Electric vehicles have experienced explosive growth in recent years. In my investigation of the current transportation landscape, I have observed that the rapid expansion of the electric-vehicle market has placed unprecedented pressure on the charging infrastructure. The number of charging piles is far behind the number of electric vehicles on the road, and this imbalance is particularly severe in urban and highway service areas. My research is motivated by the fact that owners of modern EV batteries still suffer from range anxiety and charging anxiety, even though battery energy density has improved steadily. In this context, fast charging technology has become one of the most critical technologies for further promoting the adoption of electric vehicles. The key issue is not only to shorten the charging time but also to maintain the health of the battery system. Therefore, my research focus is on the development of a stepwise fast charging strategy for power batteries and the accurate estimation of their state of health, which directly affects the reliability and safety of EV batteries.
During the past five years, EV battery development has diversified into several chemical systems. Lithium-ion batteries, especially ternary lithium batteries, dominate the current market due to their high specific energy and excellent charge/discharge performance. Sodium-ion batteries, on the other hand, are attracting increasing attention because of their lower raw-material cost, better low-temperature tolerance, and improved safety. These two chemistries are both important candidates for future EV batteries. In my experimental work, I selected a commercial 18650 ternary lithium cell and a commercial 18650 sodium-ion cell as representative power batteries. By performing aging-cycle experiments under different stepwise charging strategies, I investigated how the charging protocol influences capacity fade, resistance increase, and structural degradation. I also investigated the feasibility of extracting health factors from stepwise fast-charging curves for online state-of-health estimation of EV batteries.
The main contribution of my research is threefold. First, I propose a voltage-cut-off stepwise fast-charging strategy based on differential voltage analysis and direct-current internal resistance evaluation. Second, I compare this strategy with several capacity-cut-off stepwise strategies and with conventional constant-current constant-voltage charging by means of long-term cycling experiments. Third, I construct a feature engineering framework adapted to the proposed fast-charging protocol and apply particle swarm optimization combined with back-propagation neural networks and convolutional neural networks with long short-term memory to estimate the state of health of EV batteries under fast-charging conditions. The results demonstrate that the voltage-cut-off stepwise strategy can significantly reduce charging time while preserving battery capacity.
1 Background and research motivation
The number of electric vehicles in China exceeded ten million in 2022, twenty million in 2023, and more than thirty million in 2024. The average annual growth rate of new-energy vehicles over the last five years was around 37.7%, and in the last three years it approached 60%. However, the construction of charging infrastructure has not kept pace with this growth. Even though the total number of charging piles increased by more than 40% annually, the ratio between public fast-charging piles and electric vehicles remains unfavorable. A large proportion of public charging piles are alternating-current piles, and direct-current fast-charging piles constitute only about 40% of public piles. In busy locations such as highway service areas and commercial centers, fast-charging piles are heavily overloaded. This asymmetry contributes directly to the slow-charging experience and hinders the popularization of electric cars.
Most modern EV batteries are lithium-ion systems. Their advantages include high specific energy, low self-discharge, high open-circuit voltage, and long cycle life. Ternary lithium batteries, with a positive-electrode material such as LiNiCoMnO₂ and a graphite negative electrode, have become one of the mainstream technologies in the pure electric vehicle market. Their nominal voltage is about 3.6-3.65 V, and the operating voltage window is usually 2.5-4.2 V. The specific energy can be as high as 200-300 Wh/kg, which enables a longer driving range. On the other hand, sodium-ion batteries have attracted attention as a potential complementary technology for EV batteries because of their low cost, high safety, and good low-temperature behavior. A typical sodium-ion cell in my study has a nominal voltage of 3.0 V, an operating window of 1.5-4.0 V, and a specific energy of about 100-160 Wh/kg. Table 1 summarizes the basic properties of several power-battery systems, including lead-acid, nickel-metal hydride, nickel-cadmium, sodium-ion, ternary lithium, and lithium iron phosphate cells.
| Battery type | Positive electrode | Negative electrode | Nominal voltage (V) | Voltage range (V) | Specific energy (Wh/kg) | Cycle life |
|---|---|---|---|---|---|---|
| Lead-acid | PbO₂ | Pb | 2.0 | 1.8-2.4 | 35-40 | 400-600 |
| Ni-MH | Ni(OH)₂ | metal hydride | 1.2 | 1.0-1.4 | 50-80 | 800-1000 |
| Ni-Cd | Ni(OH)₂ | Cd | 1.2 | 1.0-1.4 | 40-45 | 600-1000 |
| Sodium-ion | layered oxide | hard carbon | 3.0 | 1.5-4.0 | 100-160 | 2000-5000 |
| Ternary lithium | LiNiCoMnO₂ | graphite | 3.65 | 2.5-4.2 | 200-300 | 800-2000 |
| LFP | LiFePO₄ | graphite | 3.2 | 2.0-3.65 | 120-160 | 2000-5000 |
Conventional constant-current constant-voltage charging is the most widely used protocol in practice. In the constant-current stage, a fixed current is applied until the upper voltage limit is reached; then the voltage is maintained while the current exponentially decays to a low cut-off current. Although the method is simple and robust, it is not optimized from the perspective of fast charging. If the constant current is increased, the charging time is reduced but the battery is exposed to high current over a large portion of the charging process. This high current may accelerate lithium plating, increase the internal temperature, and cause faster capacity loss. A high constant current in the low-temperature region or at high state-of-charge is particularly harmful to the EV battery.
Several alternative charging methods have been reported in the literature. Boost charging uses a higher current in the low SOC range and then switches to a normal constant-current stage. Pulse charging applies periodic positive or negative current pulses to reduce concentration polarization and heat generation. Multi-stage constant-current charging, known as stepwise charging or MSCC, divides the total charging process into several intervals with decreasing current levels. Stepwise charging has shown promising results in reducing charging time and mitigating battery degradation. In my work, I focus on the multi-stage constant-current concept and further improve the design by using voltage thresholds as the transition condition instead of relying only on accumulated capacity. This voltage-cut-off stepwise strategy is based on the physical interpretation of the differential voltage curve and the direct-current resistance evolution, and it provides a particularly beneficial charging pattern for EV batteries in practical applications.
