*3.1. Temperature Estimation*

Figure 7 presents more than 4000 temperature estimations by the neural network for the EIS data of eight Samsung INR18650 15L1 cells at five different SoCs, eight superimposed DC currents during the EIS measurements and different SoHCs. The SoHC of each cell is shown in Table 2. For each investigated SoHC, two cells were selected for testing. By including various DC currents, SoHCs and SoCs, the realistic performance of a cell was simulated. The overall root mean square error (RMSE) was less than 1 K. The maximum RMSE for a single temperature estimation was about 5 K.

**Figure 7.** More than 4000 temperature estimations by a neural network using EIS data as input are presented. The estimated temperature is plotted over the measured temperature. The black line is a guide to the eye for the target values. The overall RMSE for 8 Samsung INR18650-15L1 cells was less than 1 K. The legend shows the RMSE for each cell.

Only the real part of the impedance was used as an input parameter, since the influence of the temperature was much greater on the real part than on the imaginary part, as shown in Figure 1. A combination of real and imaginary parts and only using imaginary part as input were investigated. However, as expected, the temperature estimation became more inaccurate when using the imaginary part. For the presented results, the actual SoHCs, the SoCs and the applied DC current during the EIS measurement were used as input parameters. The ANN was able to estimate the cell temperature without additional input parameters. However, the time required to train the ANN increased significantly, and the overall RMSE increased to 1.5 K. During the measurements, there were no temperature sensors attached to the cells. The temperature was taken directly from the thermal chamber. The self heating effect in Samsung 18650-15L cells by applying DC current was not taken into account for the state estimation. When using temperature sensors within the cells, center measurements showed that the temperature difference between the center and the surface was less than 2 K. For our proof of concept study, this temperature discrepancy was assumed to be negligible, since every measurement was performed in the same way. The results show that in general the ANN was able to estimate the temperature from a corresponding EIS spectrum.

Figure 8 shows the evolution of the ANN during training. After 230 epochs, the RMSE reached its lowest value and was stabilized. The RMSE evolution was used to find a suitable configuration of the ANN. For the presented results, only one hidden layer consisting of 11 neurons within the hidden layer was used in a Bayesian regularization-backpropagation neural network. Using more than 11 neurons in the hidden layer had a tendency to overfit.

**Figure 8.** RMSE evolution during training epochs of the temperature estimation of the Samsung INR18650-15L.

Additional investigations were performed on prismatic Panasonic NCA 103450 high energy cells and cylindrical Sony US18650VTC6 high energy cells to show that the presented method is independent of cell geometry and cell type (high energy/high power).

Figure 9a presents the results for the Panasonic NCA 103450 cells with an overall RMSE of 0.7 K, and Figure 9b shows the results for the Sony VTC6 cells with an overall RMSE of 0.5 K. Each figure shows 360 temperature estimations at different SoCs and different applied superimposed DC currents. For both cells, a Bayesian regularizationbackpropagation neural network with one hidden layer consisting of five neurons was used. As input parameters, both the real and the imaginary parts of the impedance were fed to the ANN; the information about SoHC, applied DC current and SoC were restrained.

The SoHC of every Sony and Panasonic cell was about 100%. The temperature estimations for different SoHCs were performed only for the Samsung cells.

Nevertheless, we showed that the temperature estimation by an ANN using EIS data can be realized for different cell types. However, it is necessary to create an individual ANN for each cell type with individual hyperparameters. To finally generalize this method, more cells with different aging profiles and higher DC currents need to be investigated. Due to the self heating effect of the cells through high currents, temperature sensors should be installed within the cells to measure the exact temperature in real time.

The superimposed DC current affects the EIS spectrum, especially at low temperatures. Therefore, the presented ANN method is a powerful tool, especially at high temperatures where the influence of the DC current is reduced. This makes it perfectly suitable for real-life applications. In comparison to other sensorless temperature estimation methods, the main advantages are that there is no need for storing time series data, and that the computational effort is reduced, since there is no need to solve complicated equations, such as partial differential equations. It is suitable for different cell types, and it also takes a superimposed DC current and actual SoHC into account.

