9 and 10) that are equivalent to the interpolation, where the R

^{2}values were greater than 0.98. The prediction performances of the extrapolated conditions at 20 °C (cases No. seven and eight) and 45 °C (cases No. 11 and 12) were good, but not as good as those at 30 °C. The three algorithms performed well in the interpolation cases of the HGR prediction, whereas extrapolation may require more input parameters and may not achieve ideal results. Therefore, the boundary of the test conditions must be broadened as much as possible and extrapolation should be avoided in the regression.

^{2}increased (Table 2) after adding the discharge voltage to the input parameters, indicating that the prediction performance improved, whereas the performance of the ANN for 0.5 C and 1 C discharge decreased slightly. The performance of the SVM in predicting the discharge at 0.5 C and 1 C was improved after adding the discharge voltage to the input parameters (Figure 3a,c), wherein the minimum R

^{2}values increased from 0.53 to 0.82 (Table 2), and the maximum value reached 0.98 (Table 2). The RMSE of the SVM for 1.5 C discharge prediction slightly increased (Figure 3e) and R

^{2}slightly decreased (Table 2). However, the comparison between Figure 2e,f demonstrated that the accuracy of the prediction in 0.2–0.8 DOD increased after adding the input. The GPR performance significantly improved for all cases of discharge currents (Figure 3), wherein the minimum R

^{2}increased from 0.67 to 0.82 (Table 2) and the maximum value reached 0.98 (Table 2). In the prediction of the cases of different ambient temperatures, the RMSE of the ANN increased slightly after adding the discharge voltage to the inputs (Figure 5), and R

^{2}changed slightly (Table 3), demonstrating slight decreases in the prediction performance. The performance of the SVM and GPR both improved (Figure 5a), wherein the minimum R

^{2}values increased from 0.90 to 0.96 (Table 3) and the maximum value reached 0.99 (Table 3). In summary, the accuracy of the ANN was less affected by the added input parameters, and the number of neurons used exhibited no evident change pattern (Table 2 and Table 3). The SVM and GPR have a high probability of obtaining better predictions with more input parameters. Adding the discharge voltage to the inputs can slightly increase the accuracy of the prediction. However, additional tests must be conducted to obtain the discharge voltage data of the predicted conditions, which increases the test time. Therefore, the prediction accuracy and time consumption must be weighed when selecting the input parameters.

^{2}values (Table 2) all demonstrate that the performance of the GPR for the 1.5 C discharge was improved after adding the discharge voltage as an input, whereas the absolute value of δ in Figure 3f did not decrease. However, δ can still supplement the RMSE, reflecting the relative error magnitude of the prediction. For the predictions of different discharge rates, the maximum absolute value of δ was 7.17% (Figure 3b), except for the prediction of the 0.5 C discharge by the SVM and GPR without discharge voltage being used as an input (Figure 3b). The maximum absolute value of δ was 3.19% (Figure 5f) for the predictions of different ambient temperatures. These results demonstrate that the three algorithms can be effectively applied to predict the battery HGR.

^{2}was 0.89 and the maximum was 1.00 (Table 2 and Table 3). Additionally, in the predictions for different discharge rates by the ANN, the peaks of the HGR at the initial stages of 1 C and 1.5 C discharge were simulated after adding the discharge voltage as an input, as shown in Figure 2d,f. This detail was also effectively simulated in the prediction for different ambient temperatures by the ANN (Figure 4). Conversely, the SVM and GPR could effectively predict the HGR of 0.5 C and 1.5 C discharge only after the discharge voltage was added to the inputs (Table 2). The performances of the SVM and GPR in the predictions for different ambient temperatures were also inferior to those of the ANN (Table 3). The GPR performed better than the SVM in the predictions except for the 1.5 C discharge (Table 2). Moreover, it better simulated the peak at the initial stage of discharge in the predictions for different ambient temperatures. The SVM exhibited the worst detail-simulation performance among the three algorithms. Figure 3 and Figure 5 also demonstrate that the training RMSE values of the SVM were typically higher than those of the ANN and GPR, and both the training and testing performance were not ideal, which indicates that the SVM models may be underfitting. Conversely, the training RMSE values of the GPR were far lower than those of the ANN and SVM, and also far lower than their testing RMSE (Figure 3 and Figure 5), indicating that the generalization ability of the GPR models was poor and that there was overfitting. The training and testing RMSE values of the ANN were relatively close and small (Figure 3 and Figure 5), indicating that both the learning and generalization abilities of the ANN models were satisfactory. The total training time cannot be directly compared since the search for the optimal architecture or kernel function and optimization of the hyperparameters were all performed when applying the three algorithms; however, a single training time can be considered as a reference. In this study, the single training times of the ANN, SVM, and GPR were within the ranges of 8–25 s, 1–13 s, and 69–383 s, respectively. Essentially, among the three algorithms, the computation cost of the GPR was the highest, whereas that of the SVM was the lowest, and that of the ANN was relatively low.