2 Experimental platform and test methodology
My experimental platform consisted of battery testers, climate chambers, cell fixtures, and an upper computer for data acquisition. For the sodium-ion cells, I used a Neware CT-4008 five-channel battery tester with a maximum current of ±100 A, maximum voltage of ±5 V, and accuracy of ±0.05% of full scale. The climate chamber was a DGBEL BT-150C with temperature control from -40 °C to 150 °C and an accuracy of ±0.5 °C. For the lithium-ion cells, I used Arbin BT2000 testers with 16 channels per unit, current ranges of ±1 A, ±5 A, and ±20 A, and voltage ranges of ±5 V. The environmental chamber for these tests was a DGBEL BTT-1000C with a volume of 1000 L. All experiments were carried out at a controlled temperature of 25 °C unless otherwise stated. The battery Fixtures used a four-wire connection to eliminate cable resistance errors.
Two groups of commercial 18650 cells were used. The first group was sodium-ion cells with a positive electrode of sodium nickel iron manganese layered oxide, a hard-carbon negative electrode, and an electrolyte based on sodium hexafluorophosphate. The nominal capacity was 1.4 Ah. The second group was ternary lithium-ion cells with LiNiCoMnO₂ positive electrode and graphite negative electrode; the nominal capacity was 1.5 Ah. Detailed geometric and electrical parameters are listed in Table 2.
| Parameter | Sodium-ion cell | Ternary lithium-ion cell |
|---|---|---|
| Diameter (mm) | 18.40 ± 0.10 | 18.00 ± 0.10 |
| Height (mm) | 65.30 ± 0.10 | 65.00 ± 0.15 |
| Nominal capacity (mAh) | 1400 | 1500 |
| Nominal voltage (V) | 3.1 | 3.6 |
| Operating voltage (V) | 1.5 – 4.0 | 2.5 – 4.2 |
| Mass (g) | 38 | 46 ± 2 |
| AC impedance at 1 kHz (mΩ) | ≤ 18 | ≤ 13 |
Before the long-term aging test, I screened every cell to ensure a consistent initial state. I first stored all new cells in the 25 °C climate chamber for 24 hours. Then I measured the alternating-current internal resistance and rejected cells with resistance outside a narrow window. After that, each cell underwent three preliminary charge-discharge cycles at 0.5 C. Only cells with a final discharge-capacity difference smaller than 0.5% and selected resistance ranges were used. For the sodium-ion cells, the selected discharge capacity was in the range of 1370-1375 mAh and the alternating-current resistance was between 18.2 and 18.4 mΩ. For the lithium-ion cells, the capacity range was 1505-1510 mAh and the alternating-current resistance was between 12.2 and 12.4 mΩ.
Rate-charging tests were performed to characterize the response of the cells to different charging currents. The sodium-ion cells were charged from 0% SOC to 100% SOC with constant-current constant-voltage charging at current levels of 0.5, 1.0, 1.2, 1.4, 1.6, and 1.8 A. The discharge current was always 1C, which corresponded to 1.4 A. Table 3 shows the measured charging capacity, charging time, and coulombic efficiency. With increasing current, the charging capacity temporarily increased beyond the nominal value because of the broader voltage plateau, but the coulombic efficiency decreased when the current exceeded 1.4 A. The 1C constant-current constant-voltage protocol was selected as the baseline for the sodium-ion battery aging experiments.
| Charging current (A) | Charging time | Charging capacity (mAh) | Coulombic efficiency (%) |
|---|---|---|---|
| 0.5 | 1:59:14 | 1372.9 | 99.59 |
| 1.0 | 1:02:59 | 1359.6 | 99.32 |
| 1.2 | 1:11:22 | 1373.2 | 99.16 |
| 1.4 | 1:03:11 | 1404.6 | 96.99 |
| 1.6 | 0:56:33 | 1427.1 | 95.48 |
| 1.8 | 0:51:13 | 1442.7 | 94.45 |
The lithium-ion cells were tested with constant-current charging at current values between 0.5 and 12 A, while the discharge current was maintained at 1.5 A. As shown in Table 4, the lithium-ion cells had excellent rate capability. The charge acceptance remained close to 100% up to very high current values, although the constant-current portion of the constant-current constant-voltage charge decreased significantly when the current was larger than 3 A. This observation indicates that high current caused strong polarization and forced the battery to enter the constant-voltage stage earlier.
| Charging current (A) | Charging time | Charging capacity (mAh) | Efficiency (%) | CC portion (%) |
|---|---|---|---|---|
| 0.5 | 3:04:05 | 1512.0 | 99.80 | 98.37 |
| 1.0 | 1:36:40 | 1507.8 | 100.34 | 95.41 |
| 1.5 | 1:09:12 | 1510.2 | 99.96 | 92.16 |
| 2.0 | 0:55:06 | 1513.2 | 100.14 | 89.90 |
| 2.5 | 0:47:28 | 1511.0 | 99.95 | 86.99 |
| 3.0 | 0:41:26 | 1515.6 | 100.04 | 85.49 |
| 3.5 | 0:38:07 | 1511.2 | 99.94 | 82.93 |
| 4.0 | 0:34:29 | 1516.7 | 100.01 | 81.98 |
| 4.5 | 0:32:52 | 1511.3 | 99.95 | 79.56 |
| 5.0 | 0:30:10 | 1517.5 | 99.98 | 78.98 |
| 5.5 | 0:29:29 | 1511.8 | 99.93 | 76.60 |
| 6.0 | 0:27:09 | 1518.2 | 99.97 | 76.41 |
To evaluate the direct-current resistance evolution as a function of state of charge, I applied a charging-oriented hybrid pulse power characterization test. The pulse test consisted of a 10-second constant-current charging pulse followed by a 30-second or 40-second discharge pulse to return the state of charge to its previous value. By measuring the instantaneous voltage response at the beginning and end of the pulse, the ohmic resistance and polarization resistance can be calculated as
$$R_\Omega = \frac{V_2 – V_1}{I}$$
$$R_p = \frac{V_3 – V_2}{I}$$
where \(V_1\) is the voltage before the pulse, \(V_2\) is the voltage immediately after the pulse starts, \(V_3\) is the voltage at the end of the pulse, and \(I\) is the pulse current. For sodium-ion cells, I tested the charging resistance every 10% SOC over the full range and every 2% SOC in the first 30% SOC because the resistance changed rapidly at low states of charge. The results showed that the sodium-ion cell had a much higher direct-current resistance below about 20% SOC. At 10% SOC, the value was roughly 1.5 times the value at 50% SOC. This high resistance is particularly important for fast charging of EV batteries because high current in this initial region can generate excessive heat and accelerate degradation.
