3.2.1. Consideration of Cell Temperature

The change in temperature of a battery cell mainly depends on the C-rate, discharge time, relaxation time, and ambient temperature [59]. Figure 6 shows the cell temperature that rises when the cell is discharged at 1C current from DoD 0% to 100% in a temperature chamber set to 25 ◦C (same experimental conditions as shown in Table 2).

**Figure 6.** The temperature change of the battery cell during discharging at 1C current.

To measure the cell impedance at each constant temperature, cell impedances are measured through an EIS instrument (IM6ex from Zahner-Elektrik GmbH & CoKG) in a temperature chamber set to 20, 25, 30, and 35 ◦C. The measurement conditions are shown in Table 3.


**Table 3.** Parameters for EIS measurement (Galvanostatic).

The measurement results are shown in Figure 7 as a Nyquist plot and a Bode plot.

**Figure 7.** Cell impedance at each frequency at 20, 25, 30, and 35 ◦C. (**a**) Nyquist plot (O: 1 Hz, X: 1 kHz); (**b**) Bode plot.

As shown in Figure 7, the higher the cell temperature, the lower the cell impedance, and the 1 Hz impedance is more affected by temperature than the 1 kHz impedance. Figure 8 shows the 1 Hz impedance measured at each temperature and a curve fitted with a quadratic equation.

**Figure 8.** 1 Hz impedance measured at 20, 25, 30, and 35 ◦C (dotted line) and fitted curve (solid line).

Equation (8) is the quadratic equation fitted in Figure 8, where R<sup>2</sup> is 1.00 and RMSE is 0.19 mΩ.

$$\mathbf{Z(T)}\_{\text{fitted}} = -0.03717 \cdot \mathbf{T}^2 - 3.217 \cdot \mathbf{T} + 133.3 \tag{8}$$

where T is the cell temperature, and Z(T)fitted is the impedance obtained through the equation at the cell temperature T.

The measured 1 Hz impedance at various cell temperatures is uniformly adjusted to the cell impedance at 25 ◦C through Equation (9).

$$Z(\mathbf{T})\_{\text{adjusted}} = Z(\mathbf{T})\_{\text{measured}} - (Z(\mathbf{T})\_{\text{fitted}} - Z(25)\_{\text{filtered}}) \tag{9}$$

where Z(T)adjusted is the adjusted impedance to 25 ◦C and Z(T)measured is the measured impedance at temperature T.

The result of Equation (9) applied to the 1 Hz impedance in Figure 3 is shown in Figure 9.

**Figure 9.** 1 Hz impedance adjusted to 25 ◦C.

In Figure 9, the 1 Hz impedance increases with increasing cell DoD. The higher the cell DoD, the higher the cell temperature (Figure 6), and as the cell temperature increases, the cell impedance decreases (Figure 8). Figure 9 shows the impedance when the impedance of Figure 3 is adjusted to a lower temperature of 25 ◦C. Therefore, the temperature-adjusted 1 Hz impedance in Figure 9 becomes higher than the 1 Hz impedance in Figure 3.

### 3.2.2. Consideration of Cell SoH

As mentioned above, cell 1 Hz impedance is affected by cell SoH as well as cell temperature. The cell SoH can be estimated via the 1 kHz impedance even if the cell SoC is unknown (Section 3.1). The cell SoH obtained in Equation (7) is used for normalizing the 1 Hz impedance. The measured cell 1 Hz impedance (Zmeasured) can be normalized to Znorm by Equation (10).

$$Z\_{\text{norm}} = (Z\_{\text{measured}} - Z\_{\text{min}}) / (Z\_{\text{max}} - Z\_{\text{min}}) \times 100 \tag{10}$$

In Equation (10), Zmax and Zmin represent the maximum and minimum values of the impedance between DoD 10% and 90% and can be obtained by Equations (11) and (12), respectively. In Equation (11), R2 is 1.00 and RMSE is 0.12 mΩ, and in Equation (12), R2 is 1.00 and RMSE is 0.26 mΩ.

$$Z\_{\text{max}} = -0.00593 \cdot \text{SoH}^2 + 0.79922 \text{ SoH} + 63.09 \tag{11}$$

$$Z\_{\rm min} = -0.2969 \cdot \text{SoH} + 103.1\tag{12}$$

The result of applying Equation (10) to the 1 Hz impedance in Figure 9 is shown in Figure 10.