## 5. Conclusions

- The prediction performances of the three algorithms for the extrapolation cases were not as good as those for the interpolation cases. Particularly, ideal results may not be obtained for the predictions of the 0.5 C and 1.5 C discharge even after the discharge voltage was added to the inputs. For example, the R
^{2}values of the interpolation cases were greater than 0.96, whereas that of the GPR for the 1.5 C discharge after adding the discharge voltage as an input was only 0.82 (Table 2). Therefore, in practical applications, the boundary of the test conditions must be broadened and extrapolation regression must be avoided as much as possible. - The prediction accuracy of the SVM and GPR can be improved by adding the discharge voltage to the input parameters of the DOD and discharge current/ambient temperature. For example, in the prediction of different discharge currents, the minimum R
^{2}value increased from 0.53 to 0.82, and the maximum reached 0.98 (Table 2). The effect of adding the input parameter on the accuracy of the ANN was minimal. However, more tests are required to obtain the discharge voltage data under the conditions to be predicted when the input is added, which increases the time consumption. - The absolute values of the relative error of the average HGRs predicted by the three algorithms were mostly within 5%, indicating that all three algorithms can be applied to predict the battery HGR. The ANN exhibited the best performance among the three algorithms and accurately predicted the interpolation and extrapolation cases without additional input parameters. The R
^{2}values were within the range of 0.89–1.00 (Table 2 and Table 3), the architectures used were simple, and the computation cost was relatively small. Therefore, the ANN is the most preferred among the three machine learning algorithms for similar battery HGR prediction problems.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

ANN | Artificial neural network; |

DOD | Depth of discharge; |

GPR | Gaussian process regression; |

HGR | Heat generation rate; |

LSTM | Long short-term memory; |

NARX | Non-linear autoregressive exogenous; |

NN | Neural network; |

R^{2} | R-squared, or the coefficient of determination; |

RBF | Radial basis function; |

RMSE | Root mean square error; |

RUL | Remaining useful life; |

SOC | State of charge; |

SOH | State of health; |

SVM | Support vector machine; |

TMS | Thermal management system. |

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**Figure 2.**Comparison of the estimated and actual HGRs: (

**a**) 0.5 C discharge without discharge voltage as an input; (

**b**) 0.5 C discharge with discharge voltage as an input; (

**c**) 1 C discharge without discharge voltage as an input; (

**d**) 1 C discharge with discharge voltage as an input; (

**e**) 1.5 C discharge without discharge voltage as an input; (

**f**) 1.5 C discharge with discharge voltage as an input.

**Figure 3.**RMSE of training and testing, and the relative error of the average HGR, δ: (

**a**) RMSE values of 0.5 C discharge; (

**b**) δ of 0.5 C discharge; (

**c**) RMSE values of 1 C discharge; (

**d**) δ of 1 C discharge; (

**e**) RMSE values of 1.5 C discharge; (

**f**) δ of 1.5 C discharge.

**Figure 4.**Comparison of the estimated and actual HGRs during discharge: (

**a**) at 20 °C without discharge voltage as an input; (

**b**) at 20 °C with discharge voltage as an input; (

**c**) at 30 °C without discharge voltage as an input; (

**d**) at 30 °C with discharge voltage as an input; (

**e**) at 45 °C without discharge voltage as an input; (

**f**) at 45 °C with discharge voltage as an input.

**Figure 5.**RMSE of training and testing, and the relative error of the average HGR, δ: (

**a**) RMSE values of discharge at 20 °C; (

**b**) δ of discharge at 20 °C; (

**c**) RMSE values of discharge at 30 °C; (

**d**) δ of discharge at 30 °C; (

**e**) RMSE values of discharge at 45 °C; (

**f**) δ of discharge at 45 °C.

No. | Operation Conditions | Do the Inputs Contain Discharge Voltage? | Training Data | Number of Training Samples | Number of Testing Samples | Interpolation/Extrapolation |
---|---|---|---|---|---|---|

