Coordinated Lane-Level Variable Speed Limits and Ramp Metering for Successive Weaving Segments Considering Merging/Diverging Risks: A Hybrid Model Predictive Control and Multi-Agent Reinforcement Learning Approach

1. Introduction

• Forced lane changes amid merging and diverging flows can cause turbulence, capacity drops, and high collision risks at weaving bottlenecks.The interacting streams include mainline through traffic, on-ramp merging flows, and off-ramp diverging flows.Source: §1 · p. 2
• Uniform segment-level control can waste inner-lane capacity and fail to ensure outer-lane safety where weaving creates lane-level heterogeneity.Outer lanes may congest from merging and exiting traffic while inner lanes remain smooth.Source: §1 · p. 2
• Merging risks at entrances and diverging risks at exits have distinct triggers that efficiency-focused control may insufficiently address.Simplifying or neglecting these risks can lead to overly aggressive commands and safety hazards.Source: §1 · p. 2
• MPC handles constraints but depends on prediction accuracy, whereas DRL handles uncertainty but can converge slowly, oscillate, and lack safety guarantees.The paper therefore motivates a hybrid approach combining model-based and data-driven control.Source: §1 · p. 3
• The proposed framework combines L-METANET, XGBoost-SHAP-RPBL risk prediction, MPC baseline commands, and MARL residual compensation for coordinated lane-level VSL and ramp metering.MPC operates under physical constraints, while MARL adjusts its reference policies.Source: §1 · p. 3

2.1. The models of coordinated variable speed limit and ramp metering

• Coordinated VSL and ramp metering can create upstream merging space and outperform isolated strategies, but conventional models often assume homogeneous lanes.The review describes VSL-RM coordination as suppressing backward shockwave propagation and improving throughput and collision-risk outcomes in cited studies.Source: §2.1 · pp. 3–4
• Weaving segments combine through, merging, and diverging flows, creating X-shaped conflicts that make merging-zone control models non-transferable without further investigation.Lane-level heterogeneity also means uniform control can waste inner-lane capacity or compromise outer-lane safety.Source: §2.1 · p. 4
• Macroscopic risk models must balance accuracy, controllability, observability, and site-specific heterogeneity, while microscopic metrics and end-to-end learning each hinder real-time control integration.The review notes that vehicle-level indicators are difficult to collect network-wide and may not align directly with VSL and ramp-metering controls.Source: §2.1 · p. 5

2.2. The algorithms of coordinated variable speed limit and ramp metering

• MPC enforces physical constraints but depends on model accuracy and incurs online optimization costs, whereas RL offers rapid inference but requires difficult offline training.In lane-level weaving control, high-dimensional state-action spaces and non-stationarity can make MARL convergence slow or unstable.Source: §2.2 · pp. 6–7
• A hybrid controller can use MPC for physics-based baseline safety and RL for residual compensation that mitigates model mismatch.The review presents this integration as a response to the complementary limitations of isolated model-driven and data-driven methods.Source: §2.2 · p. 7

2.3. Research gaps and main contributions

• The identified gaps are lane-level model mismatch, weak closed-loop integration of collision risk, and limited adaptability of isolated MPC or DRL.The review attributes the latter gap to MPC's model mismatch and computation burden and DRL's convergence and transfer limitations.Source: §2.3 · p. 7
• L-METANET incorporates free and forced lane changes to reproduce cross-lane flow redistribution and capacity drops for weaving-segment prediction.The model is intended to support fine-grained lane-level active traffic management.Source: §2.3 · p. 8
• The proposed risk model combines XGBoost-SHAP and RPBL to estimate merging and diverging risks, while the hierarchical controller pairs MPC baselines with MARL residual compensation.The risk formulation is embedded in the MPC cost and MARL reward functions; the lower layer uses spatiotemporal attention-reinforced MAPPO.Source: §2.3 · p. 8