For the lithium-ion cells, the charging direct-current resistance was measured between 0% and 80% SOC, while the discharging resistance was measured between 100% and 0% SOC. The charging resistance and discharge resistance were almost identical at a given SOC when the same pulse current was used. The lithium-ion cell resistance followed a U-shape curve: high at very low SOC, low in the middle region, and slightly high again near high SOC. The resistance at 0-20% SOC ranged from about 24 to 34 mΩ. This reinforced my decision to use a moderate current at the beginning of the stepwise charging protocol rather than starting with a maximum current.
Incremental capacity analysis and differential voltage analysis are powerful non-destructive tools for battery diagnostics. The incremental capacity curve is calculated as
$$\frac{dQ}{dV} = \frac{I}{dV/dt}$$
and the differential voltage curve is
$$\frac{dV}{dQ} = \frac{1}{dQ/dV}$$
I charged fully discharged cells at 0.1 C and sampled the voltage every 10 s to obtain smooth curves. The resulting IC and DV curves showed characteristic peaks that correspond to phase transitions inside the electrodes. For the sodium-ion cell, the DV curve had a deep valley in the region between 3.2 V and 3.5 V, indicating that the battery can accept a relatively high charging current in that region. Above 3.8 V, the differential voltage increased rapidly and the acceptable current decreased. For the lithium-ion cell, the DV curve showed a minimum in the 3.3-3.6 V region and a pronounced central peak around 3.9 V. The central peak is associated with the transition of graphite to the fully lithiated stage and is close to the lithium-plating potential. If high charging current is used beyond that peak, lithium plating is likely to occur. Therefore, stepwise charging strategies should deliberately reduce the current after the central DV peak.
3 Stepwise fast-charging strategy design
One of the fundamental theories that supports stepwise charging is Mas’s law. Mas formulated three laws for the acceptable charging current of a battery. The first law states that for a given discharge current, the current acceptance coefficient \(a\) is inversely proportional to the square root of the discharged capacity:
$$a = \frac{K_1}{\sqrt{C}}$$
$$I_0 = a C = K_1$$
The second law relates the current acceptance coefficient to the discharge current \(I_d\) through a logarithmic relationship:
$$a = K_2 \log_k I_d$$
The third law states that the total acceptable charging current after multiple discharge rates is the linear sum of the acceptable currents for each discharge step:
$$I_t = I_1 + I_2 + I_3 + \cdots$$
Mas’s curve implies that the battery can accept a relatively large current in the middle SOC region, while the acceptable current is small when the battery is deeply discharged or nearly fully charged. Stepwise fast charging approximates Mas’s ideal curve by using a set of discrete currents. In practice, the steps are usually determined by capacity thresholds, voltage thresholds, or time thresholds. In order to compare different transition conditions, I formulated several stepwise strategies for two EV battery chemistries under a unified set of charging constraints.
Table 5 presents the constraints used for both systems. For the sodium-ion EV battery, the time required to charge from 10% SOC to 80% SOC should be less than 45 minutes, the temperature rise should be smaller than 5 °C, and the charge input ratio, defined as the ratio of charged capacity to the capacity obtained under a 1 C constant-current constant-voltage charge, should exceed 99%. For the lithium-ion EV battery, the corresponding time was set to less than 22 minutes, the temperature rise limit was set to 8 °C, and the reference charge was a 2 C constant-current constant-voltage charge.
| Constraint | Sodium-ion battery | Lithium-ion battery |
|---|---|---|
| Time for 10%-80% SOC | < 45 min | < 25 min |
| Temperature rise | < 5 °C | < 8 °C |
| Charge input ratio | > 99% | > 99% |
3.1 Capacity-cut-off stepwise charging
For the sodium-ion cells, I first designed three capacity-cut-off strategies with three, five, and ten SOC intervals. These strategies are denoted 3SOC-MSCC, 5SOC-MSCC, and 10SOC-MSCC, respectively. The current in each interval was chosen to increase the charging speed in the middle SOC region and to reduce the current in the low and high SOC regions. Table 6 lists the current levels for the three strategies. The charging process was switched to constant-voltage charging when the cell voltage reached 4.0 V, and the constant-voltage stage was terminated when the current fell below about 0.1 C.
| Strategy | 0-10% SOC | 10-20% | 20-30% | 30-40% | 40-50% | 50-60% | 60-70% | 70-80% | 80-90% | 90-100% |
|---|---|---|---|---|---|---|---|---|---|---|
| 3SOC-MSCC | 1.8 A | 1.8 A | 1.8 A | 1.4 A | 1.4 A | 1.4 A | 1.0 A | 1.0 A | 1.0 A | 1.0 A |
| 5SOC-MSCC | 1.0 A | 1.8 A | 1.6 A | 1.6 A | 1.4 A | 1.4 A | 1.2 A | 1.2 A | 1.0 A | 1.0 A |
| 10SOC-MSCC | 1.0 A | 1.7 A | 1.6 A | 1.5 A | 1.4 A | 1.3 A | 1.2 A | 1.1 A | 1.0 A | 0.9 A |
For the lithium-ion EV battery, I designed a broader family of capacity-cut-off strategies to investigate the influence of the number of stages and the current amplitude of the first stage. These strategies are named 3SOC-MSCC-1, 3SOC-MSCC-2, 5SOC-MSCC-1, 5SOC-MSCC-2, 5SOC-MSCC-3, and 10SOC-MSCC. Tables 7 and 8 summarize the current profiles. The two three-step strategies share the same capacity transition point at 33% and 66% SOC but use different current values. 3SOC-MSCC-1 uses 5 A, 3 A, and then 1.5 A; 3SOC-MSCC-2 uses 4 A, 3 A, and then 2 A. The five-step strategies use 20% increments with different current sequences. In addition, I considered a boost-charging strategy that behaved like a two-stage voltage-cut-off profile, whose current is 4 A before 3.9 V and 2.5 A after 3.9 V, followed by a constant-voltage stage.
