I have focused my research on the design, experimental validation, and health-state estimation of stepwise fast charging strategies for an electric vehicle battery pack. In the context of rapid electrification, the electric vehicle battery pack is no longer only an energy storage component; it determines driving range, charging experience, safety, and the lifetime cost of electric vehicles. I therefore investigated two important battery chemistries: sodium-ion batteries and ternary lithium-ion batteries. I built dedicated test platforms, performed pre-experiments, proposed both capacity-cutoff and voltage-cutoff stepwise charging protocols, conducted long-term aging cycles, and finally estimated state of health (SOH) under the most promising voltage-cutoff fast-charging conditions. Throughout this work, I treated the electric vehicle battery pack as a complex electrochemical system whose charging strategy must simultaneously balance charging time, temperature rise, capacity retention, internal resistance growth, and safety.
Background and motivation. The growth of electric vehicles has created a strong mismatch between vehicle adoption and charging infrastructure. Many users still worry about long charging times and charging safety. Increasing the size of the electric vehicle battery pack can extend range, but it also increases cost, mass, and thermal-management burden. A more practical path is to improve charging protocols so that the existing electric vehicle battery pack can accept energy faster without accelerated degradation. The conventional constant-current constant-voltage (CCCV) method is mature and simple, but it often becomes a bottleneck in fast charging. High current in the low-SOC region can be limited by high direct-current internal resistance, while high current in the middle-to-high SOC region can promote lithium plating, sodium plating, electrolyte decomposition, and separator damage. I therefore designed stepwise fast charging strategies that adapt current to internal resistance and differential voltage characteristics.
Battery chemistry comparison. I compared major rechargeable battery systems and selected sodium-ion and ternary lithium-ion cells for this study. The electric vehicle battery pack must satisfy different requirements in different market segments, so sodium-ion batteries are attractive as a lower-cost and safer alternative, while ternary lithium-ion batteries remain strong in high specific energy and high-rate capability.
| Battery type | Negative electrode | Positive electrode | Nominal voltage (V) | Specific energy (Wh/kg) | Specific power (W/kg) | Cycle life |
|---|---|---|---|---|---|---|
| Lead-acid | Lead | PbO2 | 2.0 | 35-40 | 150 | 400-600 |
| Nickel-metal hydride | Hydride | Ni(OH)2 | 1.2 | 50-80 | 200 | 800-1000 |
| Nickel-cadmium | Cadmium | Ni(OH)2 | 1.2 | 40-45 | 150 | 600-1000 |
| Sodium-ion | Hard carbon | Layered oxide | 3.0 | 100-160 | 100-200 | 2000-5000 |
| Ternary lithium-ion | Graphite | LiNiCoMnO2 | 3.65 | 200-300 | 800-1500 | 800-2000 |
| Lithium iron phosphate | Graphite | LiFePO4 | 3.2 | 120-160 | 1500-2000 | 2000-5000 |
The electric vehicle battery pack based on ternary lithium-ion cells offers high energy density, but it requires careful thermal and charging management. The electric vehicle battery pack based on sodium-ion cells offers cost and safety advantages, but its low-SOC internal resistance can be high, so current limitation at low SOC is important. I used 18650 sodium-ion cells with a nominal capacity of 1.4 Ah and 18650 ternary lithium-ion cells with a nominal capacity of 1.5 Ah. Both cell types were screened for consistency before aging tests.
Definitions of state of health. In my work, I used capacity-based SOH as the main health indicator, while also checking internal resistance and incremental capacity features. The capacity-based definition is:
$$
SOH_Q = \frac{Q_i}{Q_0} \times 100\%
$$
where \(Q_i\) is the current measured capacity and \(Q_0\) is the initial or rated capacity. The resistance-based definition is:
$$
SOH_R = \frac{R_0}{R_i} \times 100\%
$$
where \(R_i\) is the current internal resistance and \(R_0\) is the initial internal resistance. A cycle-life definition can also be written as:
$$
SOH_N = \frac{N_{used}}{N_{EOL}} \times 100\%
$$
For a more complete view, I considered a composite form:
$$
SOH_{composite} = \alpha SOH_Q + \beta SOH_R + \gamma f(\Delta V, ICA, \ldots) \times 100\%
$$
For the electric vehicle battery pack, capacity fade and resistance rise are both important, but capacity fade is the most direct indicator of remaining driving range. Therefore, I used capacity-based SOH as the primary target in the estimation models and used resistance, incremental capacity, and differential voltage features to interpret degradation.