3.1. Scenario definition and problem description

• Each control unit has four consecutive zones, and detectors provide lane-specific traffic states, boundary demands, and ramp queues for centralized VSL and metering.Commands reach connected vehicles through V2X and human-driven vehicles through dynamic speed displays and ramp signals.Source: §3.1 · p. 9
• Successive weaving segments couple efficiency and safety trade-offs with backward-propagating congestion waves and interacting VSL and ramp-metering actions.Downstream breakdowns can trigger cascading failures upstream, while poorly coordinated controls risk excessive mainline delay or ramp deadlocks.Source: §3.1 · p. 9
• The control problem jointly sets lane-level VSL and ramp-metering rates to balance mainline speed, ramp queueing time, and merging and diverging risks.The stated objectives include maximizing average mainline speed, minimizing total ramp queueing time, and suppressing macroscopic collision risks.Source: §3.1 · p. 9
• Control sequences must respect L-METANET traffic dynamics, ramp queue capacity, spatial variation bounds, and actuator limits on rates and changes.The constraints cover both VSL and ramp-metering signals.Source: §3.1 · p. 9

3.2. Overall research framework

• The framework links data foundation, state perception, risk assessment, MPC baseline control, and MARL residual compensation into distinct operational layers.Risk estimates provide the MPC cost and MARL reward, connecting risk assessment to both control layers.Source: §3.2 · p. 10
• At the longer macro-control period, MPC uses L-METANET and rolling-horizon optimization to compute safe baseline commands under physical constraints.The baseline is intended to establish a robust safety floor for system operations.Source: §3.2 · p. 10
• At the shorter micro-control step, distributed ST-MAPPO agents use local states to produce rapid residual actions that address disturbances and model mismatch.The residual layer is designed to compensate for MPC's sluggish response to microscopic disturbances.Source: §3.2 · p. 11
• The controller superimposes MARL adjustments on MPC commands, then clips the combined signals to physical and regulatory boundaries before dispatch.The resulting commands are sent to traffic actuators.Source: §3.2 · p. 11

4.1. L-METANET: a lane-level macroscopic traffic flow model

• The section extends classic METANET to lane-level urban expressway modeling and presents a parameter calibration approach.Source: §4.1 · p. 11

4.1.1 Formulation of L-METANET

• L-METANET extends METANET with lane-specific states, asymmetric lateral-flow speed impacts, VSL and HDV-compliance effects, lane-addition or drop dependencies, and free and forced lane-change flow estimation.The section also presents a parameter-calibration approach for the extended model.Source: §4.1.1 · p. 11
• The model tracks lane-specific traffic states across spatial segments, with speed evolution incorporating relaxation, upstream-flow convection, and downstream-density anticipation.This granularity addresses conventional METANET’s homogeneous multi-lane assumption, which misses lane differences and localized ramp and weaving effects.Source: §4.1.1 · pp. 11–12
• Lane-level METANET captures merging and diverging speed effects separately: vehicle insertions prompt stronger braking than vehicle extractions, which have a less severe impact on lane speed.The model accounts for these lateral effects in its modified dynamic speed equation.Source: §4.1.1 · pp. 13–14
• L-METANET models VSL effects in mixed traffic using CAV penetration and HDV compliance, reflecting CAVs’ full command compliance and HDVs’ partial compliance.The formulation uses these compliance characteristics to adjust the desired speed under VSL.Source: §4.1.1 · pp. 14–15
• L-METANET modifies speed evolution to account for downstream congestion waves and cross-lane friction, and estimates flow changes from free and forced lane changes.Its convection term uses a geometric mean to represent upstream–downstream interdependencies under varying cross-sectional geometry; the model also incorporates VSL constraints on desired speed.Source: §4.1.1 · p. 16

4.1.2 Calibration method of L-METANET

• Calibration separates global dynamic parameters from local fundamental-diagram parameters, minimizes speed RMSE, and uses a GA-VNS hybrid to search the nonlinear, non-convex objective.The calibration assumes uniform fundamental-diagram parameters across the network to limit over-parameterization.Source: §4.1.2 · p. 17