| Strategy | 0-33% SOC | 33-66% SOC | 66-100% SOC |
|---|---|---|---|
| 3SOC-MSCC-1 | 5.0 A | 3.0 A | 1.5 A |
| 3SOC-MSCC-2 | 4.0 A | 3.0 A | 2.0 A |
| Strategy | 0-10% | 10-20% | 20-30% | 30-40% | 40-50% | 50-60% | 60-70% | 70-80% | 80-90% | 90-100% |
|---|---|---|---|---|---|---|---|---|---|---|
| 5SOC-MSCC-1 | 5.0 A | 5.0 A | 4.0 A | 4.0 A | 3.0 A | 2.0 A | 2.0 A | 1.0 A | 1.0 A | 1.0 A |
| 5SOC-MSCC-2 | 4.0 A | 4.0 A | 3.5 A | 3.5 A | 3.0 A | 2.5 A | 2.5 A | 2.0 A | 2.0 A | 2.0 A |
| 5SOC-MSCC-3 | 2.0 A | 4.0 A | 3.5 A | 3.5 A | 3.0 A | 2.5 A | 2.5 A | 2.0 A | 2.0 A | 2.0 A |
| 10SOC-MSCC | 2.0 A | 3.6 A | 3.4 A | 3.2 A | 3.0 A | 2.8 A | 2.6 A | 2.4 A | 2.2 A | 2.0 A |
The comparison between 3SOC-MSCC-1 and 3SOC-MSCC-2 reveals the effect of the initial-stage current, while the comparison between 5SOC-MSCC-2 and 5SOC-MSCC-3 isolates the effect of a low current in the first 10% SOC range. These distinctions are important for understanding how the charging current in the high-resistance region affects the long-term health of EV batteries.
3.2 Voltage-cut-off stepwise charging
The voltage-cut-off strategy uses the characteristic valleys and peaks of the DV curve as transition boundaries. In the low-voltage or low-SOC region, the direct-current resistance is large and therefore the initial current should be limited. In the middle-voltage region where the DV curve forms a valley, the battery has the highest current acceptance, and thus the current can be increased. Near the end of charging, the DV curve increases sharply, and the current must be reduced gradually to avoid damaging the electrode structure and preventing lithium plating.
For the sodium-ion EV battery, I proposed a nine-stage voltage-cut-off strategy, denoted 9V-MSCC. The transitions occur at 2.6 V, 2.8 V, 3.2 V, 3.5 V, 3.8 V, 3.85 V, 3.9 V, 3.95 V, and 4.0 V. The corresponding current values are given in Table 9. A moderate 1.0 A current is applied in the lowest SOC range; then the current increases to 1.4 A after 2.6 V and to the maximum of 2.0 A between 3.2 V and 3.5 V. After 3.5 V, the current is reduced in a stepwise manner to prevent fast charging across the characteristic peak. The final current is only 0.5 A when the voltage approaches 4.0 V.
| Voltage threshold (V) | 2.6 | 2.8 | 3.2 | 3.5 | 3.8 | 3.85 | 3.9 | 3.95 | 4.0 |
|---|---|---|---|---|---|---|---|---|---|
| Charging current (A) | 1.0 | 1.4 | 2.0 | 1.3 | 1.2 | 1.1 | 0.9 | 0.7 | 0.5 |
For the ternary lithium-ion EV battery, I designed a ten-stage voltage-cut-off strategy, denoted 10V-MSCC. The current profile is shown in Table 10. The charging process begins at 2.0 A until 3.6 V, then rises to 5.0 A in the main plateau between 3.6 and 3.8 V. Around 3.8 V the current is reduced to 3.5 A, and at 3.9 V, which corresponds to the central graphite intercalation peak, it is further reduced to 2.5 A. From 4.0 V to 4.2 V, the current is reduced more gradually, with steps from 2.0 A down to 0.5 A. This protocol avoids high-rate charging after the central DV peak and limits the risk of lithium plating in the high-SOC region.
| Voltage threshold (V) | 3.6 | 3.8 | 3.9 | 4.0 | 4.1 | 4.12 | 4.14 | 4.16 | 4.18 | 4.2 |
|---|---|---|---|---|---|---|---|---|---|---|
| Charging current (A) | 2.0 | 5.0 | 3.5 | 2.5 | 2.0 | 1.5 | 1.25 | 1.0 | 0.75 | 0.5 |
For comparison, I also included a boost-charging constant-current constant-voltage protocol for lithium-ion EV batteries. That protocol uses a 4 A current up to 3.9 V, then a 2.5 A constant-current charge to 4.2 V, followed by constant-voltage charging down to 0.1 C. This can be considered a two-stage voltage-cut-off variant with a relatively high current in the middle region and no further gradual reduction between 3.9 and 4.2 V.
4 Aging-cycle experimental results
The validation program consisted of 150 aging cycles for sodium-ion cells and 800 aging cycles for lithium-ion cells. All cells were placed in a climate chamber set to 25 °C. After every 50 cycles for sodium-ion cells and every 100 cycles for lithium-ion cells, the cells were subjected to a capacity-calibration procedure. This procedure consisted of a full charge at 0.5 C to the upper voltage limit, a constant-voltage charge to 0.1 C, a two-hour rest, and a full discharge at 0.5 C to the lower voltage limit. The second discharge capacity was taken as the reference capacity. For the sodium-ion cells, I also measured the direct-current resistance at three voltage points and the alternating-current resistance after each calibration.
