In the pursuit of sustainable transportation, the integration of hydrogen storage systems with battery electric cars represents a promising avenue for achieving zero-emission and low-carbon driving. As a researcher focused on advancing energy efficiency and safety in electric mobility, I explore a novel control method that harmonizes regenerative braking and anti-lock braking systems (ABS) through game-based coordination of hydrogen storage. This approach aims to optimize energy utilization while ensuring braking safety, addressing critical challenges in modern battery electric car design. The core idea revolves around leveraging hydrogen as an auxiliary energy source to power braking systems, thereby enhancing overall performance. In this article, I present a comprehensive framework, from modeling to experimental validation, detailing how strategic energy allocation can revolutionize braking dynamics in battery electric cars.
The braking system in a battery electric car is a complex interplay between energy recovery and safety mechanisms. Regenerative braking converts kinetic energy into electrical energy during deceleration, feeding it back to the battery, while anti-lock braking prevents wheel lock-up to maintain stability. However, inefficient energy management can lead to excessive consumption or compromised safety. By incorporating a hydrogen storage system, we can supply additional power to both regenerative and hydraulic braking components, but this requires intelligent control to avoid waste. My work introduces a game-based collaborative control strategy that treats the hydrogen storage system as a player in an optimization game, balancing energy release between regenerative and anti-lock braking based on real-time demands. This not only improves energy recovery rates but also ensures robust braking performance under varying conditions.
To lay the foundation for this control strategy, I first develop mathematical models that capture the essential dynamics of a battery electric car during braking. These models include the road surface interaction, wheel behavior, and braking system components, providing critical inputs for subsequent control decisions. The road surface model describes the relationship between adhesion coefficient and slip ratio, which is vital for understanding traction limits. Using the Dugoff model, this relationship can be expressed as a piecewise function:
$$
\lambda = \begin{cases}
\frac{\lambda_h D_c}{D}, & D \leq D_c \\
\lambda_h – \frac{\lambda_g D_c}{1 – D_c} – \frac{\lambda_h – \lambda_g}{1 – D_c} D, & D > D_c
\end{cases}
$$
Here, $\lambda$ denotes the adhesion coefficient, $D$ is the slip ratio, $D_c$ is the critical slip ratio, and $\lambda_h$ and $\lambda_g$ are adhesion coefficients at $D_c$ and 100% slip, respectively. This model helps predict how much braking force can be applied without causing wheel slip, a key factor in anti-lock braking control for battery electric cars.
Next, the wheel model accounts for rotational dynamics, ignoring air and rolling resistance for simplicity. The equations governing wheel motion are:
$$
\begin{cases}
Q\dot{v} = -\lambda G_N \\
O\dot{\xi} = \lambda G_N T – Q_b \xi / \dot{v}
\end{cases}
$$
where $Q$ is the total vehicle mass, $\dot{v}$ is the vehicle acceleration, $G_N$ is the ground reaction force, $O$ is the wheel moment of inertia, $T$ is the wheel radius, $\xi$ is the wheel angular velocity, and $Q_b$ is the brake torque. This model outputs the wheel angular deceleration $\dot{\xi}$, which informs slip ratio allocation in later stages.
The braking system model comprises two subsystems: hydraulic braking and regenerative braking. The hydraulic brake actuator converts brake signals into force, with a transfer function $G(s) = 40 / (0.5s + 1)$, where $s$ is the brake signal. This subsystem requires electrical energy to operate, which can be supplied by the hydrogen storage system via a fuel cell. The regenerative braking system generates torque through motor regeneration, described by:
$$
\begin{cases}
n = n_0 – \chi Y D / G(s) \\
n_0 = I / V_e \phi \\
\chi = T_a / V_e V_Y \phi^2
\end{cases}
$$
Here, $n$ is the motor speed, $n_0$ is the ideal no-load speed, $Y$ is the torque, $V_e$ is the back EMF constant, $\phi$ is the main magnetic flux, $T_a$ is the armature resistance, and $V_Y$ is the electromagnetic constant. When $n_0$ decreases relative to wheel speed, negative torque indicates regenerative braking, recovering energy. The output is the real-time regenerative braking torque $Y_{\text{reg}}$, crucial for force distribution. These models collectively provide parameters like wheel angular deceleration and regenerative torque, enabling precise control in battery electric cars.