Test platforms and pre-experiments. I built two platforms. For sodium-ion cells, I used a multi-channel charge-discharge tester with a maximum current of 100 A per channel, a constant-temperature chamber, a four-wire cell fixture, and an AC resistance meter. For lithium-ion cells, I used a multi-channel tester with a maximum current of 10 A per channel, a constant-temperature and constant-humidity chamber, and the same type of fixture. The electric vehicle battery pack in real applications contains many cells, so cell-level consistency is essential. I screened cells by AC internal resistance and by three charge-discharge cycles. The screening ranges are shown below.
| Screening item | Sodium-ion cell | Lithium-ion cell |
|---|---|---|
| Discharge capacity (mAh) | 1370-1375 | 1505-1510 |
| AC internal resistance (mΩ) | 18.2-18.4 | 12.2-12.4 |
Rate charging tests. I tested rate capability to understand how much current each cell can accept before polarization and heating become severe. For sodium-ion cells, I charged with currents from 0.5 A to 1.8 A and discharged at 1C. The charge time decreased as current increased, but charge-discharge efficiency dropped when current exceeded about 1.2 A. For lithium-ion cells, I charged with currents from 0.5 A to 6.0 A and discharged at 1C. The constant-current fraction decreased significantly at high rates, indicating stronger polarization.
| Cell | Charge current (A) | Charge time | Charge capacity (mAh) | Efficiency (%) |
|---|---|---|---|---|
| Sodium-ion | 0.5 | 1:59:14 | 1372.9 | 99.59 |
| Sodium-ion | 1.0 | 1:02:59 | 1359.6 | 99.32 |
| Sodium-ion | 1.2 | 1:11:22 | 1373.2 | 99.16 |
| Sodium-ion | 1.4 | 1:03:11 | 1404.6 | 96.99 |
| Sodium-ion | 1.6 | 0:56:33 | 1427.1 | 95.48 |
| Sodium-ion | 1.8 | 0:51:13 | 1442.7 | 94.45 |
| Charge current (A) | Charge time | Charge capacity (mAh) | Efficiency (%) | CC fraction (%) |
|---|---|---|---|---|
| 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 |
HPPC and internal resistance. I used hybrid pulse power characterization (HPPC) to measure direct-current resistance (DCR). The ohmic resistance and polarization resistance were calculated from the voltage response:
$$
R_0 = \frac{V_2 – V_1}{I_1}
$$
$$
R_e = \frac{V_3 – V_2}{I_1}
$$
For sodium-ion cells, the DCR was high in the 0-20% SOC range and decreased toward middle SOC. At low SOC, the DCR was approximately 1.5 times the value near 50% SOC. For lithium-ion cells, the DCR followed a shallow U-shape: high at low SOC, lower in the middle, and slightly higher again near 90-100% SOC. This behavior directly motivated low-current charging at low SOC and current reduction at high SOC in my proposed stepwise protocols.
| Battery | SOC region | DCR behavior | Charging implication |
|---|---|---|---|
| Sodium-ion | 0-10% | Very high | Use low current to avoid damage |
| Sodium-ion | 10-60% | Low and stable | Increase current for fast charging |
| Sodium-ion | 60-100% | Gradually rising | Step down current |
| Lithium-ion | 0-20% | High | Limit current at start |
| Lithium-ion | 20-80% | Low | Use high current |
| Lithium-ion | 80-100% | Slightly rising | Reduce current before cutoff |
Incremental capacity and differential voltage analysis. I performed 0.1C charging tests to obtain incremental capacity (IC) and differential voltage (DV) curves. The IC curve is defined as:
$$
\frac{dQ}{dV} = \frac{d(It)}{dV} = \frac{I}{dV/dt}
$$
The DV curve is defined as:
$$
DV = \frac{dV}{dQ}
$$
These curves reveal phase transitions and electrode degradation. In my sodium-ion cells, the DV curve showed a low valley in the range around 3.2-3.5 V, where the cell could accept higher current. A characteristic peak near 3.5 V indicated the transition to a less tolerant region. In my lithium-ion cells, the central graphite peak near 3.9 V was especially important because charging beyond that region with high current can increase lithium plating risk. I therefore aligned my fast-charging steps with these features.