The state of health of an EV battery is commonly defined by the ratio of the current available capacity to the initial rated capacity:
$$SOH_Q = \frac{Q_{\text{current}}}{Q_{\text{initial}}} \times 100\%$$
Another definition based on internal resistance increase is:
$$SOH_R = \frac{R_{\text{initial}}}{R_{\text{current}}} \times 100\%$$
In this work, I primarily used the capacity-based SOH value calculated from the calibrated discharge capacity.
4.1 Sodium-ion battery results
Table 11 reports the charging time from 10% to 80% SOC for the five charging protocols used in the sodium-ion aging test. The voltage-cut-off strategy 9V-MSCC achieved the shortest charging time of 38 minutes and 10 seconds, which is 7 minutes and 16 seconds shorter than the 1C constant-current constant-voltage baseline. The 5SOC-MSCC strategy also performed well, while the 10SOC-MSCC strategy was slower than the three-step and five-step capacity-cut-off strategies because the final steps were assigned relatively small currents and the capacity thresholds were designed too conservatively.
| Charging protocol | 10%-80% SOC time | Saved time vs CCCV | Reduction |
|---|---|---|---|
| CCCV (1.4 A) | 45 min 26 s | – | – |
| 3SOC-MSCC | 41 min 54 s | 3 min 32 s | 7.8% |
| 5SOC-MSCC | 41 min 11 s | 4 min 15 s | 9.4% |
| 10SOC-MSCC | 43 min 00 s | 2 min 26 s | 5.4% |
| 9V-MSCC | 38 min 10 s | 7 min 16 s | 16.0% |
After 150 cycles, the measured SOH values are shown in Table 12. The voltage-cut-off strategy clearly outperformed the constant-current constant-voltage baseline. The SOH was 92.3% for 9V-MSCC, compared with 84.9% for CCCV, corresponding to a 7.4 percentage-point improvement. The five-step and ten-step capacity-cut-off strategies reached about 90.7% and 90.9% respectively, while the three-step capacity-cut-off strategy was substantially lower at 85.7%. The ten-step strategy was only 0.2 percentage points better than the five-step strategy, which indicates a diminishing return when the number of capacity stages is increased beyond five.
| Protocol | CCCV | 3SOC-MSCC | 5SOC-MSCC | 10SOC-MSCC | 9V-MSCC |
|---|---|---|---|---|---|
| SOH (%) | 84.9 | 85.7 | 90.7 | 90.9 | 92.3 |
The incremental capacity analysis of the aged sodium-ion cells provided the same ranking. I measured a 0.1 C charge after the conclusion of the 150-cycle test and extracted the second and third characteristic peaks of the IC curve. The voltage-cut-off cell showed the highest peak intensity and the smallest shift toward higher voltages. The CCCV cell showed severe broadening of the second peak, suggesting weak electrochemical activity in the corresponding phase transition region. Table 13 lists the peak heights and positions for the different protocols.
| Protocol | 2nd peak voltage (V) | 2nd peak height | 3rd peak voltage (V) | 3rd peak height |
|---|---|---|---|---|
| CCCV | 2.8721 | 1478.8 | 2.9251 | 2309.7 |
| 3SOC-MSCC | 2.8635 | 1598.4 | 2.9235 | 2327.7 |
| 5SOC-MSCC | 2.8578 | 1649.1 | 2.9194 | 2446.5 |
| 10SOC-MSCC | 2.8584 | 1649.5 | 2.9197 | 2460.6 |
| 9V-MSCC | 2.8580 | 1704.8 | 2.9199 | 2502.1 |
Direct-current resistance measurements were taken at voltages of 2.9 V, 3.2 V, and 3.55 V after the final calibration. The resistance growth after aging was smallest for 9V-MSCC and largest for CCCV. At 3.55 V, which corresponds roughly to a medium-high SOC region, the direct-current resistances were 116.03 mΩ, 113.07 mΩ, 105.63 mΩ, 103.55 mΩ, and 98.52 mΩ for CCCV, 3SOC-MSCC, 5SOC-MSCC, 10SOC-MSCC, and 9V-MSCC, respectively. The alternating-current resistance at 1 kHz also followed the same trend. These results indicate that the voltage-cut-off strategy prevents excessive impedance growth in the sodium-ion EV battery by avoiding high current in the high-resistance low-SOC region and in the high-SOC region after the DV valley.
4.2 Lithium-ion battery results
The lithium-ion aging test was much longer and lasted 800 cycles. Because there were many protocols, I evaluated the time-efficiency and capacity-retention results separately. Table 14 reports the total charging time from 0% to 100% SOC and the fast-charging time from 10% to 80% SOC. The voltage-cut-off 10V-MSCC protocol charged from 10% to 80% SOC in 21 min 9 s, which is slightly faster than the 3 A CCCV baseline. Several capacity-cut-off protocols were slightly faster, but they caused larger capacity fade after 800 cycles.
| Protocol | 0-100% SOC time | 10%-80% SOC time |
|---|---|---|
| CCCV-3A | 41 min 46 s | 21 min 03 s |
| BC-CCCV | 41 min 16 s | 20 min 16 s |
| 3SOC-MSCC-1 | 45 min 05 s | 22 min 18 s |
| 3SOC-MSCC-2 | 42 min 31 s | 21 min 15 s |
| 5SOC-MSCC-1 | 47 min 53 s | 21 min 30 s |
| 5SOC-MSCC-2 | 42 min 11 s | 20 min 36 s |
| 5SOC-MSCC-3 | 44 min 31 s | 20 min 54 s |
| 10SOC-MSCC | 44 min 05 s | 21 min 16 s |
| 10V-MSCC | 45 min 38 s | 21 min 09 s |
After 800 cycles, the capacity-based SOH values listed in Table 15 revealed enormous differences among the protocols. The CCCV baseline fell to 64.1%, and the three-step strategy with a 5 A initial current suffered even more, reaching only 60.1%. The voltage-cut-off strategy preserved 89.5% of its initial capacity, which is 25.4 percentage points higher than the CCCV baseline. The five-step protocols with a moderate 4 A initial current retained approximately 76-79% initial capacity, and the ten-step capacity-cut-off protocol retained 79.7%. These values are all significantly higher than the baseline. The exceptionally high SOH of 10V-MSCC demonstrates that controlling the current based on voltage thresholds, especially by reducing the current at the central DV peak and by using a gentle gradually decreasing current in the high SOC range, has a strong protective effect on EV batteries.
