An enhanced optimal velocity model for car-following with connected and autonomous vehicles

4.1. Optimal velocity model

The OVM provides a theoretical framework for determining the desired speed of a vehicle based on its spacing from the vehicle ahead. This model assumes that each vehicle seeks to maintain a speed that optimizes traffic flow and minimizes the risk of collisions. The desired velocity is represented as a function of spacing, with parameters that define the sensitivity to changes in spacing and relative speed. The OVM is particularly useful for studying traffic stability and the conditions under which traffic jams or stop-and-go waves occur. By integrating the OVM into simulations, researchers can design control strategies that enhance traffic efficiency and reduce fuel consumption. The Bando OVM provides acceleration dynamics^[13]:

where:

• $$ a(t) $$ is the acceleration of the following vehicle (m/s²);

• $$ v(t) $$ is the velocity of the following vehicle (m/s);

• $$ V(s) $$ is the optimal velocity determined by inter-vehicle spacing (m/s);

• $$ s $$ is the spacing between the leader and follower vehicles (m);

• $$ \Delta v=v_{1} -v $$ is the relative velocity between the leader and follower (m/s);

• $$ \alpha $$ and $$ \beta $$ are sensitivity parameters representing driver response to spacing and relative speed, respectively.

The optimal velocity is calculated as^[13]:

where:

• $$ V_{m} $$ is the maximum allowable velocity (m/s);

• $$ s_{0} $$ is the characteristic spacing parameter (m);

• $$ s^{*} $$ is the safety spacing threshold (dimensionless);

• $$ tanh(·) $$ denotes the hyperbolic tangent function.

This model captures the relationship between spacing and speed, ensuring that vehicles maintain safe distances while optimizing traffic flow. By tuning the parameters, the OVM can be adapted to different traffic scenarios, including high-density and free-flow conditions.

4.2. Modeling a single vehicle

Modeling the behavior of a single vehicle in a simulation involves capturing parameters such as speed (ν), inter-vehicle distance (s), and relative speed (Δν). These parameters are utilized to calculate the vehicle’s acceleration through a car-following model, which is typically represented as an ordinary differential equation (ODE). The acceleration is then integrated over time to derive the vehicle’s velocity and position. A frequently employed technique is Euler’s forward method, which iteratively updates velocity and spacing at each time step. This approach provides a simple numerical approximation of the ODE, making it well-suited for real-time simulations. However, selecting the appropriate time step (ΔT) is crucial, as larger increments might compromise accuracy and cause instability. By simulating a single vehicle, we can gain insights into its fundamental dynamics, including patterns of acceleration and deceleration across different traffic situations. The acceleration of the vehicle is modeled as:

where:

• $$ a(t_{k}) $$- acceleration of the vehicle at time $$ t_{k} $$;

• $$ f\left(v\left(t_{k}\right), s\left(t_{k}\right), \Delta v\left(t_{k}\right)\right) $$- acceleration function of the following vehicle, dependent on:

$$ v(t_{k}) $$- velocity of the following vehicle at time $$ t_{k} $$;

$$ s(t_{k}) $$- spacing (distance) between the leading and following vehicle at time $$ t_{k} $$;

$$ \Delta v(t_{k}) $$- relative velocity between the leading and following vehicle at time $$ t_{k} $$, defined as $$ \Delta v(t_{k})= v_{l}(t_{k})-v(t_{k}) $$.

This equation models how the vehicle adjusts its acceleration based on its current state and the state of the vehicle ahead. The function $$ f $$ typically depends on car-following models, such as the Optimal Velocity.

Using Euler’s forward method, the velocity and spacing are updated iteratively:

Here, $$ v(t_{k}) $$ represents the velocity at time $$ t_{k} $$, and $$ \Delta T $$ is the time step interval. These equations provide a numerical approximation of the vehicle’s dynamics, allowing us to simulate its behavior over time. This approach is particularly useful for evaluating how a single vehicle reacts to various traffic conditions, such as sudden braking or acceleration by the lead vehicle.