Understanding driver intent is essential for tailoring braking responses. I employ a fuzzy logic system to recognize braking intensity based on accelerator pedal displacement and vehicle speed. These inputs are normalized and fed into a multi-input single-output fuzzy inference model. Pedal displacement $S_b$ and speed $N_b$ are fuzzified into three subsets each, allowing the system to interpret emergency levels. The fuzzy controller outputs the motor regenerative braking torque $Y_{\text{reg-acc}}$:
$$
\begin{cases}
Y_{\text{reg-acc}} = Y_0 + \omega t \\
\omega = \text{fuzzy}(x_{cc}, dS_{cc}/dt, X_{\alpha})
\end{cases}
$$
where $Y_0$ is the initial torque, $t$ is time, $x_{cc}$ is the braking intensity demand, $dS_{cc}/dt$ is the pedal velocity, $X_{\alpha}$ is the pedal displacement, and $\text{fuzzy}(\cdot)$ denotes the fuzzy function. Rules are set such that lower $x_{cc}$ or $dS_{cc}/dt$ yields smaller $\omega$, aligning torque with driver needs. This intent recognition ensures that the battery electric car responds appropriately to different driving states, optimizing energy recovery and safety.
With models and intent recognition in place, I propose a hierarchical game-based collaborative control strategy for the hydrogen storage system. This strategy operates across three layers: upper, middle, and lower control. The upper layer allocates braking force and slip ratio between front and rear wheels based on vehicle dynamics. The braking force distribution is derived from the regenerative torque output:
$$
\begin{cases}
G_{x,f}(s) = \left[ N_f + \frac{x j_g \chi Y_{\text{reg}}}{Z (n_0 – n)} Q D_f \right] \\
G_{x,r}(s) = \left[ N_r + \frac{x j_g \chi Y_{\text{reg}}}{Z (n_0 – n)} Q D_r \right]
\end{cases}
$$
where $G_{x,f}$ and $G_{x,r}$ are front and rear braking forces, $N_f$ and $N_r$ are static axle loads, $x$ is the braking level, $j_g$ is the center of mass position, $Z$ is the wheelbase, and $D_f$ and $D_r$ are slip ratios. Slip ratio allocation follows a curve divided into segments: OA, AB, and BC. For segment OA, the front slip ratio is:
$$
D_{f,A} = \frac{N_f \dot{\xi} + x_{\text{req}} j_g}{x_{\text{req}} Z}, \quad D_r = 0
$$
For segment AB:
$$
\begin{cases}
D_{f,A,B} = \frac{N_f \dot{\xi} + x_{\text{req}} j_g}{D_{f,A} Z} \\
D_{r,A,B} = \frac{N_r \dot{\xi} + x_{\text{req}} j_g}{x_{\text{req}} Z}
\end{cases}
$$
For segment BC, $D_f = D_r = 1 / x_{\text{req}}$, where $x_{\text{req}}$ is the required braking intensity from intent recognition. The braking intensity for segment AB, $x_{AB}$, is calculated as:
$$
x_A = \frac{\lambda Y_{\text{max}}}{(D_{f,A,B} + D_{r,A,B}) Q N_f (G_{x,f}(s) + G_{x,r}(s))}
$$
with $\lambda$ defined based on road peak adhesion coefficient $\alpha$:
$$
\lambda = \begin{cases}
0, & 0 \leq \alpha \leq 0.4 \\
5\alpha – 0.4, & 0.4 < \alpha < 0.6 \\
1, & \alpha \geq 0.6
\end{cases}
$$
Here, $Y_{\text{max}}$ is the maximum regenerative torque. This yields the motor regenerative braking torque $Y_{\text{reg-AB}}$ for the middle layer.
The middle layer adjusts regenerative braking intensity and anti-lock braking parameters using $Y_{\text{reg-AB}}$. Regenerative intensity $Q_p$ is scaled by a coefficient $\epsilon$: $Q_p = \epsilon \cdot Y_{\text{reg-AB}}$. For anti-lock braking, the hydraulic pressure $O_c$ is modulated to control front wheel slip ratio:
$$
O_c = \frac{R}{\mu Y_{\text{reg-AB}}}
$$
where $R$ is the brake disc radius and $\mu$ is the friction coefficient. A sliding mode control ensures robustness, with a sliding surface $d_f = D_f – D_{f,\text{req}}$, where $D_{f,\text{req}}$ is the desired slip ratio. The control law adjusts the slip ratio $u_m$:
$$
u_m = b_m^{-1} \left[ -O_c (\mu – d_f) – T_{\epsilon} l_m \text{sat}(d_f) \right]
$$
where $b_m^{-1}$ is the front wheel mass inverse, $l_m$ is a proportionality coefficient, and $\text{sat}(\cdot)$ is a saturation function. These adjustments dictate how the hydrogen storage system supplies gas to the braking subsystems.
