Modeling Counterflow and Crossflow Conflicts in Crowd Evacuation

Modeling Counterflow and Crossflow Conflicts in Crowd Evacuation

Problem Description
During evacuation processes in large public spaces (such as subway stations, stadiums), pedestrian flows from different directions can form counterflows (opposing movements) or crossflows (intersecting movements), leading to path conflicts, reduced speeds, and even congestion. This problem requires analyzing the causes and dynamic impacts of such conflicts and establishing a mathematical model to quantify their effect on evacuation efficiency.

Key Knowledge Points

  1. Conflict Types:
    • Counterflow: Two groups of people moving in opposite directions along the same passageway (e.g., evacuating crowds encountering rescue personnel).
    • Crossflow: Multiple groups of people converging at the same node from different directions (e.g., a crossroads).
  2. Impact of Conflicts: Speed reduction, increased probability of congestion, delays in individual decision-making.

Problem-Solving Steps

Step 1: Analysis of the Physical Mechanisms of Conflict

  • Root Cause: Individuals frequently adjust their movement direction and speed to avoid oncoming pedestrians, disrupting the continuity of the overall flow.
  • Key Parameters:
    • Density Threshold: When the crowd density within a passage exceeds a critical value (e.g., 2 persons/m²), the impact of conflict significantly intensifies.
    • Speed Difference: Greater speed differences between opposing flows lead to larger avoidance maneuvers and more pronounced energy loss (reflected in the "friction" term of the social force model).

Step 2: Establishing a Microscopic Model of Conflict (Based on the Social Force Model)
In the social force model, the force on an individual is given by:

\[\vec{f}_i = m_i \frac{d\vec{v}_i}{dt} = \vec{f}_{goal} + \sum_{j \neq i} \vec{f}_{ij} + \sum_{w} \vec{f}_{iw} \]

  • Extended Conflict Term: Add a counterflow conflict force \(\vec{f}_{conflict}\), related to the relative velocity \(\vec{v}_{ij} = \vec{v}_i - \vec{v}_j\) and distance to oncoming pedestrians:

\[\vec{f}_{conflict} = -k \cdot \frac{\vec{v}_{ij}}{\|\vec{r}_{ij}\|} \quad (\text{where } k \text{ is the conflict intensity coefficient}) \]

  • Physical Meaning: Individuals tend to decelerate or change direction to reduce their relative speed to oncoming pedestrians.

Step 3: Macroscopic Flow Analysis

  • Basic Formula: Passage flow rate \(Q = \rho \cdot v \cdot A\) (\(\rho\) is density, \(v\) is speed, \(A\) is passage area).
  • Speed Correction under Conflict: Use the empirical formula \(v_{conflict} = v_0 \cdot e^{-\beta \cdot \rho_{reverse}}\), where \(\rho_{reverse}\) is the density of the opposing flow and \(\beta\) is the decay coefficient.
  • Example Calculation: If the normal speed \(v_0 = 1.5\text{m/s}\), \(\beta=0.3\), and the opposing flow density \(\rho_{reverse}=1.5\text{persons/m²}\), the actual speed reduces to \(1.5 \cdot e^{-0.45} \approx 0.96\text{m/s}\).

Step 4: Dynamic Conflict Mitigation Strategies

  • Spatiotemporal Flow Separation: Use controlled gates or temporary barriers to allocate passage to different flow directions at different times (e.g., "tidal lanes").
  • Guidance Signal Optimization: Install dynamic indicator lights at intersections to prioritize passage rights for higher-density flow directions.
  • Simulation Verification: Use tools like Anylogic or Viswalk to compare evacuation times with and without strategies (e.g., in a case shown in the right figure, the separation strategy can reduce congestion time by 15%-20%).

Step 5: Discussion on Model Limitations

  • Individual decision-making differences (e.g., risk-taking behavior) are not fully covered.
  • Sudden environmental changes (e.g., smoke obscuration) may require the introduction of random terms.

Summary
By modeling microscopic forces and correcting macroscopic flow rates, the impact of conflicts on evacuation can be quantified. Combined with dynamic management strategies, overall efficiency can be optimized. In practical applications, model coefficients need to be adjusted according to specific scenario parameters (e.g., passage width, crowd composition).