Maas law and stepwise fast charging. I used the Maas law as a conceptual foundation. The first law states that the acceptable current ratio is inversely proportional to the square root of discharged capacity:
$$
a = \frac{K_1}{\sqrt{C}}
$$
$$
I = aC = K_1
$$
The second law relates the acceptance ratio to discharge current:
$$
a = K_2 \log_k I_d
$$
The third law states that the total acceptable current is the linear superposition of contributions from different discharge rates:
$$
I_t = I_1 + I_2 + I_3 + \cdots
$$
In practice, the ideal Maas curve cannot be followed exactly, so I approximated it with a finite number of current steps. The electric vehicle battery pack must operate under charger constraints, thermal constraints, and user expectations, so my stepwise design also included explicit charging constraints. I set the following limits.
| Constraint | Sodium-ion target | Lithium-ion target |
|---|---|---|
| 10%-80% SOC charging time | Less than 45 min | Less than 25 min |
| Temperature rise | Less than 5 °C | Less than 8 °C |
| Charge input ratio | Greater than 99% | Greater than 99% |
Capacity-cutoff stepwise strategies. I first divided the SOC range into three, five, and ten equal intervals for both chemistries. The transition condition was the charge capacity corresponding to each SOC boundary. These capacity-cutoff strategies are useful for understanding how the number of steps affects aging. For sodium-ion cells, I designed 3SOC-MSCC, 5SOC-MSCC, and 10SOC-MSCC. For lithium-ion cells, I designed 3SOC-MSCC-1, 3SOC-MSCC-2, 5SOC-MSCC-1, 5SOC-MSCC-2, 5SOC-MSCC-3, and 10SOC-MSCC. I also included a boost-charging CCCV (BC-CCCV) case for lithium-ion cells.
| Sodium-ion strategy | 0%-10% | 10%-20% | 20%-33% | 33%-40% | 40%-50% | 50%-66% | 66%-70% | 70%-80% | 80%-90% |
|---|---|---|---|---|---|---|---|---|---|
| 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 |
| 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 |
| 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 |
| Lithium-ion strategy | 0%-10% | 10%-20% | 20%-33% | 33%-40% | 40%-50% | 50%-66% | 66%-70% | 70%-80% | 80%-90% |
|---|---|---|---|---|---|---|---|---|---|
| 3SOC-MSCC-1 | 5.0 A | 5.0 A | 5.0 A | 3.0 A | 3.0 A | 3.0 A | 1.5 A | 1.5 A | 1.5 A |
| 3SOC-MSCC-2 | 4.0 A | 4.0 A | 4.0 A | 3.0 A | 3.0 A | 3.0 A | 2.0 A | 2.0 A | 2.0 A |
| 5SOC-MSCC-1 | 5.0 A | 5.0 A | 4.0 A | 4.0 A | 3.0 A | 3.0 A | 2.0 A | 2.0 A | 1.0 A |
| 5SOC-MSCC-2 | 4.0 A | 4.0 A | 3.5 A | 3.5 A | 3.0 A | 3.0 A | 2.5 A | 2.5 A | 2.0 A |
| 5SOC-MSCC-3 | 2.0 A | 4.0 A | 3.5 A | 3.5 A | 3.0 A | 3.0 A | 2.5 A | 2.5 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 |
Voltage-cutoff stepwise strategies. The most important contribution of my work is the voltage-cutoff stepwise fast charging strategy. Instead of switching at fixed capacity boundaries, I switched at voltage boundaries that were selected from DV features and DCR behavior. This makes the strategy more robust to capacity fade because the electric vehicle battery pack may lose capacity, but the electrode voltage signatures remain meaningful. For sodium-ion cells, I designed a nine-step voltage-cutoff strategy called 9V-MSCC. For lithium-ion cells, I designed a ten-step voltage-cutoff strategy called 10V-MSCC.