4.3. Simulating a string of vehicles

Extending the single-vehicle model to a string of vehicles provides insights into traffic flow and the collective behavior of vehicles in a platoon. Each vehicle in the string, except the lead, determines its acceleration based on the state of the vehicle directly ahead. The spacing between vehicles is influenced by factors such as the length of the vehicles and their relative velocities. This approach captures the interactions within a traffic stream, enabling the study of emergent phenomena such as traffic waves, congestion, and stability. For example, by simulating a homogeneous platoon of vehicles, researchers can observe how disturbances, such as sudden braking by the lead vehicle, propagate through the string. Such simulations are essential for designing Advanced Driver Assistance Systems (ADAS) and CAV algorithms that prioritize safety and efficiency. In a string of vehicles, each vehicle (except the lead) depends on the state of the vehicle ahead. The dynamics for the i-th vehicle are expressed as:

where:

• $$ a_{i}(t_{k}) $$- acceleration of vehicle $$ i $$ at time $$ t_{k} $$;

• $$ v_{i}(t_{k}) $$- velocity of vehicle $$ i $$ at time $$ t_{k} $$;

• $$ s_{i}(t_{k}) $$- spacing between vehicle $$ i $$ and its preceding vehicle at time $$ t_{k} $$;

• $$ \Delta v_{i}(t_{k})=v_{i-1}(t_{k})-v_{i}(t_{k}) $$- relative velocity with the preceding vehicle;

• $$ s_{i-1}(t_{k}) $$- position of the preceding vehicle $$ i-1 $$ at time $$ t_{k} $$;

• $$ L_{i} $$- length of vehicle $$ i $$;

• $$ \Delta T $$- simulation time step;

• $$ v_{i}(t_{k+1}) $$- velocity of vehicle $$ i $$ at the next time step.

By extending this model to multiple vehicles, we can study traffic phenomena such as congestion and the propagation of traffic waves. Moreover, a sudden deceleration by the lead vehicle can create a ripple effect, influencing the behavior of all following vehicles. Understanding these dynamics is critical for designing algorithms that enhance traffic stability and efficiency.

Theoretically, to assess the stability of a vehicle platoon, the linearized dynamics of the enhanced OVM around the equilibrium spacing s_e and velocity v_e can be represented by the transfer function between successive vehicles^[1,11]:

where:

• $$ G(s) $$- transfer function from vehicle $$ i-1 $$ to vehicle $$ i $$;

• $$ \tilde{v}_{i}(s) $$- Laplace transform of the velocity deviation of vehicle i from equilibrium;

• $$ \tilde{v}_{i-1}(s) $$- Laplace transform of the velocity deviation of the preceding vehicle $$ i-1 $$;

• $$ \alpha ,\beta $$- sensitivity parameters of the enhanced OVM;

• $$ V^{\prime}(s_{e}) $$- derivative of the optimal velocity function evaluated at the equilibrium spacing $$ s_{e} $$;

• $$ s $$- Laplace variable.

A sufficient condition for string stability, ensuring that small velocity or spacing disturbances do not amplify along the platoon, is given by^[1,11]:

where:

• $$ s_{e} $$ is the equilibrium spacing (m);

• $$ v_{e} $$ is the equilibrium velocity (m/s);

• $$ V^{\prime}(s_{e}) $$ is the derivative of the optimal velocity function evaluated at $$ s_{e} $$;

• $$ \alpha $$ and $$ \beta $$ are sensitivity parameters as previously defined.

This condition provides a compact analytical criterion to evaluate whether the vehicle string is expected to exhibit stable or unstable behavior under the chosen model parameters.

