An Energy-Efficient Lyapunov-Based Cooperative Adaptive Cruise Controller for Electric Vehicles

This paper proposes a novel, energy-efficient Lyapunov-based Cooperative Adaptive Cruise Control (CACC) strategy for electric vehicle platoons, utilizing a third-order dynamic model derived from real-world data to achieve significant energy savings (up to 38.5%) and improved string stability compared to conventional methods.

The Gist

Imagine a group of electric cars (EVs) driving down a highway, not just following the car in front of them, but holding hands via a digital nervous system. This is Cooperative Adaptive Cruise Control (CACC). Instead of just reacting to the car ahead, they talk to each other, sharing speed and distance data instantly to move as a single, fluid unit—a "platoon."

However, there's a problem. Most existing control systems treat electric cars like old gas-powered cars. They forget that electric cars have a superpower: Regenerative Braking. When an EV slows down, it doesn't just waste energy as heat; it acts like a generator, catching that energy and putting it back in the battery.

This paper introduces a new "brain" for these electric platoons that understands this unique superpower. Here is the breakdown in simple terms:

1. The Problem: The "Gas Car" Mistake

Think of driving a gas car like riding a bicycle with a heavy flywheel. When you pedal (accelerate), it takes effort. When you stop pedaling, it coasts. When you brake, you just stop, and the energy is gone.

Electric cars are different. They are like a hybrid bicycle with a battery. When you pedal, you charge the battery. When you stop pedaling and let the wheels spin the motor, the battery charges even more.

Previous controllers treated EVs like the heavy flywheel bike. They didn't account for the fact that slowing down is actually a way to gain energy. This led to jerky driving and wasted power.

2. The Solution: A New "Third-Order" Model

The authors built a new mathematical model of an electric car (specifically a Ford Mustang Mach-E) based on real-world experiments.

• The Analogy: Imagine trying to teach a robot to dance. If you only tell it "move forward" and "stop," it will look clumsy. But if you teach it "move forward with a specific rhythm" and "stop with a specific glide," it dances beautifully.
• The Innovation: This new model is "third-order," meaning it looks at three things at once: Position, Speed, and Acceleration. Crucially, it treats "pushing the gas" and "slowing down for regen" as two completely different dance moves, rather than just opposite sides of the same coin.

3. The Controller: The "Lyapunov" Safety Net

The paper proposes a new controller based on something called Lyapunov stability.

• The Analogy: Imagine a marble rolling in a bowl. No matter where you drop the marble, gravity (the Lyapunov function) ensures it eventually rolls to the very bottom and stays there. It can't escape the bowl.
• How it works: The controller acts like that gravity. It constantly checks the "energy" of the error (how far off the car is from the perfect distance). It uses math to prove that the cars will always settle into a smooth, stable line, no matter how the leader car speeds up or slows down.
• The Benefit: Because this "gravity" is so strong, the cars can drive much closer together (smaller "headway time") without crashing. This means more cars can fit on the road, and they move more efficiently.

4. The Results: Smoother, Safer, Cheaper

The team tested this in two ways: computer simulations and real experiments with a car on a track.

• The "String Stability" Test: Imagine a line of dominoes. If you push the first one, do they all fall smoothly, or does the line start shaking and falling apart?
  • Old Controllers: The shaking got worse as it went down the line (like a ripple effect).
  • New Controller: The line stayed perfectly smooth, even when the cars were very close together.
• The Energy Savings: Because the cars didn't have to brake and accelerate as wildly, they saved a massive amount of energy.
  • The Stat: The new system saved up to 38.5% more energy compared to the standard system.
  • The Metaphor: It's like the difference between a driver who slams on the brakes and then floors it again, versus a driver who glides smoothly. The glider gets to the destination with a full tank; the slammer arrives with an empty one.

5. Why This Matters

As more electric and self-driving cars hit the road, we need them to work together efficiently.

• Safety: They can drive closer together without crashing.
• Range Anxiety: They can drive further on a single charge because they aren't wasting energy on jerky movements.
• Traffic Flow: More cars can fit on the highway without causing traffic jams.

In a nutshell: This paper gave electric cars a new "brain" that understands how to dance with their own batteries. By treating acceleration and regenerative braking as distinct moves, the cars can drive in a tight, smooth, energy-saving formation that saves nearly 40% of their energy compared to current technology.

Technical Summary

1. Problem Statement

The paper addresses the critical need to enhance the energy efficiency of Electric Vehicle (EV) platoons operating under Connected and Automated Vehicle (CAV) frameworks. While Cooperative Adaptive Cruise Control (CACC) improves traffic flow and safety, existing algorithms often fail to account for the unique dynamics of EVs, specifically:

• Distinct Acceleration/Deceleration Dynamics: EVs utilize Regenerative Braking Systems (RBS) for deceleration, which differs fundamentally from the friction braking used in Internal Combustion Engine Vehicles (ICEVs). Conventional models often treat acceleration and deceleration as symmetric or ignore RBS entirely.
• Energy vs. Stability Trade-off: Many existing CACC strategies prioritize safety or string stability (preventing oscillation amplification down the platoon) without explicitly optimizing for energy consumption. Conversely, energy-saving controllers often lack rigorous stability guarantees.
• Lack of Practical Models: There is a gap in practical, third-order dynamic models derived from real-world experimental data that explicitly separate motoring and regenerative braking phases.