**Figure 10.** Normalized 1 Hz impedance vs. DoD.

The average of the normalized impedance at each SoH is fitted with a cubic equation and plotted as a curve in Figure 10. This fitted curve is expressed by Equation (13) and has an R<sup>2</sup> of 1.00 and an RMSE of 1.51.

$$Z\_{\rm norm} = -0.00014 \cdot \text{SoC}^3 + 0.032 \cdot \text{SoC}^2 - 0.85 \cdot \text{SoC} + 7.58 \tag{13}$$

Table 4 shows each R2 and RMSE when Equation (13) is applied to cells with different SoHs.

**Table 4.** Accuracy of SoC estimation in cells with different SoHs.


Because of the cell nonlinear response mentioned above, SoC estimation using 1 Hz impedance is made here between DoD 10% and 90%.

## 3.2.3. Battery Cell SoC Estimation at Different Initial SoCs

Section 3.2.2 shows the results of SoC estimation when a fully charged cell is fully discharged. However, battery cells are not always operated in a fully charged state. Here, battery cell SoC estimation at different cell initial SoCs is emulated. A total of 10 min of cell discharge and 60 min of cell relaxation are repeated until the cell is completely discharged. The cell impedance is measured every second while the cell is discharging, and each cell state after the relaxations represents the cell state at different initial SoCs. The experimental conditions are shown in Table 5.

**Table 5.** Experimental conditions for SoC estimation of cells that start operating at different SoCs.


Figure 11a shows the cell voltage and cell temperature for the cell DoD, and Figure 11b shows the 1 Hz and 1 kHz impedances.

As shown in Figure 11a, the cell voltage drops while the cell is discharging and increases during each relaxation. At the same time, the cell temperature increases while the cell is discharged and decreases during the relaxation time. Figure 11b shows that cell 1 Hz impedance is more affected by DoD than 1 kHz impedance. The cell 1 kHz impedance is relatively constant for DoD changes and can be used for cell SoH estimation as shown in Section 3.1, while the 1 Hz impedance is highly influenced by DoD, and it is used for SoC estimation.

**Figure 11.** (**a**) Cell voltage (solid line) and cell temperature (dotted line) vs. DoD; (**b**) 1 Hz impedance (solid line) and 1 kHz impedance (dotted line) vs. DoD.

In Figure 11b, the DoD ranges where the 1 Hz impedance cannot be measured correctly are shown in gray. As explained earlier in Chapter 3, the cell 1 Hz impedance is not measured correctly at the beginning of each discharge, especially due to the nonlinearity of the cell discharge curve due to activation polarization shown in Figure 4. Each of these gray ranges in Figure 11b corresponds to 265 s. In addition, the range from DoD 95% in Figure 11b is also included in these gray ranges, as this range corresponds to the concentration polarization region in Figure 4. However, it should be noted that in most cell operations where cells are rarely fully discharged, a range that cell impedance cannot be measured correctly only appear once in the beginning of cell discharge. In Figure 11b, multiple gray ranges are shown because the impedance at the beginning of the discharge in different initial SoCs is shown in a single figure.

The 1 Hz impedance of Figure 11b adjusted to the cell temperature is shown in Figure 12a. Equation (9) is used to take into account the effects of cell temperature. The temperature-adjusted 1 Hz impedance is then normalized to the cell SoH, and the result is shown as a dotted line in Figure 12b. Equation (10) is used to normalize the temperatureadjusted impedance to the cell SoH.

**Figure 12.** (**a**) 1 Hz impedance adjusted to cell temperature vs. DoD; (**b**) temperature-adjusted 1 Hz impedance normalized to cell SoH (dotted line) and expected value (solid line) vs. DoD.