| Sodium-ion 9V-MSCC voltage (V) | 2.6 | 2.8 | 3.2 | 3.5 | 3.8 | 3.85 | 3.9 | 3.95 | 4.0 |
|---|---|---|---|---|---|---|---|---|---|
| Current (A) | 1.0 | 1.4 | 2.0 | 1.3 | 1.2 | 1.1 | 0.9 | 0.7 | 0.5 |
| Lithium-ion 10V-MSCC voltage (V) | 3.6 | 3.8 | 3.9 | 4.0 | 4.1 | 4.12 | 4.14 | 4.16 | 4.18 | 4.2 |
|---|---|---|---|---|---|---|---|---|---|---|
| Current (A) | 2.0 | 5.0 | 3.5 | 2.5 | 2.0 | 1.5 | 1.25 | 1.0 | 0.75 | 0.5 |
For sodium-ion cells, I used a low current of about 1.0 A below 2.6 V, then increased to 2.0 A in the low-DV valley around 3.2-3.5 V. After the characteristic peak near 3.5 V, I reduced the current to 1.3 A and then gradually stepped down to 0.5 A at 4.0 V. For lithium-ion cells, I started at 2.0 A to protect the high-DCR low-SOC region, increased to 5.0 A in the low-DV region around 3.6-3.8 V, reduced to 3.5 A after 3.9 V, and then stepped down to 0.5 A at 4.2 V. I also included a BC-CCCV case with 4.0 A up to 3.9 V and 2.5 A up to 4.2 V, followed by constant voltage.
Aging cycle design. I aged sodium-ion cells for 150 cycles and lithium-ion cells for 800 cycles. Sodium-ion cells were calibrated every 50 cycles, and lithium-ion cells were calibrated every 100 cycles. Each calibration included capacity measurement, internal resistance measurement, and, for sodium-ion cells, HPPC tests at selected voltage points. I also performed 0.1C charging for IC and DV analysis after calibration. The electric vehicle battery pack in service rarely experiences perfectly identical cycles, but controlled aging tests are necessary to isolate the effect of charging protocol.
Sodium-ion aging results. I compared CCCV, 3SOC-MSCC, 5SOC-MSCC, 10SOC-MSCC, and 9V-MSCC. The 9V-MSCC strategy achieved the shortest 10%-80% charging time, reducing it by 7 minutes 16 seconds relative to CCCV, a reduction of about 16%. After 150 cycles, the SOH values differed significantly. The 9V-MSCC cells reached 92.3% SOH, which was 7.4 percentage points higher than CCCV. The 5SOC-MSCC and 10SOC-MSCC cells reached 90.7% and 90.9%, respectively, while 3SOC-MSCC reached 85.7%. This showed that increasing the number of steps from three to five improved health, but increasing from five to ten produced only marginal benefit.
| Sodium-ion strategy | 10%-80% time | Time saved | Reduction |
|---|---|---|---|
| CCCV | 45 min 26 s | 0 | 0% |
| 3SOC-MSCC | 41 min 54 s | 3 min 32 s | 7.78% |
| 5SOC-MSCC | 41 min 11 s | 4 min 15 s | 9.35% |
| 10SOC-MSCC | 43 min 00 s | 2 min 26 s | 5.36% |
| 9V-MSCC | 38 min 10 s | 7 min 16 s | 16.0% |
| Sodium-ion strategy | SOH after 150 cycles | Relative improvement vs CCCV |
|---|---|---|
| CCCV | 84.9% | 0 |
| 3SOC-MSCC | 85.7% | +0.8% |
| 5SOC-MSCC | 90.7% | +5.8% |
| 10SOC-MSCC | 90.9% | +6.0% |
| 9V-MSCC | 92.3% | +7.4% |
I also examined IC peaks. The second and third peaks were higher and shifted less for 9V-MSCC than for CCCV, indicating better active-material retention. The DCR at 3.55 V and the AC internal resistance were also lowest for 9V-MSCC. The table below summarizes the sodium-ion internal resistance after aging.