| Protocol | CCCV | 3SOC-1 | 3SOC-2 | 5SOC-1 | 5SOC-2 | 5SOC-3 | 10SOC | 10V |
|---|---|---|---|---|---|---|---|---|
| SOH (%) | 64.1 | 60.1 | 74.9 | 76.7 | 78.6 | 79.5 | 79.7 | 89.5 |
The alternating-current resistance of lithium-ion cells was measured before and after the 800-cycle test. The resistance increase was smallest for the 10V-MSCC protocol, with a final value of 18.57 mΩ. The CCCV protocol reached 21.32 mΩ, and 3SOC-MSCC-1 reached 21.53 mΩ. The other capacity-cut-off protocols were in the range of 19.0-20.0 mΩ. These results are consistent with the capacity retention data and further confirm that avoiding high current at low SOC and after the central graphite peak reduces interface degradation, SEI growth, and loss of active material in EV batteries.
After the aging test, I disassembled selected lithium-ion cells in an argon-filled glove box and observed the separator and graphite negative electrode by scanning electron microscopy. The cells examined were a fresh unused cell, a cell cycled with CCCV, a cell cycled with 5SOC-MSCC-2, and a cell cycled with 10V-MSCC. The fresh separator exhibited a uniform fibrous network with high porosity. The separator of the CCCV cell was heavily degraded, with the fiber network collapsed and many pores blocked. The 5SOC-MSCC-2 separator showed occasional fiber fracture and partial blockage, while the 10V-MSCC separator retained a well-defined fibrous structure with only a few locally loose regions. On the graphite negative electrode side, the CCCV cell showed substantial structural damage, surface deposits, and the presence of lithium dendrites. Some dendrites were longer than 3 micrometers. In contrast, the graphite particles of the 10V-MSCC cell still displayed clear edges and a compact structure similar to that of the fresh electrode. The 5SOC-MSCC-2 cell had only a few short dendrites around 1 micrometer in length. This postmortem evidence directly links the charging protocol to the microstructure of the electrode and separator, and it explains why the voltage-cut-off strategy produced the best capacity retention for EV batteries.
5 State-of-health estimation based on stepwise charging data
The second major objective of my study was to provide an accurate state-of-health estimate for EV batteries under the proposed voltage-cut-off stepwise charging protocols. Traditional SOH estimation methods are often built on data obtained from constant-current constant-voltage cycles at moderate rates. However, fast charging with multiple stages creates different voltage and current trajectories that do not conform to conventional assumptions. Therefore, I developed a feature-engineering procedure specific to the 9V-MSCC and 10V-MSCC protocols.
5.1 Feature extraction and correlation analysis
In the voltage-cut-off stepwise charging process, the battery is charged at a constant current until a pre-defined voltage threshold is reached, and then the current is changed. Since the current values are known and fixed, the duration of each stage is directly related to the amount of charge that can be stored in the battery at the current aging level. When a battery degrades, its voltage curve shifts for a given charged capacity, and the time required to reach each voltage threshold changes accordingly. Therefore, I selected the charging time of early stages as the primary degradation-related features.
For the sodium-ion cell under 9V-MSCC, I examined capacity-voltage curves sampled every 30 cycles. The first three charging stages showed the highest differentiation among different cycle numbers. The third stage, where the charging current was 2.0 A between 3.2 V and 3.5 V, displayed the largest decrease in duration as aging progressed. The later stages with lower currents had much smaller time shifts and were less sensitive to aging. Thus I chose the first three stage times as ladder-specific factors \(H_1,H_2,H_3\). I also selected the total charging time \(H_4\), the initial charging direct-current resistance \(H_5\), and the median charging voltage \(H_6\) as additional health factors. Table 16 summarizes the selected features.
| Symbol | Description |
|---|---|
| H1 | Charging duration of the first stage |
| H2 | Charging duration of the second stage |
| H3 | Charging duration of the third stage |
| H4 | Total charging time |
| H5 | Direct-current resistance at the beginning of charging |
| H6 | Median charging voltage |
I quantified the relationship between the extracted factors and the capacity using Pearson’s correlation coefficient:
$$r_{XY} = \frac{\sum_{i=1}^n (X_i – \bar{X})(Y_i – \bar{Y})}{\sqrt{\sum_{i=1}^n (X_i – \bar{X})^2}\sqrt{\sum_{i=1}^n (Y_i – \bar{Y})^2}}$$
The absolute values of the correlations between all selected factors and the capacity were above 0.9, indicating strong monotonic relationships. This validated the use of these features as inputs for SOH estimation.