1 | 0.5 C | No | 0.75 C, 1 C, 1.25 C, and 1.5 C | 3820 | 978 | extrapolation |

2 | Yes | |||||

3 | 1 C | No | 0.5 C, 0.75 C, 1.25 C, and 1.5 C | 3836 | 962 | interpolation |

4 | Yes | |||||

5 | 1.5 C | No | 0.5 C, 0.75 C, 1 C, and 1.25 C | 3856 | 942 | extrapolation |

6 | Yes | |||||

7 | 20 °C | No | 25, 30, 35, 40, and 45 °C | 4930 | 933 | extrapolation |

8 | Yes | |||||

9 | 30 °C | No | 20, 25, 35, 40, and 45 °C | 4887 | 976 | interpolation |

10 | Yes | |||||

11 | 40 °C | No | 20, 25, 30, 35, and 40 °C | 4856 | 1007 | extrapolation |

12 | Yes |

**Table 2.**ANN architectures and covariance functions of GPR that were used, and R

^{2}of the regressions.

No. | Operation Conditions | Do the Inputs Contain Discharge Voltage? | ANN Architecture | Covariance Function of GPR | R^{2} | ||
---|---|---|---|---|---|---|---|

ANN | SVM | GPR | |||||

1 | 0.5 C | No | 1 hidden layer–5 neurons | Matern 3/2 | 0.95 | 0.53 | 0.67 |

2 | Yes | 1 hidden layer–8 neurons | Matern 3/2 | 0.95 | 0.82 | 0.88 | |

3 | 1 C | No | 1 hidden layer–3 neurons | Matern 5/2 | 0.99 | 0.96 | 0.97 |

4 | Yes | 1 hidden layer–5 neurons | Exponential | 0.98 | 0.98 | 0.98 | |

5 | 1.5 C | No | 1 hidden layer–10 neurons | Matern 3/2 | 0.89 | 0.94 | 0.72 |

6 | Yes | 1 hidden layer–4 neurons | Matern 3/2 | 0.94 | 0.93 | 0.82 |

**Table 3.**ANN architectures and covariance functions of GPR that were used, and R

^{2}of the regressions.

No. | Operation Conditions | Do the Inputs Contain Discharge Voltage? | ANN Architecture | Covariance Function of GPR | R^{2} | ||
---|---|---|---|---|---|---|---|

ANN | SVM | GPR | |||||

7 | 20 °C | No | 1 hidden layer–15 neurons | Matern 3/2 | 0.99 | 0.90 | 0.98 |

8 | Yes | 1 hidden layer–5 neurons | Matern 3/2 | 0.99 | 0.96 | 0.97 | |

9 | 30 °C | No | 1 hidden layer–9 neurons | Matern 3/2 | 1.00 | 0.98 | 1.00 |

10 | Yes | 1 hidden layer–5 neurons | Rational quadratic | 1.00 | 0.99 | 1.00 | |

11 | 45 °C | No | 1 hidden layer–6 neurons | Matern 3/2 | 0.99 | 0.92 | 0.96 |

12 | Yes | 1 hidden layer–7 neurons | Matern 3/2 | 0.99 | 0.98 | 0.98 |

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## Share and Cite

**MDPI and ACS Style**

Cao, R.; Zhang, X.; Yang, H.
Prediction of the Heat Generation Rate of Lithium-Ion Batteries Based on Three Machine Learning Algorithms. *Batteries* **2023**, *9*, 165.
https://doi.org/10.3390/batteries9030165

**AMA Style**

Cao R, Zhang X, Yang H.
Prediction of the Heat Generation Rate of Lithium-Ion Batteries Based on Three Machine Learning Algorithms. *Batteries*. 2023; 9(3):165.
https://doi.org/10.3390/batteries9030165

**Chicago/Turabian Style**

Cao, Renfeng, Xingjuan Zhang, and Han Yang.
2023. "Prediction of the Heat Generation Rate of Lithium-Ion Batteries Based on Three Machine Learning Algorithms" *Batteries* 9, no. 3: 165.
https://doi.org/10.3390/batteries9030165