4.2. Accuracy- and controllability-driven risk prediction and assessment model for weaving segments

• The risk model bridges microscopic conflict mechanisms and macroscopic control variables by identifying macroscopic precursors of merging and diverging risks.XGBoost-SHAP-RPBL addresses black-box prediction and multicollinearity while accounting for unobserved heterogeneity and producing a closed-form risk probability for optimization.Source: §4.2 · pp. 17–18

4.2.1 The XGBoost-SHAP-RPBL framework and implementation steps

• The framework selects core traffic features with SHAP before fitting RPBL, which yields a smooth, differentiable analytical crash-risk equation for the control system.Its candidate features span lane, segment, and traffic-factor dimensions; SHAP selection uses an importance threshold of 0.2 and retains at most eight variables.Source: §4.2.1 · pp. 18–19

4.2.2 Risk prediction and assessment model based on random parameters binary Logit

• RPBL models the probability of a weaving segment entering a high-risk conflict state while allowing core-feature coefficients to vary and capture unobserved heterogeneity.A parameter is treated as random only when its estimated standard deviation is statistically significant at p = 0.05; otherwise, it remains fixed.Source: §4.2.2 · pp. 19–20
• Because integration lacks a closed-form analytical solution, SML uses 500 Halton-sequence Monte Carlo draws for high-precision calibration; the hybrid XGBoost-SHAP-RPBL model distills heterogeneous discrete traffic-flow data into a continuous crash-probability equation bounded between 0 and 1.Source: §4.2.2 · p. 20
• The crash-probability function serves as the core objective for coordinated VSL-RM optimization to manage expressway states and mitigate merging and diverging crash risks.Source: §4.2.2 · p. 20

• MPC supplies a rigid baseline safeguard, while MARL adds residual compensation for joint lane-level VSL and ramp-metering control.The hybrid design addresses model mismatch that can challenge MPC stability and the risk that standalone MARL may trigger breakdowns in unseen scenarios.Source: §5 · p. 20

5.1. Overall architecture

• The control loop distinguishes macroscopic state updates, MPC optimization cycles, and MARL execution cycles, with the latter providing finer-grained adjustment.The lower-level MARL actions are sampled and applied at a higher frequency than the MPC baseline.Source: §5.1 · pp. 20–21
• Lane-level VSL gantries and on-ramp meters define the physical actuators, while MPC and MARL states map to L-METANET outputs at their respective timescales.The architecture anchors the reference command to the latest MPC solution at cycle boundaries and transitions smoothly between executions.Source: §5.1 · pp. 21–22
• MPC predicts traffic evolution with L-METANET and issues a low-frequency baseline; lane and ramp agents produce high-frequency residual actions.The combined command is generated through a chronological rolling process rather than static action addition.Source: §5.1 · p. 22
• The superposition mechanism keeps issued controls within acceptable ranges and avoids abrupt changes in speed limits and metering rates.At MPC cycle boundaries, the reference is anchored to the latest global optimum; between boundaries, it transitions to the previous executed MARL command.Source: §5.1 · p. 22

5.2. Upper-level baseline controller based on model predictive control

• The upper-level MPC uses L-METANET and dynamic crash-risk equations to establish a long-horizon baseline balancing safety and efficiency.It predicts traffic-flow evolution and issues a baseline command for the coordinated control system.Source: §5.2 · p. 22

5.2.1 State-space and prediction model formulation

• L-METANET predicts future traffic states from current traffic conditions, boundary demands, and a baseline command held constant during each MPC cycle.The controller rolls predictions forward over its specified prediction horizon.Source: §5.2.1 · p. 23

5.2.2 Multi-objective comprehensive cost function formulation

• The MPC cost combines merge/diverge crash risk, mainline efficiency, ramp queue delay, and penalties for temporal and spatial actuator changes.The risk term uses the XGBoost-SHAP-RPBL crash-probability equations, while the efficiency metric uses average speed in target segments.Source: §5.2.2 · pp. 23–24

5.2.3 Traffic physical boundaries and control execution constraints

• Physical constraints enforce L-METANET traffic dynamics, safe and smooth VSL commands, ramp-storage limits, and bounded ramp-metering rate changes.VSL limits address both changes over time and differences between adjacent gantries on the same lane.Source: §5.2.3 · pp. 24–25