The lower layer executes gas supply decisions based on $Y_{\text{reg-AB}}$. The hydrogen release rates for regenerative braking $P_w$ and hydraulic braking $P_v$ are:
$$
\begin{cases}
P_w = \frac{O_P}{\eta} \cdot Y_{\text{reg-AB}}, & Y_{\text{reg-AB}} > Y_0 \\
P_v = \frac{u_m O_c \xi}{\eta Y_{\text{reg-AB}}}, & Y_{\text{reg-AB}} \leq Y_0
\end{cases}
$$
where $O_P$ is a pressure constant and $\eta$ is a conversion coefficient. When regenerative torque is high, more hydrogen is allocated to regenerative braking to boost energy recovery; otherwise, hydraulic braking receives more to enhance safety. This game-based allocation optimizes hydrogen usage, ensuring efficient energy release for the battery electric car.
To validate this control method, I conducted experiments in a realistic setting using a solar-powered fast-charging battery electric car. The test platform included a hydrogen storage system and fuel cell, with key parameters summarized in Table 1.
| Parameter | Value |
|---|---|
| Hydrogen storage capacity (kg) | 10 |
| Storage pressure (MPa) | 70 |
| Pressure test cycle (years) | 4 |
| Input hydrogen purity (%) | 99.9 |
| Efficiency (%) | 60 |
| Hydrogen absorption rate (kg/s) | 0.0333 |
| Hydrogen supply rate (kg/s) | 0.0167 |
| Power density (kW/L) | 4 |
| Energy density (kWh/L) | 2 |
| Fuel cell response time (s) | 1 |
I evaluated the method based on braking intent recognition accuracy, energy recovery rate, and slip ratio performance. For intent recognition, tests under mild, moderate, and emergency braking scenarios from an initial speed of 60 km/h showed that the fuzzy system accurately identified driver intentions, with smooth and stable braking trajectories. This confirms the reliability of torque estimation for the battery electric car.
Energy recovery rates were measured across varying speeds, as shown in Table 2. Compared to existing methods from literature, my approach consistently achieved higher recovery rates, with minimal fluctuation, demonstrating superior energy utilization for the battery electric car.
| Braking speed (km/h) | Proposed method (%) | Literature method 1 (%) | Literature method 2 (%) | Literature method 3 (%) |
|---|---|---|---|---|
| 20 | 60.25 | 40.25 | 42.88 | 51.08 |
| 30 | 59.77 | 38.88 | 41.37 | 49.74 |
| 40 | 59.02 | 37.48 | 40.58 | 48.25 |
| 50 | 58.77 | 36.78 | 39.88 | 47.01 |
| 60 | 58.12 | 36.03 | 39.23 | 45.18 |
| 70 | 57.89 | 35.55 | 38.79 | 44.02 |
| 80 | 57.22 | 35.01 | 38.24 | 42.08 |
| 90 | 56.96 | 34.74 | 37.68 | 41.17 |
| 100 | 56.59 | 34.33 | 37.05 | 40.01 |
The proposed method maintained recovery rates above 56.59%, with variations under 3.7 percentage points, indicating stability. In contrast, other methods showed larger fluctuations, up to 11.07 percentage points, highlighting their inefficiency in managing energy for the battery electric car.
For emergency braking, slip ratio control was tested. The results, illustrated in Figure 9, reveal that my method quickly stabilized the slip ratio near the optimal value within 0.2 seconds, minimizing speed deviations and preventing wheel lock-up. This contrasts with slower responses and oscillations in comparative methods, proving the robustness of the game-based control in ensuring safety for the battery electric car.
The effectiveness of this approach stems from its holistic integration of models, intent recognition, and game-theoretic allocation. By treating the hydrogen storage system as an active participant, I achieve a balance between energy recovery and braking force, tailored to real-time conditions. This is particularly beneficial for battery electric cars, where energy efficiency directly impacts range and performance. The hierarchical control structure allows for precise adjustments, from high-level force distribution to low-level gas release, optimizing every aspect of the braking process.
However, challenges remain. The models rely on assumptions that may be affected by factors like tire wear or road condition changes, potentially impacting control accuracy. Future work could involve adaptive calibration using sensor data or machine learning to enhance model precision. Additionally, expanding the game-based framework to include predictive elements, such as traffic patterns or terrain, could further improve energy management for battery electric cars.
In conclusion, my game-based collaborative control strategy for hydrogen storage systems in battery electric car regenerative anti-lock braking demonstrates significant advantages in energy recovery, stability, and robustness. Through detailed modeling, intent recognition, and strategic energy allocation, I have developed a method that not only safeguards braking safety but also maximizes hydrogen utilization. Experimental validation confirms its superiority over existing approaches, with high recovery rates and rapid slip ratio stabilization. This work contributes to the advancement of battery electric car technology, offering a pathway toward more efficient and secure electric mobility. As the automotive industry evolves, such integrated control systems will play a crucial role in realizing sustainable transportation goals.