| Sodium-ion strategy | DCR at 3.55 V (mΩ) | ACR after 150 cycles (mΩ) |
|---|---|---|
| CCCV | 116.03 | 27.81 |
| 3SOC-MSCC | 113.07 | 27.62 |
| 5SOC-MSCC | 105.63 | 25.63 |
| 10SOC-MSCC | 103.55 | 25.41 |
| 9V-MSCC | 98.52 | 23.51 |
The sodium-ion results led me to conclude that the electric vehicle battery pack using sodium-ion cells should avoid high current below 10% SOC and above 60% SOC. In the middle SOC range, higher current is acceptable because the DV valley and lower DCR allow efficient ion transport. The voltage-cutoff design is also easier to implement in a real electric vehicle battery pack because it does not require recalibrating capacity thresholds as the cells age.
Lithium-ion aging results. I aged lithium-ion cells for 800 cycles. The CCCV reference reached 64.1% SOH. The 10V-MSCC strategy reached 89.5% SOH, which was 25.4 percentage points higher than CCCV. The 10SOC-MSCC strategy reached 79.7%, and the 5SOC-MSCC variants reached 76.7%, 78.6%, and 79.5%. The 3SOC-MSCC-1 strategy with 5.0 A in the first third reached only 60.1% SOH, slightly worse than CCCV, showing that excessive current at low SOC can accelerate degradation. The 3SOC-MSCC-2 with 4.0 A in the first third reached 74.9%, better than CCCV but worse than the five- and ten-step strategies.
| Lithium-ion strategy | 0-100% time | 10%-80% 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 |
| Lithium-ion strategy | SOH after 800 cycles | ACR after 800 cycles (mΩ) |
|---|---|---|
| CCCV-3A | 64.1% | 21.32 |
| BC-CCCV | Not superior to 10V-MSCC | 21.11 |
| 3SOC-MSCC-1 | 60.1% | 21.53 |
| 3SOC-MSCC-2 | 74.9% | 19.76 |
| 5SOC-MSCC-1 | 76.7% | 19.95 |
| 5SOC-MSCC-2 | 78.6% | 19.65 |
| 5SOC-MSCC-3 | 79.5% | 19.17 |
| 10SOC-MSCC | 79.7% | 19.08 |
| 10V-MSCC | 89.5% | 18.57 |
The lithium-ion results confirmed the same principle: the electric vehicle battery pack benefits from low current at the beginning, high current in the low-DV valley, and reduced current after the central graphite peak. The 10V-MSCC strategy maintained high SOH until about 700 cycles, after which capacity fade accelerated. The 5SOC-MSCC-3, which started at 2.0 A instead of 4.0 A, performed slightly better than 5SOC-MSCC-2, showing again that low-SOC current limitation matters. The 3SOC-MSCC-1 results showed that 5.0 A at the start is too aggressive for this cell design.
Post-mortem and SEM analysis. I disassembled aged lithium-ion cells in an argon-filled glovebox and examined separator and graphite anode samples by scanning electron microscopy. The fresh separator had a regular fiber network with uniform pores. The CCCV-aged separator showed severe collapse, blocked pores, and a disrupted fiber network. The 5SOC-MSCC-2 separator showed occasional fiber breakage and uneven pores. The 10V-MSCC separator remained mostly regular with only slight local loosening. On the anode side, the fresh graphite particles were rounded and loosely packed. The CCCV anode showed irregular particles, dense surface deposits, and many lithium dendrites longer than about 3 μm. The 5SOC-MSCC-2 anode showed a glued-like structure with reduced porosity and some dendrites around 1 μm. The 10V-MSCC anode remained closer to the fresh state, with clearer edges and tighter but stable packing. These observations support the conclusion that voltage-cutoff stepwise charging is gentler on the electric vehicle battery pack.