For the lithium-ion cell under 10V-MSCC, I selected the charged capacity of each early stage because the data acquisition during those stages was performed at a higher current and the charging-time differences were more visible in the capacity-domain representation. Specifically, for stage 1, stage 2, and stage 3, I extracted both the charged capacity and the end-of-stage temperature. I also extracted the voltage-gradient at the beginning of stage 3. That resulted in seven features \(F_1\) through \(F_7\), as listed in Table 17. Since the 800-cycle capacity decay of the lithium-ion cell was not perfectly linear, I used Spearman’s rank correlation coefficient to assess feature relevance:
$$\rho = \frac{\sum_{i=1}^n (R(x_i)-\bar{R}_x)(R(y_i)-\bar{R}_y)}{\sqrt{\sum_{i=1}^n (R(x_i)-\bar{R}_x)^2 \cdot \sum_{i=1}^n (R(y_i)-\bar{R}_y)^2}}$$
The Spearman coefficients were all above 0.9, confirming that these first-stage features carry strong aging information.
| Symbol | Description |
|---|---|
| F1 | Charged capacity in the first stage |
| F2 | End-of-stage temperature in the first stage |
| F3 | Charged capacity in the second stage |
| F4 | End-of-stage temperature in the second stage |
| F5 | Voltage rise rate at the beginning of the third stage |
| F6 | Charged capacity in the third stage |
| F7 | End-of-stage temperature in the third stage |
To test whether adding later-stage features could further improve the estimation, I constructed a second feature set that included the charged capacities and end-of-stage temperatures of stages four, five, and six. This expanded set is referred to as Scheme 2, whereas the original seven factors are Scheme 1. By comparing the performance under the two schemes, I could evaluate whether the first three stages are sufficient for an early SOH prediction during the stepwise fast-charging process of an EV battery.
5.2 PSO-BP for sodium-ion SOH estimation
In the first SOH estimation task, I applied a hybrid algorithm that combines particle swarm optimization with a back-propagation neural network, denoted PSO-BP. The back-propagation neural network is a multi-layer feed-forward network that updates its weights by gradient descent. However, BP networks are sensitive to the choice of initial weights and may converge to local minima. Particle swarm optimization can search the parameter space globally in order to determine an improved set of initial weights and biases.
The PSO velocity and position update equations are given by:
$$v_{id}^{k+1} = w v_{id}^{k} + c_1 r_1 (p_{id}^{k} – x_{id}^{k}) + c_2 r_2 (p_{gd}^{k} – x_{id}^{k})$$
$$x_{id}^{k+1} = x_{id}^{k} + v_{id}^{k+1}$$
where \(w\) is the inertia weight, \(c_1\) and \(c_2\) are acceleration coefficients, \(r_1\) and \(r_2\) are random numbers in \([0,1]\), \(p_{id}\) is the personal-best position, and \(p_{gd}\) is the global-best position. The fitness function used in the optimization was the mean square error between the network output and the target capacity. After the optimization loop, the best-obtained weights and thresholds were assigned to the BP neural network for fine-tuning.
In my estimation, I used the first 100 cycles of sodium-ion data, with 70% or 50% of the samples used for training and the remaining samples used for testing. The performance metrics were the root-mean-square error and the mean absolute error:
$$RMSE = \sqrt{\frac{1}{n}\sum_{i=1}^{n} (y_i – \hat{y}_i)^2}$$
$$MAE = \frac{1}{n}\sum_{i=1}^{n} |y_i – \hat{y}_i|$$
I compared the PSO-BP method with a genetic-algorithm-optimized BP network (GA-BP), a standard BP network, a convolution neural network, an extreme learning machine, and a support vector machine. The results under both 70% and 50% training ratios are summarized in Table 18. PSO-BP achieved the lowest RMSE of 0.610% and MAE of 0.450% under the 70% training condition. When the training ratio was reduced to 50%, its RMSE was still only 0.678%, while standard BP, CNN, ELM, and SVM all degraded substantially. This demonstrates that the PSO-based initialization makes the neural network more robust and less dependent on the amount of supervised data. This is an important advantage for real-world EV battery management, where historical fast-charging data may be limited.
| Method | Training set | RMSE (%) | MAE (%) |
|---|---|---|---|
| PSO-BP | 70% | 0.610 | 0.450 |
| PSO-BP | 50% | 0.678 | 0.583 |
| GA-BP | 70% | 0.873 | 0.647 |
| GA-BP | 50% | 0.910 | 0.739 |
| BP | 70% | 1.326 | 1.113 |
| BP | 50% | 2.182 | 2.043 |
| CNN | 70% | 2.150 | 2.076 |
| ELM | 70% | 2.571 | 2.024 |
| SVM | 70% | 1.445 | 1.220 |
5.3 CNN-LSTM for lithium-ion SOH estimation
For the lithium-ion EV battery under the 10V-MSCC protocol, I built a hybrid deep-learning model composed of a convolutional neural network and long short-term memory layers. The convolutional layers are effective at extracting local patterns from input features, while the LSTM layers capture the long-term temporal dependencies between cycles. This hybrid architecture is particularly suitable for monitoring the aging trajectory of EV batteries because the degradation process has both smooth long-term trends and nonlinear stage-dependent behavior.
The convolution operation in the feature extraction layer can be expressed as:
$$y_{i,j}^l = f\left(\sum_{m=0}^{M-1} \sum_{n=0}^{N-1} w_{m,n}^l x_{i+m,j+n}^{l-1} + b^l\right)$$
where \(w_{m,n}^l\) is the kernel weight, \(x\) is the input feature map, \(b\) is the bias term, and \(f(\cdot)\) is the activation function. I used the rectified linear unit activation for the convolutional layers. After the convolutional operations, a max-pooling layer reduces the dimensions:
$$\gamma(c_i,c_{i-1}) = \max(c_i, c_{i-1})$$
The LSTM unit consists of a forget gate, an input gate, and an output gate. The forget gate is given by:
$$f_t = \sigma(W_f \cdot [h_{t-1}, x_t] + b_f)$$
where \(\sigma\) is the sigmoid activation. The input gate updates the cell state with new information:
$$i_t = \sigma(W_i \cdot [h_{t-1}, x_t] + b_i)$$
$$\tilde{C}_t = \tanh(W_C \cdot [h_{t-1}, x_t] + b_C)$$
$$C_t = f_t \cdot C_{t-1} + i_t \cdot \tilde{C}_t$$
Finally, the output gate decides which memory content is transmitted to the next time step:
$$o_t = \sigma(W_o \cdot [h_{t-1}, x_t] + b_o)$$
$$h_t = o_t \cdot \tanh(C_t)$$
I trained the CNN-LSTM model using 800-cycle data from the 10V-MSCC cell. I again used a 70%/30% train/test split. The input features followed Scheme 1 (three stages) or Scheme 2 (six stages), as described previously. Table 19 reports the RMSE and MAE for the CNN-LSTM model, the standard LSTM model, and the PSO-BP model. Under Scheme 2, CNN-LSTM achieved an RMSE of 0.856% and an MAE of 0.681%, almost identical to PSO-BP. However, under Scheme 1, CNN-LSTM retained a low RMSE of 0.929%, whereas PSO-BP degraded to 1.304%. The standard LSTM model was not as accurate, with RMSE values of about 1.98% under Scheme 2 and 2.36% under Scheme 1. This finding indicates that the CNN-LSTM hybrid network makes better use of the limited first-stage features, likely because the convolutional layers can extract subtle spatial correlations between the capacity and temperature inputs.
