**3. Experiment Verification**

## *3.1. Introduction of the Experiment Bench*

A Samsung INR18650-33 G battery (Cell Business Division, Samsung SDI Co., Ltd., Yongin, Korea) with nominal capacity 2700 mAh (0.2C, 2.50 V discharge), nominal voltage 3.6 V, charging end voltage 4.1 V, and discharge cut-off voltage 2.5 V, was adopted as the sample battery. An Arbin BT-5HC (Arbin Instruments, LLC, College Station, TX, USA) with voltage range 0–5 V DC, and maximum current ±30 A was adopted for calibration, driving schedule simulation, and temperature monitoring. A Sanwood SC-80-CC-2 (Sanwood, Dongguan, China) thermal cabinet provided a controlled temperature and humidity environment during experiments. Arbin Mits Pro Software (v7 PV.202103) and MATLAB R2019b (MathWorks, Natick, MA, USA) were adopted for driving schedule file editing and application, data logging, model establishing, and data processing [47,48]. The configuration of the offline test bench is shown in Figure 2.

**Figure 2.** Outline of the test bench: offline calibration and schedule condition application.

In addition, to test the online application performance of the proposed method, a hardware platform for real-time quantitative characterization on polarization voltage of lithium-ion batteries was built, as shown in Figure 3. A Chroma DC electronic load 63206E and programmable DC power supply 62050H (Chroma Electronics (Shenzhen) Co., Ltd., Shenzhen, China) were adopted for schedule condition application. A battery fixture (homemade) was used to hold battery cells and connect the circuit. Batteries were connected in series in this case. A data acquisition board (homemade) was used to acquire current and voltage signals, where the sampling frequency was 1 Hz. MATLAB R2019b (MathWorks, Natick, MA, USA) was adopted for driving schedule file editing and application, data communication, logging, and data processing.

**Figure 3.** Outline of the hardware platform: online characterization of polarization voltage.

#### *3.2. Experiment Configuration*

The experimental flow was divided into two main parts: the offline validation of the proposed method and the online hardware implementation of the polarization characterization, as shown in Figure 4.

In preliminary work, the battery was calibrated for relevant parameters, including the actual capacity of the battery, *SOC-OCV* curve, and offline identified model parameters. The temperature dependence was not considered, and all experiments were conducted in the thermal cabinet at 25 ◦C and temperature fluctuations on the cell surface were monitored using thermocouples.

Battery capacity was obtained using standard capacity testing methods at 25 ◦C. The battery was first fully charged using the Constant current Constant voltage (CC-CV) method and rest for 2 h, then discharged to the lower cut-off voltage at 0.2 C constant rate (1 C is 2.7 A in this case). The above steps were repeated three times and the average of three discharge capacities was used as the exact value of the actual battery capacity. The actual capacity of the cell in this case was obtained by the above method is 2.5907 Ah.

**Figure 4.** Schedule of experiments.

The *SOC-OCV* curve was obtained by a series of discharge pulses with different spacing to calibrate a series of points and then fit. The specific steps were: (1) use the CC-CV method to fully charge the battery, and rest for 2 h, record the end voltage as the open-circuit voltage (*OCV*) of the battery at 100% *SOC*; (2) discharge the battery to 98% *SOC* using a constant current rate of 0.2 C and rest for two hours, and record the end voltage as the open-circuit voltage at *SOC* level of 98%; (3) repeat step (2) and measure the open-circuit voltage of *SOC* at levels 95%, 90%, 80%, 70%, 60%, 50%, 40%, 30%, 20%, 10%, 8%, 5%, 3%, 1% and 0%. The *SOC-OCV* points obtained at different *SOC* levels were recorded and the relationship between *OCV* and *SOC* was described using a sixth-order polynomial. And the recorded result is shown in Table 1 and the recorded data points and fitted curve are shown in Figure 5.

$$OCV = \sum\_{n=0}^{6} a\_n SOC^n$$

**Table 1.** Polynomial coefficients of *SOC-OCV* curve.

**Figure 5.** The measured data points and fitted curve of *SOC* versus *OCV*.

#### *3.3. Offline Verification of Proposed Method*

A CITY driving cycle was applied to the battery at an ambient temperature of 25 ◦C to obtain realistic battery cycle data to verify that the model could accurately describe the battery behavior or not. Part of the battery cycle data was intercepted as input data for the improved PSO algorithm, L-M method, and the joint algorithm, respectively. In this case, *SOC*<sup>0</sup> = 0.69692 was used as the starting point, and 1408 subsequent data points (i.e., one CITY cycle) were intercepted. The current curve is shown in Figure 6 as the input data for the parameter identification algorithm. The test data were fed into those three algorithms separately, and the identification results of the three model parameters were obtained, as shown in Table 2.

**Figure 6.** Current profile of the CITY operating condition test.

**Table 2.** Model parameter identification results of three types of algorithm.


To verify the superiority of the parameter identification algorithm proposed in this paper, the model parameters identified by PSO algorithm alone, L-M method alone, and the joint algorithm were substituted into the model, and the model output voltage was compared with the sampling voltage data, as shown in Figure 7. The mean error (ME) and root mean square error (RMSE) were used to describe the deviation between the model output voltage and the sampling voltage. The results are listed in Table 3. The joint algorithm significantly improved the fit accuracy of the model to the sampling voltage, in terms of voltage ME or voltage RMSE, compared to the PSO algorithm or L-M method alone for the intercepted cycle data. Specifically, the joint algorithm reduced voltage ME by 86.3% compared to the PSO algorithm and 83.2% compared to the L-M method, and reduced voltage RMSE by 77.1% compared to the PSO algorithm and 72.3% compared to the L-M method.

