*2.1. Electrochemical Impedance Spectroscopy*

EIS involves a non-destructive technique for characterizing electrochemical systems by applying a sinusoidal excitation and measuring the corresponding response, as shown in Figure 3.

**Figure 3.** Schematic figure of a sinusoidal current excitation and the voltage response of the system with the phase shift.

The impedance was calculated using the complex voltage, complex current and the phase shift between those values:

$$\underline{Z} = \frac{\underline{W}}{\underline{i}} = R + \mathbf{j}X \; ; \tag{1}$$

and from the complex impedance equation, the real part *R* and the imaginary part j*X* were calculated. Applying AC currents at different frequencies (usually between 10 kHz and 10 mHz) creates an electrochemical impedance spectrum. The results of an EIS measurement are usually plotted in a Nyquist plot, as shown in Figure 4.

**Figure 4.** Nyquist plot of a single impedance spectrum for a lithium ion cell.

In order to test the performance in different loading conditions and guarantee charge conservation, the galvanostatic mode was chosen. All EIS measurements were performed using a Gamry reference 3000 AE potentiostat multiplexed on a Basytec CTS. Each internal process can be allocated a characteristic time constant. Therefore, the operating frequency range was varied between 1 Hz and 10 kHz with 15 frequencies per decade (61 frequencies in total). Moreover, the frequency band is limited to the change of charge while running a possibly superimposed current. As a compromise, the AC current was set to a C-rate of 1/10 C for all measurements, in order to guarantee linearity while achieving a good noise-to-signal ratio. The temperature was controlled by a Memmert ICP 110 thermal chamber (Δ*T* = ±0.1 K). The SoC adjustment was performed by charging/discharging with constant current/constant voltage (CC/CV) (current limit: C/30). The selected voltage corresponds to the open circuit potential. For measurements with superimposed DC current during the EIS measurements, the SoC was set to be 5% higher than the SoC of interest by CC/CV. From there, the cell was discharged by CC with the same value of current, which was used for the superimposed EIS measurement. After reaching the target SoC, the EIS measurement was performed without relaxation, to simulate dynamic working conditions.

To make sure that the presented results can be generalized, cells of different types, such as cylindrical high power and prismatic high energy cells, were investigated. To evaluate the impact of the cell to cell variance, at least nine cells per cell type were measured using the same load. The investigated cell types are shown in Table 1.

**Table 1.** Cells investigated for state estimation.


#### 2.1.1. Temperature Estimation

Investigations were performed on the Samsung INR18650-15L1 1500 mAh lithium ion cells. Therefore, 36 cells at different SoHCs (SoHC = state of health regarding actual capacity to nominal capacity *C*/*C*N) were used. The cells were aged by cycling (at *T* = 40◦C, constant current charge at 2C, discharge at 3C) using a PEC ACT0550. More detailed information about the SoHC of all 36 cells is shown in Table 2.

**Table 2.** State of health with respect to nominal capacity (*C*/*C*N). Thirty-six cells were investigated. For each SoHC range, the data of 2 cells were defined as test data; the data of the other cells were used as the training dataset for the neural network (ANN). This guaranteed that each SoHC range would be taken into account. The labels Hx refer to the investigated cells.


Figure 5 shows the measurement procedure. EIS measurements were performed at 14 temperatures (*T* in ◦C: 10, 15, 20, 25, 30, 35, 40, 45, 50, 52, 54, 56, 58, 60), with 5 different SoCs for each temperature (SoC: 90%, 70%, 50%, 30%, 10%).

**Figure 5.** Procedure of data acquisition.

For each temperature and SoC state, EIS measurements were performed with 8 different superimposed DC-currents (C-rate: 0C, −1/4C, −1/2C, −3/4C, −1C, −3/2C, −7/8C, +1C). For the training and testing process, a multi-dimensional dataset was created by performing more than 20,000 EIS-measurements at different states. A relaxation time of 1 hour was used after changing the temperature to ensure that the cells were at the same temperature as the thermal chamber. To simulate a real application, there was no relaxation time between charging/discharging and EIS measurements.

A shorter series of measurements for the temperature estimation were created on the Panasonic NCA 103450 (9 cells SoHC = 100%, *C*<sup>N</sup> = 2350 mAh, prismatic high energy) and on the Sony US18650VTC6 (9 cells SoHC = 100%, *C*<sup>N</sup> = 3000 mAh, cylindrical high energy). The temperature was varied from 10 to 50 ◦C in 5 K steps. For each temperature, the SoC was varied from 10% to 90% in steps of 20%. For each temperature setting and SoC state, EIS measurements were performed with 4 different superimposed DC-currents (C-rate: 0C, −1/4C, −1/2C, −1C).

#### 2.1.2. State of Charge Estimation

The SoC estimation was performed for all cell types shown in Table 1. The SoHC of every cell was nearly 100%. EIS measurements were performed at 4 different temperatures (20, 25, 30, and 35 ◦C). To take the charging/discharging history (hysteresis) of the cells into account, the cells were discharged in steps from 95% to 5% (in total 36 SoCs: 95%, 92%, 90%, 87%, 85%, 82%, . . . , 5%) and then went through the same SoCs for charging (again 36 SoCs). During the EIS measurement, there was no DC applied, but only an AC of C/10.

## 2.1.3. State of Health Estimation

The SoHC estimation with respect to total capacity (SoHC = *C*/*C*N) was performed only for the Samsung INR18650-15L1 cells. To generate the training data, three lithium ion cells were aged by cycling (at *T* = 40 ◦C, constant current: charge at 2C, discharge at 3C, 1400 cycles) using a Basytec CTS. After every hundredth cycle, the capacity was determined (charge constant current C/10 constant voltage C/20, discharge constant current C/10) and EIS measurement was performed (at 25 ◦C, at 70% SoC, 0C DC-current). The data of the investigated cells are described in Section 2.1.1 (at 25 ◦C, at 70% SoC, 0C DC-current).