4.4. Model calibration

Model calibration ensures that the simulation accurately reflects real-world behavior by tuning model parameters to minimize discrepancies between simulated and observed data. This process typically involves defining an objective function, such as the root-mean-square error (RMSE), which quantifies the difference between simulated and measured velocities or spacings. Optimization techniques are then used to find the parameter set that minimizes this error. Calibration is particularly important for validating car-following models, as it allows for the adjustment of parameters such as sensitivity to spacing and relative speed. Calibration adjusts model parameters to minimize the error between simulated and observed data. The objective function is^[20]:

where:

• $$ e(\theta) $$- root-mean-square error of vehicle velocities;

• $$ \theta $$- vector of model parameters to be calibrated (e.g., sensitivity to spacing, sensitivity to relative velocity, reaction time);

• $$ v_{m}(t) $$- measured velocity of the vehicle at time $$ t $$;

• $$ v(t) $$- simulated velocity of the vehicle at time $$ t $$;

• $$ T $$- total simulation period;

• $$ dt $$- infinitesimal time increment used in the integral.

Alternatively, calibration can focus on spacing^[20]:

where:

• $$ e(\theta) $$- root-mean-square error of vehicle spacing;

• $$ \theta $$- vector of model parameters to be calibrated (e.g., sensitivity to spacing, sensitivity to relative velocity, reaction time);

• $$ s_{m}(t) $$- measured spacing between the following and preceding vehicles at time $$ t $$;

• $$ s(t) $$- simulated spacing between the following and preceding vehicles at time $$ t $$;

• $$ T $$- total simulation period;

• $$ dt $$- infinitesimal time increment used in the integral.

Here, $$ s_{m}(t) $$ represents the measured spacing. This approach is particularly useful when precise spacing data is available, allowing for more accurate modeling of vehicle interactions. Calibration is an iterative process that ensures the model accurately captures the dynamics of real-world traffic.

The parameters of the enhanced OVM were estimated using the HighD dataset, which provides detailed time-series information on vehicle positions, speeds, and spacings. Initial parameter values, including sensitivity coefficients (α, β), optimal velocity parameters, and reaction time characteristics, were derived from statistical analysis of vehicle spacing, speed, and relative velocity across multiple driving scenarios. These initial values provided a physically meaningful starting point for calibration. Parameters were then refined through an iterative simulation-based process, where multiple combinations of parameters were systematically evaluated. For each combination, the RMSE between simulated and observed vehicle trajectories, including velocities and spacings, was computed. The parameter set that minimized the RMSE was selected as the final calibrated model. This approach ensures that the calibrated parameters not only reproduce realistic driving behavior but also maintain physical interpretability and robustness across different traffic conditions.

4.5. Communication delay

It is a key consideration in connected vehicle systems, where information from the lead vehicle or other sources may not be instantaneously available. This delay introduces a lag in the response of the following vehicles, affecting their acceleration and spacing decisions. The delay is modeled by incorporating a time lag (t_d) into the equations governing acceleration and spacing. Understanding the impact of communication delay is crucial for designing robust control algorithms that maintain stability and prevent collisions, even in the presence of latency. Simulating communication delay also helps evaluate the performance of V2V and V2I communication systems under different network conditions. Communication delay introduces a lag in the transmitted information, affecting acceleration and spacing:

where:

• $$ a(t_{k}) $$- acceleration of the following vehicle at time $$ t_{k} $$;

• $$ s(t_{k}-t_{d}) $$- spacing between the following and preceding vehicles using the delayed leader information;

• $$ \Delta v(t_{k}-t_{d})=v_{l}(t_{k}-t_{d})-v(t_{k}-t_{d}) $$- relative velocity using the delayed leader information;

• $$ v_{l}(t_{k}-t_{d}) $$- velocity of the lead vehicle received after a communication delay $$ t_{d} $$;

• $$ t_{d} $$- communication delay from the lead vehicle.

Here, $$ t_{d} $$ represents the delay in receiving information from the lead vehicle. This delay can cause a lag in the vehicle’s response, potentially affecting stability and safety.