| Sample | Separator morphology | Anode morphology | Dendrite observation |
|---|---|---|---|
| Fresh cell | Uniform fiber network, regular pores | Rounded particles, loose packing | None |
| CCCV | Severe collapse, blocked pores | Irregular particles, dense deposits | Many, often longer than 3 μm |
| 5SOC-MSCC-2 | Occasional fiber breakage, uneven pores | Glued-like structure, lower porosity | Few, around 1 μm |
| 10V-MSCC | Mostly regular, slight local loosening | Clear edges, stable packing | Rare and short |
SOH estimation under voltage-cutoff stepwise charging. The standard SOH estimation methods are often developed for low-rate CCCV data. For an electric vehicle battery pack operating under stepwise fast charging, the current, voltage, and thermal signatures are different. I therefore extracted new fast-charging features from the step curves. For sodium-ion 9V-MSCC cells, I analyzed capacity-voltage curves under different cycle numbers. The first three charging steps showed clear separation as aging progressed. The third step had the strongest discrimination, followed by the second and first. The later steps had small current and weak discrimination. I selected the charging times of the first three steps as stepwise features, and I added total charging time, initial DC resistance, and median charging voltage as general features.
| Symbol | Feature for sodium-ion 9V-MSCC |
|---|---|
| H1 | First step charging time |
| H2 | Second step charging time |
| H3 | Third step charging time |
| H4 | Total charging time |
| H5 | Initial charging DC internal resistance |
| H6 | Median charging voltage |
I used Pearson correlation to evaluate the relationship between these features and capacity. The Pearson coefficient is:
$$
p_{X,Y} = \frac{cov(X,Y)}{\sigma_X \sigma_Y} = \frac{E((X-\mu_x)(Y-\mu_y))}{\sigma_X \sigma_Y}
$$
The selected sodium-ion features had absolute correlation values above 0.9 with capacity. I then used a particle swarm optimization back-propagation (PSO-BP) model. The PSO updates velocity and position as follows:
$$
v_{id}^{(t+1)} = u v_{id}^{(t)} + c_1 r_1 (p_{id}^{(t)} – x_{id}^{(t)}) + c_2 r_2 (p_{gd}^{(t)} – x_{id}^{(t)})
$$
$$
x_{id}^{(t+1)} = x_{id}^{(t)} + v_{id}^{(t+1)}
$$
I compared PSO-BP with GA-BP, BP, CNN, ELM, and SVM. The evaluation metrics were RMSE and MAE:
$$
MAE = \frac{1}{n} \sum_{i=1}^{n} |y_i – \hat{y}_i|
$$
$$
RMSE = \sqrt{\frac{1}{n} \sum_{i=1}^{n} (y_i – \hat{y}_i)^2}
$$
For sodium-ion 9V-MSCC, PSO-BP achieved the best results. With a 70% training set, its RMSE was 0.610 and MAE was 0.450. With a 50% training set, its RMSE was 0.678 and MAE was 0.583. The optimized models clearly outperformed standalone BP, CNN, ELM, and SVM.
| 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 |
| CNN | 70% | 2.150 | 2.076 |
| ELM | 70% | 2.571 | 2.024 |
| SVM | 70% | 1.445 | 1.220 |
For lithium-ion 10V-MSCC cells, I extracted features from the first several voltage steps. The first three steps were especially informative. I defined scheme 1 using first-step capacity, first-step ending temperature, second-step capacity, second-step ending temperature, third-step voltage rise, third-step capacity, and third-step ending temperature. I defined scheme 2 by adding the capacities and ending temperatures of steps four, five, and six. I used Spearman correlation because the 800-cycle aging trend was monotonic but not strictly linear. The Spearman coefficient is:
$$
p(X,Y) = \frac{\sum_{i=1}^{n}(x_i-\bar{x})(y_i-\bar{y})}{\sqrt{\sum_{i=1}^{n}(x_i-\bar{x})^2 \sum_{i=1}^{n}(y_i-\bar{y})^2}}
$$
| Symbol | Feature for lithium-ion 10V-MSCC |
|---|---|
| F1 | First step capacity |
| F2 | First step ending temperature |
| F3 | Second step capacity |
| F4 | Second step ending temperature |
| F5 | Third step voltage rise |
| F6 | Third step capacity |
| F7 | Third step ending temperature |
I used a CNN-LSTM model for lithium-ion SOH estimation. The CNN extracts local features from the stepwise charging data, and the LSTM captures temporal dependencies. The LSTM gates are:
$$
f_t = \sigma(W_f [h_{t-1}, x_t] + b_f)
$$
$$
i_t = \sigma(W_i [h_{t-1}, x_t] + b_i)
$$
$$
\tilde{C}_t = \tanh(W_C [h_{t-1}, x_t] + b_C)
$$
$$
C_t = f_t C_{t-1} + i_t \tilde{C}_t
$$
$$
o_t = \sigma(W_o [h_{t-1}, x_t] + b_o)
$$
$$
h_t = o_t \tanh(C_t)
$$
The convolution operation in the CNN can be written as:
$$
y_{i,j}^{l} = f\left(\sum_{m=0}^{M-1}\sum_{n=0}^{N-1} w_{m,n}^{l} x_{i+m,j+n}^{l-1} + b^l\right)
$$
I compared CNN-LSTM, PSO-BP, and standalone LSTM under scheme 1 and scheme 2. CNN-LSTM performed best overall. Under scheme 2, CNN-LSTM achieved RMSE 0.856 and MAE 0.681. Under scheme 1, it achieved RMSE 0.929 and MAE 0.829. PSO-BP was close under scheme 2, with RMSE 0.848 and MAE 0.571, but its accuracy dropped more when only the first three steps were used. Standalone LSTM was clearly worse. The results show that using only the first three steps is sufficient for an early SOH estimate, and the accuracy loss is small, especially for CNN-LSTM. This means the electric vehicle battery pack can obtain a useful SOH estimate within the first part of a fast-charging session.
| Method | 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 |
Synthesis and implications. My experiments led to several clear conclusions for the electric vehicle battery pack. First, stepwise charging is superior to conventional CCCV when it is designed around internal resistance and differential voltage features. Second, voltage-cutoff stepwise charging is more robust than capacity-cutoff stepwise charging because it does not depend on fixed capacity boundaries that shift with aging. Third, the beginning and end of charge are the most sensitive regions. At low SOC, high DCR makes large current harmful. At high SOC, high current across the central DV peak promotes plating and structural damage. Fourth, increasing the number of capacity steps from three to five helps, but increasing from five to ten gives diminishing returns and can increase charging time. Fifth, the extracted stepwise features from the first few charging steps are highly correlated with SOH, and data-driven models such as PSO-BP and CNN-LSTM can estimate SOH accurately under fast-charging conditions.
The electric vehicle battery pack is a safety-critical and cost-critical subsystem. Fast charging must not be treated as a simple current increase. My work shows that a well-designed stepwise protocol can reduce charging time while also improving capacity retention. For sodium-ion cells, 9V-MSCC reduced 10%-80% charging time by 16% and improved 150-cycle SOH by 7.4 percentage points compared with CCCV. For lithium-ion cells, 10V-MSCC improved 800-cycle SOH by 25.4 percentage points compared with CCCV and maintained better separator and anode morphology. These gains are significant for the electric vehicle battery pack because they directly affect range, warranty cost, second-life value, and user confidence.
Limitations and future work. I also note several limitations. The DV curve shifts as the electric vehicle battery pack ages, so a fixed voltage-step table may eventually become suboptimal. Future work should update the voltage thresholds online using impedance or differential voltage monitoring. Sodium-ion cells were tested only up to 150 cycles in my study, and some cells showed abnormal behavior when charged to 4.0 V. A lower upper cutoff may be preferable. Temperature was not fully monitored for sodium-ion cells, and post-mortem analysis was performed mainly on lithium-ion cells. Future work should include thermal imaging, more cycle life, and disassembly of sodium-ion cells. In addition, I estimated SOH only for the voltage-cutoff strategies. The capacity-cutoff strategies also deserve SOH modeling, especially for fleet-level electric vehicle battery pack management. Finally, the electric vehicle battery pack in real vehicles experiences variable temperature, variable depth of discharge, and cell-to-cell inconsistency. These factors should be integrated into future charging and SOH estimation frameworks.
Overall, I have developed and experimentally validated stepwise fast charging strategies for two battery chemistries and demonstrated accurate SOH estimation under the best voltage-cutoff protocols. The results provide a practical path for improving the electric vehicle battery pack charging experience without simply increasing battery size or charger power. By combining internal-resistance-aware current steps, differential-voltage-guided transitions, and data-driven health estimation, I believe this work contributes to safer, faster, and longer-lived electric vehicle battery pack operation.