5.2.4 Model optimization and solution

• A hybrid GA-VNS algorithm solves the constrained, nonlinear optimization, and only the first element of its optimal sequence is dispatched as the baseline command.At the next control step, rolling optimization restarts using fresh lane-level state feedback.Source: §5.2.4 · p. 25

5.3. Lower-level residual compensator based on multi-agent reinforcement learning

• The lower level models lane and ramp agents as a fully cooperative system that uses local information to coordinate high-frequency residual actions.Agents sample local macroscopic states more frequently than the low-frequency baseline cycle to smooth model-prediction mismatches and adaptively control macroscopic traffic flow.Source: §5.3 · pp. 25–26

5.3.1 State-space formulation incorporating upper-level MPC outputs

• Agents use local traffic-state observations, the MPC baseline command, and the previously applied control as inputs for high-frequency residual decisions.The prior action supports temporal memory and, with a residual-smoothing penalty, helps prevent fluctuating commands and supports smooth vehicle operations.Source: §5.3.1 · pp. 26–27

5.3.2 Joint action space design

• Agents output bounded continuous residuals rather than absolute controls; residuals are added to baseline commands and physically truncated before actuator dispatch.The residuals provide fine-tuning to absorb disturbances from model mismatch or sudden demand surges while preserving physical operating bounds.Source: §5.3.2 · p. 27

5.3.3 Fully cooperative shared global reward function formulation

• All VSL and ramp-metering agents receive the same global reward combining weaving safety, mainline travel efficiency, ramp queuing, and control smoothness.The first three reward components match corresponding terms in the upper-level MPC cost function.Source: §5.3.3 · pp. 27–28

5.3.4 ST-MAPPO: MAPPO algorithm architecture with spatiotemporal attention mechanisms

• ST-MAPPO combines Mamba temporal memory in decentralized actors with graph causal topological attention in a centralized critic under centralized training and decentralized execution.The design addresses local partial observations and the coupled spatiotemporal structure of traffic states.Source: §5.3.4 · p. 28
• ST-MAPPO uses a clipped Actor policy objective with entropy regularization to encourage early exploration of high-frequency residual actions.The policy objective uses an importance-sampling ratio and proxy policy clipping; the entropy coefficient decays during training.Source: §5.3.4 · pp. 29–30
• Training preserves chronological trajectory slices for Mamba learning and applies moving-average feature scaling, mini-batch advantage normalization, orthogonal initialization, and linear learning-rate decay.The Actor and Critic learning rates and policy entropy coefficient decay across epochs to balance early exploration with stable late convergence.Source: §5.3.4 · pp. 29–30
• The Critic estimates true returns using a dual-clipped mean-squared-error loss designed to keep value estimation robust over long horizons.The passage also describes the Critic loss as approximating the true return value.Source: §5.3.4 · pp. 29–30

6.1. Experimental setup

• The testbed replicated the 18-km Eastern Expressway at 1:1 scale with 9 VSL gantries and 3 ramp meters, using empirical flows and UAV trajectories for model calibration and evaluation.The trajectories came from weaving segments WS3, WS4, and WS6; the control evaluation used the same 13-hour empirical flow as network boundary demand.Source: §6.1 · pp. 30–31

6.2. Calibration and validation of L-METANET

• Other-lane flow MAPE fell from 24.9% with METANET to 12.1% with L-METANET, while RMSE fell from 256 to 133.For speed prediction, MAPE was 6.7% for Lane 1 and 19.4% for other lanes with L-METANET, versus 10.8% and 27.5% with METANET.Source: Table 2 · p. 34, §6.2 · p. 34
• L-METANET reproduces forced-lane-change flow redistribution and capacity drops, and captures lane-asymmetric speed transitions from free flow to congestion in alignment with ground truth.The model's performance is contrasted with METANET's cross-sectional homogeneity assumption and omission of lane heterogeneity and lateral vehicle physics.Source: Figure 7 · p. 33, Figure 8 · p. 33, §6.2 · pp. 34–35