Therefore, the spacing dynamics should be expressed using the delayed leader velocity while keeping the follower’s own velocity at the current time. The physically consistent discrete form is given by:

This formulation aligns with the physical definition $$ \dot{s}(t)=v_{1}(t)-v(t) $$, ensuring that the change in spacing accurately represents the instantaneous difference between the leader and follower velocities. The delay term $$ t_{d} $$ only affects the leader’s information, as it is received through V2V communication, while the follower’s speed is measured locally without delay. When the delay duration does not correspond exactly to the simulation time step, $$ v_{1}(t_{k}-t_{d}) $$ is obtained through linear interpolation between available samples. This corrected formulation maintains the physical validity of the model and prevents inconsistencies that could arise from using a fully delayed relative velocity term.

4.6. Energy efficiency

Energy consumption is a critical factor in evaluating the efficiency of vehicle systems. It is influenced by the vehicle’s velocity, acceleration, and resistive forces such as rolling resistance and aerodynamic drag. The energy consumption model integrates these factors over time to compute the total energy required for a given driving scenario. In practical terms, this involves accounting for the energy expended during acceleration, deceleration, and maintaining a constant speed. By incorporating energy consumption into the simulation, we can assess the impact of driving behaviors and control strategies on fuel efficiency and emissions. This is particularly relevant for autonomous and connected vehicles, which aim to optimize energy use while maintaining safety and performance. Energy consumption is influenced by resistive forces and inertial effects. It is modeled as^[21]:

where:

• $$ E_{c} $$ is the cumulative energy consumption (J);

• $$ C_{r} $$ is the rolling resistance coefficient;

• $$ C_{a} $$ is the aerodynamic drag coefficient;

• $$ m $$ is the vehicle mass (kg);

• $$ v $$ is vehicle velocity (m/s);

• $$ a $$ is vehicle acceleration (m/s²).

Equation (16) integrates the energy required to overcome resistive forces and maintain motion over time. In discrete form, it can be approximated as:

where:

• $$ v_{k} $$- velocity of the vehicle at the $$ k $$-th time step;

• $$ a_{k} $$- acceleration of the vehicle at the $$ k $$-th time step;

• All other parameters $$ (C_{r},C_{a},m) $$ are as defined above.

This formulation is useful for practical simulations, where continuous integration may not be feasible. By analyzing energy consumption, we can evaluate the efficiency of different driving strategies and their impact on fuel economy.

5.1. Optimal velocity model analysis

The optimal velocity (OV) model exhibits persistent instability in car-following behavior, producing clear stop-and-go waves throughout the simulation. As shown in Figure 1 (0-100 s), the system quickly evolves into a limit cycle, with vehicle velocity oscillating around 28-30 m/s in a regular 20-25 s period. This reflects the sensitivity parameter exceeding the critical stability threshold, causing smooth traffic flow to collapse into sustained oscillations. Figure 2 (150-240 s) confirms that these fluctuations are not transient; the amplitude and period remain consistent, demonstrating stable, recurring velocity waves. By the long-term stage (Figure 3, 300-440 s), the identical oscillatory pattern continues without decay or divergence, indicating both the robustness of the OV model and the reliability of the numerical integration scheme. In summary, the simulation reproduces phantom traffic jams with high accuracy, showing a persistent oscillatory regime where vehicle velocity cycles within the 28-30 m/s range, never settling into a steady state.

Figure 1. Emergence of velocity instability in the OVM from initial conditions. OVM: Optimal velocity model.

Figure 2. Time Evolution of OVM demonstrating sustained stop-and-go waves. OVM: Optimal velocity model.

Figure 3. Long-term behavior of OVM showing persistent oscillatory instability. OVM: Optimal velocity model.

Figure 4. Comparison of actual and simulated velocity showing high-fidelity tracking. RMSE: Root-mean-square error.

Figure 5. Velocity tracking with introduced error under dynamic conditions. RMSE: Root-mean-square error.

Figure 6. Sustained velocity tracking error during prolonged operation. RMSE: Root-mean-square error.

Figure 7. Stable car-following with damped oscillatory decay.

