Figure 1.
Fractional-order equivalent circuit model of LIBs in various forms, (a) n- Model, (b) fractional-order PNGV model, (c) fractional-order Randles model, and (d) three-orders fractional-order model ().
Figure 1.
Fractional-order equivalent circuit model of LIBs in various forms, (a) n- Model, (b) fractional-order PNGV model, (c) fractional-order Randles model, and (d) three-orders fractional-order model ().
Figure 2.
Fractional-order recurrent neural network encoding with physics-informed battery knowledge.
Figure 2.
Fractional-order recurrent neural network encoding with physics-informed battery knowledge.
Figure 3.
Fractional-order state feedback in the forward propagation of recurrent neural network, (a) integer-order state feedback, and (b) fractional-order state feedback.
Figure 3.
Fractional-order state feedback in the forward propagation of recurrent neural network, (a) integer-order state feedback, and (b) fractional-order state feedback.
Figure 4.
Fractional-order partial differential equation constraint.
Figure 4.
Fractional-order partial differential equation constraint.
Figure 5.
Simplifie FOM of LIBs as fractional-order PDE constraint.
Figure 5.
Simplifie FOM of LIBs as fractional-order PDE constraint.
Figure 6.
Fractional-order gradient descent methods, (a) integer-order GD method, (b) fractional-order GD method (FOGD), (c) integer-order GD method with momentum (GDm), and (d) fractional-order GD method with momentum (FOGDm).
Figure 6.
Fractional-order gradient descent methods, (a) integer-order GD method, (b) fractional-order GD method (FOGD), (c) integer-order GD method with momentum (GDm), and (d) fractional-order GD method with momentum (FOGDm).
Figure 7.
Collected dataset, (a) current in FUDS operation condition; (b) voltage in FUDS operation condition.
Figure 7.
Collected dataset, (a) current in FUDS operation condition; (b) voltage in FUDS operation condition.
Figure 8.
Sensitivity of the fractional order in FOGD and FOGDm method, = 0.1:0.1:1, (a) testing loss, (b) outputs of all dataset including training, validation, and testing data, (c) outputs of testing dataset, (d) errors of all dataset, (e) errors of testing dataset.
Figure 8.
Sensitivity of the fractional order in FOGD and FOGDm method, = 0.1:0.1:1, (a) testing loss, (b) outputs of all dataset including training, validation, and testing data, (c) outputs of testing dataset, (d) errors of all dataset, (e) errors of testing dataset.
Figure 9.
Sensitivity of the momentum of fractional-order gradient, = 0.1:0.1:1, (a) testing loss, (b) testing outputs, and (c) testing errors.
Figure 9.
Sensitivity of the momentum of fractional-order gradient, = 0.1:0.1:1, (a) testing loss, (b) testing outputs, and (c) testing errors.
Figure 10.
Sensitivity of the learning rate, = 0.08:0.02:0.26, (a) testing loss, (b) testing outputs, and (c) testing errors.
Figure 10.
Sensitivity of the learning rate, = 0.08:0.02:0.26, (a) testing loss, (b) testing outputs, and (c) testing errors.
Figure 11.
Sensitivity of the fractional order in PDE constraint encoded into loss function, = 0.1:0.1:1, (a) testing loss, (b) testing outputs, and (c) testing errors.
Figure 11.
Sensitivity of the fractional order in PDE constraint encoded into loss function, = 0.1:0.1:1, (a) testing loss, (b) testing outputs, and (c) testing errors.
Figure 12.
Sensitivity of the ratio between OCV and SOC in battery FOM for fractional-order PDE constraint, = 5:5:50, (a) testing loss, (b) testing outputs, and (c) testing errors.
Figure 12.
Sensitivity of the ratio between OCV and SOC in battery FOM for fractional-order PDE constraint, = 5:5:50, (a) testing loss, (b) testing outputs, and (c) testing errors.
Figure 13.
Sensitivity of the capacitance in battery FOM for fractional-order PDE constraint, = 2.5:2.5:25, (a) testing loss, (b) testing outputs, and (c) testing errors.
Figure 13.
Sensitivity of the capacitance in battery FOM for fractional-order PDE constraint, = 2.5:2.5:25, (a) testing loss, (b) testing outputs, and (c) testing errors.
Figure 14.
Sensitivity of the ohm resistance in battery FOM for fractional-order PDE constraint, = 5.1 :1 :6 , (a) testing loss, (b) testing outputs, and (c) testing errors.
Figure 14.
Sensitivity of the ohm resistance in battery FOM for fractional-order PDE constraint, = 5.1 :1 :6 , (a) testing loss, (b) testing outputs, and (c) testing errors.
Figure 15.
Sensitivity of the loss weights and in loss function calculation, = 0.1:0.1:1, (a) testing loss, (b) testing outputs, and (c) testing errors.
Figure 15.
Sensitivity of the loss weights and in loss function calculation, = 0.1:0.1:1, (a) testing loss, (b) testing outputs, and (c) testing errors.
Table 1.
LIB 18650 cell parameters.
Table 1.
LIB 18650 cell parameters.
Parameters | Values | Parameters | Values |
---|
Rated capacity (0.5A) | 2000 mAh | Rated voltage | 3.7 V |
Max charge voltage | 4.2 V | Discharge cut-off voltage | 2.75 V |
Working temperature (charge) | 0 –45 | Working temperature (discharge) | −20 –60 |
Table 2.