6.3. Fitting and performance verification of XGBoost-SHAP-RPBL

• All 14 risk tasks had AUC above 0.70; all but WS1_D and WS3_D exceeded 0.80, with WS6 merging peaking at 0.889.The ROC curves cover 14 risk-identification tasks.Source: Figure 9 · p. 35, §6.3 · p. 35
• XGBoost-SHAP-RPBL outperformed Pearson-RPBL, Spearman-RPBL, and Random-RPBL, and slightly surpassed the fixed-effects XGBoost-SHAP-FPBL baseline.The authors attribute the latter comparison to random parameters capturing unobserved heterogeneity.Source: §6.3 · pp. 35–36, Figure 10 · p. 36
• SHAP selection reduced risk-model inputs to single digits, supporting real-time estimation and closed-loop control feedback compared with the dozens of candidate variables in All Params-RPBL.The proposed model's accuracy was slightly lower than All Params-RPBL in certain scenarios.Source: §6.3 · p. 36

6.4.1 Baseline control strategies

• The 11 baselines span model-driven, data-driven, hybrid, and reference strategies, including segment- and lane-level MPC, three MARL methods, no-control, and random control.The hybrid comparisons pair the MPC baseline with ST-MAPPO, MAPPO, or MADDPG residual control.Source: §6.4.1 · p. 37

6.4.2 Training process and convergence mechanism analysis

• Without MPC guidance, ST-MAPPO converges near -25 around episode 280; MAPPO stalls near -33, while MADDPG fluctuates between -25 and -48 without converging.The reported explanation attributes the difference to Mamba's temporal context filtering and GCTA's spatial inductive bias.Source: §6.4.2 · pp. 37–38, Figure 11 · p. 39
• Among model-driven baselines, MPC-LL averages -29.81 reward, ahead of MPC-LS at -32.29 and MPC-SS at -36.58.The comparison is attributed to lane-level L-METANET capturing heterogeneous lane-changing effects and supporting differentiated speed limits.Source: §6.4.2 · p. 38, Figure 11 · p. 39
• MPC-STMAPPO reaches approximately -26 reward within 90 episodes, while MPC-MAPPO stagnates and MPC-MADDPG oscillates.MPC guidance improves initial reward, convergence speed, and volatility relative to ST-MAPPO, although ST-MAPPO's final steady-state reward is about 1 higher.Source: §6.4.2 · p. 38

6.4.3 Quantitative evaluation of multi-objective control efficacy

• Lane-level MPC-LL minimizes the WS4 queue to 0.22 veh and raises WS6 speed to 32.4 km/h, outperforming coarser model-driven control.The passage links the improvement to lane-level L-METANET and differentiated lane-by-lane speed limits.Source: §6.4.3 · p. 40
• MPC-STMAPPO ranks in the top three for 6 of 13 network-wide metrics and performs especially well at heavily congested WS6.At less congested WS3 and WS4, it remains competitive, with localized upstream sacrifices supporting downstream congestion mitigation.Source: §6.4.3 · p. 40
• MPC-STMAPPO achieves the highest WS6 speed (38.5 km/h), the shortest on-ramp queue (21.82 veh), and 25 network-wide collisions among 11 schemes.The collision count is a 91.99% reduction relative to No-Control.Source: §6.4.3 · p. 40, Table 3 · p. 41

Transfer test design

• Transfer tests deploy baseline-trained ST-MAPPO and MPC-STMAPPO without fine-tuning across low-load, oversaturated, and randomized demand scenarios.Random demand 2 independently samples each OD flow from 1 to 200, removing the historical demand topology.Source: §6.5 · p. 42

Transfer performance

• Under Random demand 2, MPC-STMAPPO raises WS3 speed by 50.37% over ST-MAPPO and reduces WS6 delay by 17.22% relative to that baseline.Under 1.5×Base_Demand, it records 16.4 km/h at WS6 and 88 network-wide collisions, versus 109 for ST-MAPPO.Source: §6.5 · p. 44
• The authors conclude that the upper-level MPC's physical conservation constraints provide a performance lower bound and improve generalization under unseen, abrupt, and oversaturated disturbances.The conclusion presents this transfer resilience as supporting real-world engineering viability.Source: §6.5 · p. 44