| Model | Feature scheme | RMSE (%) | MAE (%) |
|---|---|---|---|
| CNN-LSTM | Scheme 2 | 0.856 | 0.681 |
| CNN-LSTM | Scheme 1 | 0.929 | 0.829 |
| PSO-BP | Scheme 2 | 0.848 | 0.571 |
| PSO-BP | Scheme 1 | 1.304 | 0.988 |
| LSTM | Scheme 2 | 1.975 | 1.529 |
| LSTM | Scheme 1 | 2.360 | 1.946 |
The fact that Scheme 1, which only uses data from the first three charging stages, leads to an RMSE below 1% for the CNN-LSTM model has an important practical implication. During the 10V-MSCC fast-charging protocol, the third stage ends after approximately 13 minutes of charging. Therefore, the SOH of the EV battery can be estimated before the charging process is even one third complete. This early evaluation can enable battery management systems to adapt the remaining charging stages in real time, potentially reducing degradation and improving safety. The ability to extract health indicators from the early portion of a fast-charging session is valuable for cloud-based battery monitoring and for extending the useful life of EV batteries.
6 Discussion and conclusions
Through the systematic comparison of different stepwise fast-charging protocols on two types of power batteries, my research yields several important conclusions. First, the voltage-cut-off stepwise strategy is superior to both conventional constant-current constant-voltage charging and capacity-cut-off stepwise charging for EV batteries. The voltage thresholds are directly related to the electrochemical phase transitions that are embodied in the DV curve. If the charging current is maintained at a high level when the battery voltage approaches the main DV peak, the local negative-electrode potential may drop below the lithium-plating potential. The resulting lithium deposition consumes cyclable lithium, increases the thickness of the solid-electrolyte interphase, and degrades the separator pore structure. By lowering the current at the correct voltage thresholds, the strategy protects the battery from these degradation mechanisms while still preserving a short total charging time.
Second, the number of capacity-cut-off stages should be chosen carefully. In both sodium-ion and lithium-ion cells, increasing the number of stages from three to five produced a significant improvement in capacity retention. Increasing the number from five to ten, however, yielded only marginal gains and in the sodium-ion case even increased the charging time. Therefore, a moderate number of capacity stages around five may be a reasonable practical compromise when voltage-sensitive stage design is not available. Yet, the best solution appears to be the voltage-switched profile because it can automatically adapt to voltage thresholds that reflect the current state of the electrode materials, rather than relying on fixed SOC intervals that change when the battery capacity fades.
Third, the internal resistance behavior at low SOC must not be ignored in fast charging. Both sodium-ion and lithium-ion cells had elevated direct-current resistance in the very low SOC region. The sodium-ion cell showed a particularly high resistance below 10% SOC. If a high current is applied immediately after the start of charging, the large ohmic drop causes local heating and uneven reaction distribution. The consequence is accelerated loss of active material and impedance growth. In my proposed charging strategies, the first stage is intentionally set to a moderate current below 1.4 A for sodium-ion and below 2.0 A for lithium-ion. The current is then increased only after the battery has left the high-resistance region.
Fourth, the SOH estimation results prove that health indicators can be successfully identified from the early stages of a stepwise fast-charging process. For the sodium-ion 9V-MSCC protocol, the charging durations of the first three stages contain sufficient information about the aging state, and PSO-BP is capable of producing SOH estimates with an error below 0.7%. For the lithium-ion 10V-MSCC protocol, the CNN-LSTM network can estimate SOH with an RMSE below 1% using only the first three charging stages. These findings suggest that future battery management systems can integrate SOH estimation into the charging controller itself. Instead of waiting for a full discharge or a dedicated capacity test, the system can use the incomplete fast-charging segment to update the health status of EV batteries. This real-time information also enables the charging strategy to be adjusted from cycle to cycle, creating a feedback loop between battery state and charging policy.
Naturally, my research has some limitations that should be addressed in future studies. The DV curve and the optimum voltage thresholds change as the battery ages. A fixed voltage-threshold profile may become less optimal after 800 cycles, when the cell impedance has grown substantially. An adaptive version of the voltage-cut-off strategy that shifts the thresholds according to estimated degradation would be an interesting extension. In addition, temperature data were not recorded during the sodium-ion cycle test because of sensor limitations. Since temperature is a critical variable in fast-charging safety, future experiments should incorporate thermal measurements for both sodium-ion and lithium-ion EV batteries. Finally, my SOH estimation work focused on the two voltage-cut-off strategies. Applying similar methods to capacity-cut-off stepwise protocols would allow a more complete comparison between different charging patterns.
In summary, my work demonstrates that stepwise fast charging based on voltage-cut-off thresholds is a promising route for improving the charging experience without sacrificing the life of EV batteries. The proposed strategies reduce the 10%-80% SOC charging time from approximately 45 minutes to slightly more than 38 minutes for sodium-ion cells and from 21 minutes to about 20.5-21 minutes for lithium-ion cells under the selected battery types. After prolonged cycling, the capacity retention of EV batteries charged with the voltage-cut-off strategy is dramatically higher than that of batteries charged with constant-current constant-voltage. Moreover, the state-of-health of such batteries can be estimated accurately from the early stages of the very same fast-charging cycle using machine-learning algorithms. My hope is that these findings will contribute to the development of smarter and safer fast-charging solutions for future electric vehicles.