## 5 Validation Using Test Bench Measurements  
A comparison between the model prediction and test bench measurements was carried out, in order to validate the proposed model. The error was quantified by calculating the MAE and MSE, as described in Equations (15) and (16), respectively. The error at each step, \(err(t)\), was calculated using the ground truth \(gt(t)\) and the predicted values from the model \(pred_{model}(t)\). \(n\) describes the number of steps. \[err(t) =gt(t)-pred_{model}(t) \tag{14}\] \[MAE =\frac{\sum_{t=1}^{n}|err(i)|}{n},\] (15) \[MSE =\frac{\sum_{t=1}^{n}|err(i)|^{2}}{n}. \tag{16}\]  
Unlike the training and validation datasets, the test set was measured on a test bench, where the temperature and current were adjustable. This ensured that a wide range of power, SOC and temperature could be tested. To achieve this, a current profile was recorded during a vehicle test drive and adapted to the adjusted temperature and SOC to obtain the voltage boundaries of the battery. Batteries display non-linear behavior in the SOC and voltage limit values, due to the steepening OCV and rising internal resistances. Adapting the current profiles allows access to these hard-to-predict areas, thus ensuring more exact validation. The test bench offers a measurement setup with a climate chamber to condition the battery temperature, a power supply and electric load to apply the current profiles and a computer to control the BMS and plot the data. Before starting measurement, the batteries were first conditioned electrically to the requested state of charge by completely discharging a fully charged battery. Thermal conditioning was then carried out for at least 12 h to ensure a fully tempered battery. Concatenated current profiles for the four temperature regions are shown in Figure 6. The current is selected to meet the requirements of the test-bench and voltage limits in consideration of temperature dependent overvoltages. Figure 5: Training progress indicated by the MSE (**a**) and Mean Max Error (**b**) of GRU, LSTM and RNN architectures, based on validation of the test set. Figure 7 presents a validation of the current profile applied at an average temperature of \(-23\)\({}^{\circ}\)C in low SOC regions. The validation had difficulties predicting these operation points. As Figure 7a shows, the model was nevertheless capable of predicting the qualitative progression of the voltage with an MSE of 1.18 V\({}^{2}\). The prediction differed from the ground truth at high rates due to the gradient of inner resistances in the battery cell increasing at lower temperatures. Figure 7b shows the maximum error (of 3.9 V) occurring during a high current peak. The better voltage prediction closer to room temperature is shown in Figure 8a. The maximum error was less than 1.1 V with an MSE of 0.19 V\({}^{2}\) within this profile, as shown in Figure 8b. Better predictions resulted from a less volatile voltage course and more resilient data availability. This temperature-dependent error is similar to that occurring in common ECMs. Three different current profiles were performed at different charge states for every temperature region. The maximum and minimum current were restricted at \(-25\)\({}^{\circ}\)C, such that the lower and upper voltage barriers of 38 and 53 V were not exceeded. There was no current restriction at 25 \({}^{\circ}\)C, on account of the lower inner resistance. The measured profiles were used as the input for the prediction, in order  
Figure 6: Currents in test set at a temperatures of: \(-25\)\({}^{\circ}\)C (**a**); \(0\)\({}^{\circ}\)C (**b**); \(25\)\({}^{\circ}\)C (**c**); and \(50\)\({}^{\circ}\)C (**d**). Figure 7: Voltage (**a**) and error (**b**) validation result for a current profile at \(-23\)\({}^{\circ}\)C and a SOC range from 30% to 47%. to obtain an equivalent input profile for validation. The model predicted current and temperature, where the first 128 voltage values were used to calculate the initial \(U_{trend}\).