*3.2. Comparison of Model against Training Data*

The measurement data are given as current versus time and voltage versus time series. The current served as the external input of the model which approximated the battery voltage. Figure 7 shows the training results in the form of voltage versus SOC, which allows a better comparison for different C-rates than a voltage versus time plot. The left panel shows the measured and the learned battery voltage as a function of SOC. The right panel shows the approximation error relative to the measured voltage. Figure 7a shows the complete SOC range while Figure 7b focuses on a medium SOC. The simulation results are in good agreement with the experiments over the complete SOC range and for all investigated C-rates. The absolute value of the deviation is smaller than 1% relative to the measured voltage for a wide range of SOC. Only for very low and very high SOC values, the absolute value of the relative approximation error reaches up to around 3%, which is still acceptable. In these ranges the OCV(SOC) curve (shown in blue in Figure 7a,b) is very steep. Therefore, higher approximation errors can be expected.

**Figure 7.** Simulation results using NODEs for grey-box modelling of a lithium-ion battery in comparison to experimental data; left: charge and discharge curves for different C-rates at *T* = 25 °C. The lower branches represent discharge (time progresses from right to left), while the upper branches represent charge (time progresses from left to right); right: relative approximation error; (**a**) the whole SOC range (**b**) focus on medium SOC.

Figure 3 compares the training results for a pulsed current charge with the measured voltage. Here, we have chosen a temporal representation. The pulses in Figure 3 are in the area of a medium SOC. The model reproduces the dynamic voltage response of the battery following a current step in a qualitatively correct way. Quantitatively, the absolute voltage drop after the pulse is underestimated by the model. The characteristics of the time behaviour are also different in the simulation compared to the experiment. While the simulation shows an exponential behaviour resulting from the first-order dynamics of the RC element (Equation (12)), the experiment shows a <sup>√</sup>*<sup>t</sup>* behaviour resulting from the solidstate diffusion inside the electrode materials, also referred to as Warburg diffusion [48]. Still, given the relative simplicity of the GB model, the comparison between model and experiment is adequate. Note that we also achieved similar results for other SOC values and for the discharge branch.

In conclusion, the training results show that the GB model can reproduce the training data very well.

#### *3.3. Comparison of Model against Test Data*

After finishing the training process we wanted to test the model against data not included in the training. The first test data set consists of consecutive half cycles. The results are shown in Figure 8. Figure 8a shows the test results for the complete time series. In this complete view, the test results are very good. In Figure 8b the focus is on the last three half cycles of the time series. One can see that the dynamics of the battery voltage are modelled well on this scale, although there are deviations between simulation and experiment particularly at the beginning of each half cycle.

**Figure 8.** Test results in comparison to experimental data at *T* = 25 °C for half cycles; (**a**) the complete time series; (**b**) focus on the last three half cycles.

We tested the model against a second test data set, a synthetic load profile of a homestorage battery. The results are shown in Figure 9. Figure 9a covers the complete time series, whereas Figure 9b focuses on the segment in the middle covering faster dynamics. The simulations show good agreement with experimental data for the complete load profile. The highest relative approximation errors occur in the area of high SOC values. This was expected because the training error is high at high values of SOC. It is worth mentioning that this synthetic load profile covers the longest measuring time with *t* = 190,231 s. The longest training time series spanned only *t* = 41,846 s. Nevertheless, the test results are good for the complete time series.

**Figure 9.** Test results in comparison to experimental data at *T* = 25 °C for a synthetic load profile; (**a**) the complete time series (**b**) focus on the segment in the middle.