7. Conclusion

• L-METANET models free and forced lane changes and reproduces lane-flow redistribution and capacity drops in field validation against Changchun detector data.The reported state evolution aligns with ground-truth flow and speed more closely than conventional segment-level METANET.Source: §7 · p. 44
• XGBoost-SHAP feature selection with RPBL yields merging and diverging risk classification AUC above 0.70 across tasks, with most above 0.80, outperforming conventional Logit models..Source: §7 · p. 44
• MPC-STMAPPO stabilizes near -26 reward within 90 episodes, ranks top-three on 6 of 13 core metrics, and reduces simulated network-wide crashes from 312 to 25.The paper also reports better transfer robustness and worst-case performance protection than pure MARL in zero-shot cross-load tests.Source: §7 · pp. 44–45
• The study focuses on macroscopic lane-level VSL and ramp metering; integration with microscopic trajectory planning and longer-range aerial trajectory data remains future work.The authors propose collaborative UAV photography and finer macroscopic aggregation for that extension.Source: §7 · p. 45

Free lane-changing flow

• Free lane-changing demand is driven by adjacent-lane density differences, with mixed-traffic aggressiveness and spatial weighting shaping attractiveness and demand.The model combines CAV and HDV aggressiveness using CAV penetration and scales demand by vehicles in the current lane.Source: Appendix A · pp. 45–46
• When competing demands exceed target-lane space, actual lane-changing flows are proportionally scaled to remaining supply, preventing density from exceeding physical limits.Supply depends on target-lane jam density.Source: Appendix A · p. 46

Forced lane-changing flow

• Forced lane changes use cascading vehicle borrowing to represent lane-drop and ramp constraints while avoiding mass leakage or duplicate counting.For a flow deficit, borrowing proceeds from outer to inner lanes, and flows from inner lanes transit sequentially through intermediate lanes; surplus flow can instead be forced into an adjacent inner lane.Source: Appendix A · pp. 46–47

Appendix B. Traffic demand of the Eastern Expressway

• 06:00–19:00 traffic volumes collected on November 10, 2022, supplied boundary demands across Eastern Expressway mainline cross-sections and ramps.These empirical demands were used for the SUMO simulation.Source: Appendix B · p. 47
• Fig. 12 depicts inflows, with the mainline origin in subfigure (1) and on-ramp demands in the remaining subfigures.The figure caption describes the distribution of inflow at each entrance over time.Source: Appendix B · p. 47
• Fig. 13 depicts outflows, with the mainline destination in subfigure (10) and off-ramp demands in the remaining subfigures.The figure caption describes the distribution of flow at each exit over time.Source: Appendix B · p. 47

Appendix C. Identification of key variables and RPBL fitting results

• Table 5 lists the key feature variables extracted via the XGBoost-SHAP framework.The passage does not enumerate the variables.Source: Appendix C · p. 47
• Table 6 presents parameter values estimated through RPBL calibration.The passage does not provide the values.Source: Appendix C · p. 47

Appendix D. Data used in this study

• Fig. 12 describes inflow distributions at each entrance over time.The companion text identifies mainline origin and on-ramp demand profiles.Source: Figure 12 · p. 48
• Fig. 13 describes flow distributions at each exit over time.The companion caption repeats the exit-flow scope.Source: Figure 13 · p. 48
• Table 5 is titled as the extraction of key feature variables.The supplied passage identifies the table title but gives no variable values.Source: Table 5 · p. 49
• The notation lists weaving segments WS 1–7 with merging (M) and diverging (D) labels; variables refer to lane-segment density, speed, flow, or maximum speed.CT, FP, and RP denote constant, fixed, and random parameters, respectively.Source: Appendix D · pp. 49–50
• Table 6 is titled as a summary of risk assessment and prediction functions for all tasks.The supplied passage gives the table scope, not its values.Source: Table 6 · p. 49