The cell SoC can be estimated by applying Equation (13) to Figure 12b, where R2 is 0.99 and RMSE is 3.07 mΩ. Cell impedance in grayed-out ranges is excluded from SoC estimation. These ranges include a range of 265 s after the start of the discharge and a range of cell DoD

exceeding 95%. The purpose of this experiment is to show that SoC estimation is possible even when discharges are initiated from different cell SoCs, so it can lead to misunderstanding that there are too many grayed-out ranges where SoC estimation is impossible. Again, in most cases when battery cells are used, this grayed-out range appears only once a battery cell begins to discharge and rarely twice when the cell is completely discharged.

#### **4. Conclusions and Discussion**

This paper introduces a SoC monitoring method of Li-ion battery cells using impedance measurement. For accurate SoC estimation, the estimated cell SoH is considered with the measured temperature. Unlike traditional EIS measurement methods, the proposed method does not require impedance at wide frequencies, saving measurement time and simplifying measurement saves hardware costs. Especially, a multi-sine signal is applied to measure the cell impedance at two frequencies simultaneously. The cell impedance at 1 kHz for SoH estimation and the cell impedance at 1 Hz for SoC estimation are used. As a result, the proposed cell SoC monitoring method enables simultaneous estimation of unknown cell SoH and SoC. This is verified through an experiment in several different initial SoCs in a cell as well.

One problem to be pointed out is that at the start of battery cell discharge (ca. 265 s at 1C) and in the high DoD range (over ca. 95%), accurate cell impedance measurement is not possible due to the large nonlinearity in the cell voltage response; hence, the SoC cannot be correctly estimated. Nonetheless, the proposed SoC estimation method using impedance can be used together with the SoC estimation method using cell voltage without any additional hardware or measurement. This is because the proposed method already measures the cell voltage response to obtain cell impedance. At the same time, this SoC estimation method using cell impedance can compensate for the weaknesses of the existing SoC estimation methods using cell voltage. The voltage of the Li-ion battery cell drops significantly at the beginning and end of the discharge. This enables SoC estimation as a simple method through cell voltage measurement. However, the decrease in voltage in the middle region of the cell discharge curve, the so-called "flat plateau", is not significantly noticeable, and this is one of the factors that makes it difficult to estimate SoC simply with the measured cell voltage. In particular, this flat plateau is notorious for estimating the SoC of lithium iron phosphate (LFP) battery cells. Contrariwise, this flat plateau makes impedance measurements more accurate. This is because accurate impedance measurement is possible when the target system is linear. That is, when the proposed method is used, SoC and SoH estimation is possible on this flat plateau and even more accurate.

In this paper, only the experimental results of lithium nickel manganese cobalt oxide (NMC) type cells are shown. The proposed method is based on EIS, which has already been validated in other types of battery cells through numerous literatures. Thus, it can be expected to be applied to other types of Li-ion cells. Nevertheless, it is worth comparing the results of applying this method to other types of cells in future studies.

This paper only deals with SoC estimation during battery cell discharge. Accurate SoC estimation during cell charging is also worth studying. However, there is the problem that this method is only possible during CC charging. This is because the offset current changes frequently during CV charging, making it impossible to measure the impedance correctly. If SoC estimation is possible only during CC charging, the range of SoC that can be estimated is limited. The authors evaluate that monitoring SoC under this limited condition does not have a definite advantage; hence, this is not covered in this paper.

Lastly, the experiments in this paper consider the estimation of the state of a cell discharged at room temperature. Therefore, the experiment is conducted between 20 and 35 ◦C. However, since the cell state estimation method in this paper has the advantage of considering the effect of cell temperature on impedance, experiments will be conducted in a wider temperature range for wider application of this method in the future.

**Author Contributions:** Conceptualization, methodology, validation, software, hardware implementation, writing—original draft preparation J.K. (Jonghyeon Kim); writing—review and editing, supervision, counseling J.K. (Julia Kowal); All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by DAAD (German Academic Exchange Service), Research Grants—Doctoral Programmes in Germany.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest. The funder had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.