To further verify the effectiveness of the proposed method, the second set of test data (*SOC*<sup>0</sup>  = 0.621165046, data length 1408) was fed into the model which adopted model parameters obtained from the first set of test data, and the error of the model output voltage from the sampling voltage was compared, as shown in Figure 8. The voltage RMSE was 0.0095320142 V and the voltage ME was 0.0082487339 V. Based on the above experiments, it can be concluded that: (1) the LDM describes the nonlinear characteristics of the battery under high dynamic driving cycles, and (2) the same model parameters are used in test data from different but adjacent *SOC* stages, which can still describe, relatively well, the terminal voltage characteristics of the battery, indicating that the proposed method reflects the real physicochemical processes inside the battery to a certain extent.

**Figure 7.** Comparison of LDM output voltage curve based on three types of parameter identification algorithm during the CITY test at 25 ◦C ambient temperature. (**a**) Voltage curve; (**b**)voltage error curve.

**Table 3.** Voltage error statistics based on model parameters from three algorithms.


**Figure 8.** Model output voltage error curve based on the second group of testing data.

After solving the partial differential equation in LDM, the distribution of the local *SOC* inside the electrode particle can be obtained, as shown in Figure 9. At the particle size dimension taken as X = 1, the distribution of *SOC* on the electrode particle surface with time is obtained, as shown in Figure 10. Based on LDM, the variation curves of activation polarization voltage, ohmic polarization voltage, and concentration polarization voltage with time can be obtained, respectively, as shown in Figure 11. A conclusion can be drawn that the activation polarization and ohmic polarization respond quickly to the change of input current excitation; compared with the other two, and the concentration polarization responds more slowly to the change of input current excitation. When a non-zero current was applied to the cell system, a gradient in the concentration of the active material in the solid and liquid phases was gradually formed, and the voltage drop from concentration polarization gradually increased, while the time constant of this process was much larger than that of the ohmic and activation polarization. This conclusion is consistent with that obtained in [5] using the Pseudo-2D model under EUCAR driving conditions. The superposition of these three types of polarization phenomena is reflected in the output terminal voltage of LDM, which determines whether the proposed method can accurately describe the cell behavior. The results of the terminal-voltage accuracy comparison above justify the proposed method.

**Figure 9.** Temporal and spatial distribution of *iSOC* inside the electrode particle during the CITY test at 25 ◦C ambient temperature.

**Figure 10.** Variation curve of *SOC* on the surface of electrode particles with time.

**Figure 11.** Quantitative characterization curves of three types of polarization voltage based on LDM.

#### *3.4. Online Polarization Voltage Characterization Using a Hardware Platform*

To realize the online quantitative characterization of the polarization voltage drop based on LDM, the model parameters at different *SOC* levels need to be identified offline in advance. Discharge pulses were applied to the battery at 25 ◦C at different *SOC* levels and rest for 2 h after each discharge pulse, and current versus voltage data were recorded throughout. The *SOC* levels were selected as 98%, 95%, 90%, 80%, 70%, 60%, 50%, 40%, 30%, 20%, 10%, 8%, 5%, 3%, 1% and 0%. A portion of the data before and after each discharge pulse, containing the zero-state response and zero-input response phases, was intercepted as input data for the parameter identification algorithm. A 9th order polynomial was used to fit the parameter points. The fitting curves for three model parameters are shown in Figures 12–14.

**Figure 12.** Variation curve of diffusion time constant *τ* with *SOC*.

**Figure 13.** Variation curve of dimensionless charge exchange current *invJ*<sup>0</sup> with *SOC*.

**Figure 14.** Variation curve of ohmic overpotential at 1 C rate *ηIR*,1*C*/mV with *SOC*.

At an ambient temperature of 25 ◦C, the New European Driving Cycle (NEDC) data were used as the test data for real-time quantitative characterization of polarization voltage based on LDM. The data were fed into the characterization platform when *SOC* = 0.8468 and stopped when *SOC* = 0.7006. Input current update frequency and terminal voltage acquisition frequency were 1 Hz. The values of the model parameters at specific *SOC* level were obtained by interpolation of the previously calibrated curves. The current vs. terminal voltage curves (Figure 15), *SOC* curves (Figure 16), and polarization voltages (Figure 17) were plotted dynamically during cycling. Based on the hardware platform used, real-time characterization of polarization voltage drops at a frequency of 1 Hz could be achieved using the proposed method (actual calculation time consumption for each time step is less than 500 ms). The model output voltage maintained good tracking performance by comparing with the battery terminal voltage data obtained from the acquisition board. However, it was observed that the voltage tracking error increased when the current increased. The possible sources of error are (1) LDM does not include the electrolyte concentration polarization, (2) errors from the identification algorithm or the curve fitting, which are expected to be further improved. It can be seen that the voltage drop from all three types of polarization was positively correlated with the current applied to the cell, which is consistent with the findings of previous studies [14,17,49]. In summary, the proposed method achieves the function of quantitative characterization of polarization voltage, and the algorithm computation efficiency can meet a good real-time performance.

**Figure 15.** Accuracy test of hardware implementation platform under NEDC schedule at 25 ◦C ambient temperature. (**a**) NEDC current profile. (**b**) Comparison of model output voltage and actual voltage.

**Figure 16.** SOC curve based on ampere-hour integral method under the NEDC condition.

**Figure 17.** Curves of three kinds of polarization voltage under the NEDC condition.