Figure 8. Amplification of a deceleration pulse, indicating string instability.

Figure 9. Sustained high-amplitude limit-cycle oscillations in the vehicle string.

Figure 10. Simulated vs. actual vehicle spacing underestimation during deceleration. RMSE: Root-mean-square error.

Figure 11. Simulated and actual vehicle spacing showing strong agreement and high fidelity. RMSE: Root-mean-square error.

Figure 12. Simulated vs. actual spacing over an extended period during oscillatory traffic flow. RMSE: Root-mean-square error.

Figure 13. High-Frequency Tracking Jitter from Minimal Latency.

Figure 14. Overshoot Response from Moderate Communication Latency.

Figure 15. Large-Amplitude Oscillations from Significant Communication Delay.

Figure 16. Baseline cumulative energy consumption.

Figure 17. Cumulative energy during sustained operation.

Figure 18. Detail of a high-power transient event.

Figure 19. Performance comparison of the OVM, IDM, and Enhanced OVM models. RMSE: Root-mean-square error; OVM: optimal velocity model; IDM: intelligent driver model.

5.2. Analysis of modeling a single vehicle

The model was evaluated over three consecutive intervals using RMSE as the primary metric. In the first period (0-100 s, Figure 4), the model shows strong performance, with a low RMSE of 0.74 m/s, indicating well-calibrated parameters and minimal disturbance effects. However, accuracy declines in the second interval (150-250 s, Figure 5), where RMSE rises by 55% to 1.15 m/s. This suggests growing sensitivity to unmodeled disturbances such as wind or vehicle dynamics variations. In the final stage (300-450 s, Figure 6), RMSE reaches 1.21 m/s, a 64% increase over the initial phase, reflecting sustained and systematic error accumulation. Overall, the results reveal a clear trend of performance degradation over time: while the model performs reliably under initial conditions, it lacks adaptive capability to maintain accuracy under prolonged or varying operational scenarios.

5.3. Simulating a string of vehicles

The outcomes highlight a clear progression in system dynamics, moving from initial stability to amplification of disturbances and finally to sustained oscillations. In the initial interval (0-100 s, Figure 7), the system remains stable, with a mean acceleration close to zero and a standard deviation below 5 m/s², indicating damped oscillations and steady cruising. In the second interval (150-250 s, Figure 8), instability develops as a deceleration pulse in the lead vehicle, reaching about -160 m/s², propagates and amplifies to nearly -200 m/s² in the following vehicle. This amplification demonstrates a string instability gain factor greater than one. By the final interval (300-450 s, Figure 9), the system enters a sustained limit cycle, with acceleration oscillations consistently ranging from +30 m/s² to -30 m/s² (about 60 m/s² peak-to-peak) at a frequency of approximately 0.5 Hz. This persistent oscillatory mode highlights the transition from initial stability to amplified disturbances and ultimately to a dominant unstable oscillation.

5.4. Model calibration

Calibration of the car-following model against empirical spacing data produced RMSE values of 10.60 m [Figure 10], 14.02 m [Figure 11], and 15.35 m [Figure 12], highlighting its performance across different operational scenarios. In the first scenario (0-100 s), the low RMSE of 10.60 m indicates excellent agreement between simulated and actual spacing, capturing the overall trend and amplitude of vehicle interactions with minimal deviation. In the second scenario, the RMSE increases to 14.02 m, primarily due to minor underestimation of spacing during the 150-250 s interval, suggesting a slight overestimation of simulated vehicle acceleration in response to dynamic events. The third scenario, with the highest RMSE of 15.35 m, represents the most challenging conditions, likely involving complex maneuvers or traffic disturbances, where the simulation slightly leads the actual spacing data between 300-450 s. Despite these variations, the model consistently reproduces the key characteristics of car-following behavior, maintaining all RMSE values below 16 m. This indicates that the underlying model structure and parameterization are robust, capable of capturing real-world vehicle dynamics across a range of conditions. The calibration results validate the model’s predictive capability, demonstrating that it can reliably simulate vehicle trajectories for both stable and perturbed traffic scenarios.