**Figure 9.** Temperature estimations for (**a**) Panasonic NCA 103450 with an RMSE of 0.7 K, and (**b**) Sony US18650VTC6 with an RMSE of 0.5 K. In both figures, 360 temperature estimations are shown. The estimated temperature is plotted over the measured temperature. The black line is a guide to the eye of the target values. The legend shows the RMSE for each cell.

The use of EIS data greatly improved the estimation accuracy. Various internal processes in the lithium ion cell show different temperature dependencies. These processes can be allocated to various frequency domains measured via EIS [24]. For some electrochemical processes in lithium ion cells, the influence of the temperature predominates over the influence of the cell to cell variance. The estimation accuracy can be further improved by adding impedance data at various frequency domains in addition to the cell resistance.

#### *3.2. State of Charge Estimation*

The state estimation methodology using EIS data and ANN was investigated for the estimation of the SoC. Figure 6 shows the results for the Panasonic NCA 103450 cells. Since the influence of the SoC on the impedance varies, it was necessary to split the SoC into area 1 from 5% to 45%, as shown in Figure 10a, for which the RMSE was 1.9%; and area 2 from 48% to 95%, as shown in Figure 10b, for which the RMSE was 2.2%. In both cases, a Bayesian regularization-backpropagation neural network with one hidden layer consisting of four neurons was used. For the lower SoCs, only the imaginary part was used as input parameter, and at higher SoCs the real and the imaginary parts were used. Using only one network for the whole SoC area increased the RMSE significantly.

The RMSE shows higher deviations in the mid SoC range (30–70%), where the EIS data mostly overlap. This makes it harder for the ANN to distinguish SoCs.

For the Sony VTC6 cells it was also necessary to split the SoC in two areas. For the lower SoCs from 5% to 35% the RMSE was 3.1%, as shown in Figure 11a. Figure 11b presents the results from 40% to 95% with an overall RMSE of 2%. In both cases a Levenberg– Marquardt backpropagation artificial neural network with one hidden layer and five neurons was used. Only the real part of the impedance was used as an input parameter.

**Figure 10.** SoC estimation for Panasonic NCA 103450 cells with an RMSE of about 2%. The SoC was split into (**a**) 5% to 45% and (**b**) 47% to 95% and estimated by individual neural networks. Overall, almost 600 SoC estimations are shown. The estimated SoC is plotted over the measured SoC. The black line is a guide to the eye of the target values. The legend shows the RMSE for each cell.

**Figure 11.** SoC estimation for Sony US18650VTC6 cells with an RMSE of about 3%. The SoC was split into (**a**) 5% to 35% and (**b**) 40% to 95% and estimated by individual neural networks. Overall, more than 300 SoC estimations are shown. The estimated SoC is plotted over the measured SoC. The black line is a guide to the eye of the target values. The legend shows the RMSE for each cell.

The RMSE was increased for the estimation of lower SoCs because the impedance varies only slightly with low SoC.

The SoC estimation for the Samsung 15L cells is not shown, since the best overall RMSE was about 8%, with a maximum estimation discrepancy up to 20%, which is not sufficient for any application. This was caused by the influence of the SoC on the impedance spectra being in the same order of magnitude as the influence of the cell to cell variance. The presented results for the SoC estimation clearly show that it is necessary to create an individual ANN for each cell type. If there is a clear dependency between the SoC and the EIS spectra and only a little cell to cell variance, the signal to noise ratio is big enough to allow the state estimation, as shown for the Panasonic NCA 103450 cells.

In general, SoC estimation is much more complex than temperature estimation. The reason is the behavior of the EIS spectra depends on the state. As shown in Figure 1, an increase in temperature causes a decrease in impedance. The SoC dependency of the EIS spectra varies among different cell types. For some cell types, an increase in SoC causes a decrease in the value of impedance at a low SoC or an increase in value at a high SoC. For such a case, several ANNs for different SoC ranges are required.

The advantages of this method compared to other methods are, again, the little computational effort, since no partial differential equations to be solved and no error-prone parameterization needs to be performed. Furthermore, there is no need for storing and handling time series data. The utilizable capacity of a cell depends strongly on the cell temperature. Therefore, the SoC varies with the temperature. The SoC was calculated by the ANN using an EIS spectrum that is characteristic of the measured state.