2. Methodology

A. Vehicle Modeling

The authors developed a third-order switched dynamic model for EVs, derived from real-world experimental data using a Ford Mustang Mach-E.

• State Variables: Position ($x$), velocity ($v$), and acceleration ($a$).
• Switched Dynamics: The model distinguishes between acceleration ($u \geq 0$) and deceleration ($u < 0$) phases using signum functions. The dynamics are defined as:
  $$\dot{a}_i(t) = -\gamma_i(u_i)a_i(t) + \beta_i(u_i)u_i(t)$$
  where $\gamma$ and $\beta$ are distinct positive constants identified experimentally for motoring ($\gamma_1, \beta_1$) and regenerative braking ($\gamma_2, \beta_2$).
• Filippov Framework: Due to the discontinuity at the switching surface ($u=0$), the system is treated as a switched nonsmooth system, analyzed using Filippov differential inclusions to ensure mathematical rigor.

B. Controller Design

A novel Lyapunov-based CACC controller was designed to ensure safety, string stability, and energy efficiency.

• Error Definition: The controller minimizes distance error ($e_i$) and its derivatives ($\dot{e}_i, \ddot{e}_i$) relative to a desired headway time ($b_i$).
• Control Law: The control input is derived using a backstepping-like approach with auxiliary error signals ($r_{i1}, r_{i2}$). The control law for the filtered signal $P_i$ is:
  $$P_i = (\alpha_{i1} + \alpha_{i2})e_{i3} + \beta_i C_i r_{i2} + \beta_{i-1}u_{i-1} + (\alpha_{i1}\alpha_{i2} + 1)r_{i1} - \alpha_{i2}\alpha_{i1}^2 e_{i1} - \phi_i$$
  where $\alpha$ and $C$ are user-specified gains, and $\phi_i$ represents disturbance terms from the leader.
• Stability Proof: The authors utilized a Lyapunov function candidate and the Filippov framework to prove asymptotic stability and time-domain string stability. They demonstrated that the system avoids Zeno behavior (infinite switching in finite time) due to physical constraints on the control input.

C. Evaluation Metrics

• String Stability: Evaluated using a time-domain criterion ($||\Omega_i||_2 \leq 1$), measuring the ratio of velocity norms between followers and leaders. This was chosen over frequency-domain methods due to the difficulty of applying them to switched systems.
• Energy Efficiency: Calculated by integrating real-time power consumption over a drive cycle, accounting for aerodynamic drag, rolling resistance, acceleration, gravity, and regenerative braking recovery.

3. Key Contributions

1. Realistic EV Model: Proposed a third-order switched dynamic model explicitly separating motoring and regenerative braking dynamics, identified via extensive experimental testing (25 motoring and 40 deceleration trials).
2. Novel Lyapunov-Based CACC: Designed a controller that guarantees string stability and safety while requiring lower control gains than conventional methods. It ensures stability at smaller headway times ($b=0.5s$) compared to typical CACC platoons ($0.6-1.5s$).
3. Filippov Stability Analysis: Provided a rigorous stability proof for the switched EV dynamics using the Filippov framework, addressing the discontinuities inherent in regenerative braking switching.
4. Experimental Validation: Validated the controller in both simulation and a Vehicle-in-the-Loop (VIL) environment, demonstrating practical feasibility.

4. Results

The proposed controller was tested against a baseline PID-based CACC using the aggressive US06 drive cycle and a custom test scenario.

• String Stability:

  • The proposed controller maintained string stability (criterion $\leq 1$) at a headway time of 0.5 seconds.
  • In contrast, the baseline controller showed unstable behavior (criterion increasing down the platoon) and required larger headways to stabilize.
  • At $b=0.5s$, the proposed controller achieved an average string stability criterion of 0.9999, whereas the baseline was 1.3638 (unstable).

• Energy Efficiency:

  • The proposed controller reduced total energy consumption by 38.5% compared to the baseline controller in a 10-vehicle platoon.
  • This improvement is attributed to smoother velocity profiles, reduced unnecessary braking, and optimized utilization of regenerative braking.
  • The reduction in velocity fluctuations directly correlated with lower energy expenditure.

• Headway Time:

  • The controller successfully stabilized a platoon at 0.5s headway, significantly lower than the 0.6–1.5s typically required for stable CACC. This implies a potential increase in traffic capacity.

5. Significance

This paper bridges a critical gap between theoretical control design and practical EV deployment. By explicitly modeling the asymmetric nature of EV acceleration and regenerative braking, the proposed controller achieves a level of energy efficiency (38.5% savings) that significantly outperforms existing literature (which typically reports 10–20% savings for EVs).

The ability to maintain string stability at lower headway times suggests that this approach can increase road capacity without compromising safety. Furthermore, the use of the Filippov framework provides a robust mathematical foundation for controlling switched systems, which is essential for the safe deployment of autonomous EVs in mixed traffic environments. The experimental validation confirms that these theoretical gains are achievable in real-world hardware configurations.