Unchanged parameters of the proposed FORNN with PIBatKnow.
Table 2.
Unchanged parameters of the proposed FORNN with PIBatKnow.
Name | Value/Range | Name | Value/Range |
---|
Hidden layers | 1 | Hidden neurons | 12 |
max epoch | 300 | Performance function | MSE |
train:valid:test | 0.75:0.048:0.202 | Training goal | 1.6 |
Table 3.
Sensitivity categories of the fractional-order constraint and the fractional-order gradient in the proposed algorithm.
Table 3.
Sensitivity categories of the fractional-order constraint and the fractional-order gradient in the proposed algorithm.
Type | Name | Value/Range | Attribution |
---|
Fractional-order Gradient sensitivity | Fractional order | | FOGDm in (41) |
Momentum weight | |
Learning rate | |
Impedance sensitivity | Fractional order | | FO PDE in (32) |
Ratio of OCV-SOC | |
Capacitance (unit: C) | |
ohm resistance (unit: ) | [5 , 6 ] |
Loss weight sensitivity | Loss weight | | final loss in (20) |
Loss weight | |
Table 4.
Default values of the nine main fractional-order parameters.
Table 4.
Default values of the nine main fractional-order parameters.
| | | | | | | | |
---|
0.9 | 0.75 | 0.18 | 0.9 | 40 | 20 | 0.005 | 0.8 | 0.2 |
Table 5.
Ten groups of radom values of the nine parameters in the three categories for correlation analysis.
Table 5.
Ten groups of radom values of the nine parameters in the three categories for correlation analysis.
No. | | | | | | | | | | | | | |
---|
1 | 1 | 0.72 | 0.125 | 0.5 | 11.75 | 22.75 | 5.24 | 0.89 | 0.11 | 0.001349 | 0.001158 | 0.002434 | 0.001799 |
2 | 0.47 | 0.6 | 0.251 | 0.23 | 39.2 | 18.025 | 5.34 | 0.66 | 0.34 | 0.001327 | 0.001177 | 0.001309 | 0.001889 |
3 | 0.54 | 0.98 | 0.152 | 0.16 | 23.45 | 24.325 | 5.45 | 0.38 | 0.62 | 0.218343 | 0.232547 | 0.005316 | 0.216220 |
4 | 0.08 | 0.66 | 0.1484 | 0.32 | 33.35 | 11.725 | 5.12 | 0.3 | 0.7 | 0.002358 | 0.002098 | 0.003900 | 0.002954 |
5 | 0.97 | 0.15 | 0.2186 | 0.6 | 8.15 | 23.425 | 5.52 | 0.29 | 0.71 | 0.029044 | 0.032181 | 0.003943 | 0.023360 |
6 | 0.62 | 0.13 | 0.1088 | 0.47 | 36.95 | 9.925 | 5.85 | 0.89 | 0.11 | 0.010673 | 0.009013 | 0.011097 | 0.016737 |
7 | 0.45 | 0.91 | 0.233 | 0.78 | 45.5 | 3.625 | 5.67 | 0.15 | 0.85 | 0.004387 | 0.004279 | 0.000754 | 0.005652 |
8 | 0.91 | 0.92 | 0.1394 | 0.82 | 16.25 | 16.675 | 5.93 | 0.91 | 0.09 | 0.000861 | 0.000894 | 0.000692 | 0.000782 |
9 | 0.81 | 0.14 | 0.1934 | 0.28 | 45.5 | 6.775 | 5.45 | 0.52 | 0.48 | 0.024046 | 0.025149 | 0.003720 | 0.024783 |
10 | 0.12 | 0.04 | 0.0854 | 0.66 | 12.2 | 2.95 | 5.19 | 0.34 | 0.66 | 1.044450 | 1.235187 | 0.046351 | 0.573336 |
Table 6.
Correlation coefficients of performance with nine main fractional-order parameters.
Table 6.
Correlation coefficients of performance with nine main fractional-order parameters.
MSE | | | | | | | | | |
---|
| 0.5103 | 0.3907 | 0.5173 | 0.1686 | 0.3843 | 0.3870 | 0.3737 | 0.2855 | 0.2855 |
| 0.5104 | 0.3977 | 0.5161 | 0.1794 | 0.3839 | 0.3957 | 0.3744 | 0.2821 | 0.2821 |
| 0.5108 | 0.5532 | 0.5948 | 0.1963 | 0.3357 | 0.4748 | 0.3196 | 0.1887 | 0.1887 |
| 0.5019 | 0.3272 | 0.5174 | 0.0812 | 0.3818 | 0.3115 | 0.3632 | 0.3091 | 0.3091 |
Table 7.
Positive and negative correlation of the nine main fractional-order parameters to the algorithm performance.
Table 7.
Positive and negative correlation of the nine main fractional-order parameters to the algorithm performance.
index | | | | | | | | | |
---|
speed 1 | ↗ | ↘ | ↘ | ↘ | ↗ | middle 4 | ↘ | ↗ | ↘ |
loss 2 | ↗ | ↗ | - | ↗ | - | - | ↘ | - | - |
accuracy 3 | ↗ | ↗ | - | ↗ | - | ↗ | ↗ | - | - |
stability | ↘ | ↘ | ↘ | ↗ | ↘ | ↗ | ↘ | ↗ | ↘ |