5.5. Communication delay

The performance of inter-vehicle communication critically affects string stability in a vehicle platoon. Analysis of relative speed (ΔV) under varying communication latencies reveals a clear progression in destabilizing effects. With minimal delays (< 100 ms, Figure 13), ΔV exhibits low-amplitude, high-frequency oscillations around ±0.1 m/s, maintaining overall stability with minor tracking error. Moderate delays (100-500 ms, Figure 14) induce under-damped behavior, with deviations peaking at -0.6 m/s and overshooting to +0.4 m/s before settling, reflecting over-correction due to delayed state information. Significant delays (> 500 ms, Figure 15) produce large, sustained oscillations between -1.5 m/s and +1.5 m/s over extended periods, demonstrating string instability as phase lags amplify downstream disturbances and violate the stability condition. These results indicate a nonlinear relationship between communication latency and platoon performance, establishing critical bounds for acceptable delay to ensure safe and effective vehicle coordination.

5.6. Energy efficiency

The energy efficiency of the system was assessed by examining cumulative energy consumption over the operational timeline. During the initial 0-100 s [Figure 16], energy consumption increased linearly, with a slope of approximately 5 kW, indicating steady-state operation with consistent power draw and no significant spikes. This behavior persists in the subsequent 150-250 s interval [Figure 17], where cumulative energy rises from 100 kJ to 500 kJ while maintaining the same 5 kW average, confirming sustained baseline efficiency. A localized high-frequency transient [Figure 18] shows a temporary steep increase in power, reflecting a short-duration computational task. Importantly, the system immediately returns to the baseline trend afterward, demonstrating robust management of intensive events without long-term efficiency loss. Overall, the system exhibits stable, predictable energy consumption with efficient handling of transient high-power operations, validating its energy-conscious design for prolonged operation under dynamic conditions.

5.7. Comparative performance with baseline models

The simulation results highlight key dynamics of car-following behavior across single vehicles and vehicle platoons. The OVM exhibits persistent stop-and-go oscillations in all intervals, confirming its inherent instability under the chosen sensitivity parameters. These results demonstrate the model’s inability to maintain smooth traffic flow without additional control strategies. For a single vehicle, the RMSE values indicate accurate velocity tracking initially (0-100 s), but performance gradually declines in later intervals (150-250 s and 300-450 s), reflecting cumulative error under dynamic conditions. This underscores the need for adaptive mechanisms to maintain tracking accuracy over prolonged operation, summarized in Table 1. Extending to a string of vehicles reveals amplification of disturbances, with deceleration pulses growing in magnitude and eventually producing sustained oscillatory limit cycles. This string instability highlights the critical influence of vehicle interactions on platoon dynamics. Communication delay analysis further demonstrates the sensitivity of platoon stability to latency. Minimal delays result in stable tracking, while moderate to large delays cause under-damped or high-amplitude oscillations, quantifying practical bounds for V2V and V2I systems. Energy efficiency analysis shows consistent cumulative energy consumption across all intervals, even during transient high-power events, validating the framework’s effectiveness in maintaining sustainable operation under dynamic conditions.

In addition, to evaluate the improvement of the proposed Enhanced OVM, a performance comparison among the Classical OVM, IDM, and Enhanced OVM is presented in Figure 19. The results clearly show that the Enhanced OVM achieves lower velocity and spacing RMSE values and higher stability delay limits compared to both baseline models. These findings confirm that the Enhanced OVM not only mitigates instability but also enhances adaptability and control robustness under communication delays and dynamic traffic scenarios. Energy efficiency analysis similarly indicates smoother power variation and reduced fluctuation, validating the potential of the proposed model for sustainable and stable car-following control. Overall, the results emphasize that while classical models capture fundamental traffic phenomena, their performance is limited under prolonged, high-density, or delayed communication scenarios, highlighting the importance of adaptive control and delay-aware strategies.