# Analysis of Cascading Failures and Recovery in Freeway Network Under the Impact of Incidents
## 3 Methodology
### Modeling the Cascading Failures of the Freeway Network
#### 3.2.1 Preliminaries  
In this section, we first define the fundamental concepts necessary to understand the simulation of cascading failures in a network. Cascading failures occur when an initial disruption in a network propagates, leading to a sequence of failures that can compromise the network's overall functionality. To model and analyze such events, we first define three fundamental concepts: initial edge load definition, edge capacity, and edge state. Initial Edge Load Definition: Before the cascading process, a traffic assignment method is employed to distribute travel demand across the road network, establishing the initial load on the roads [12]. We calculate the flow of each path based on the stochastic user equilibrium rule. The initial flow on a road segment is determined by summing the path flows that pass through it. The initial load on the network \(L_{a}(0)\) is expressed as  
\[L_{a}(0)=\sum_{i,j\in N}f_{ij}\cdot\delta_{a}^{ij} \tag{1}\]  
where \(f_{ij}\) represents the path flow from \(i\) to \(j\), expressed as the sum of flows distributed across multiple possible paths. \(\delta_{a}^{ij}\) indicates whether the path from \(i\) to \(j\) passes through segment \(a\). If the path passes through segment \(k\), then \(\delta_{a}^{ij}=1\); otherwise, \(\delta_{a}^{ij}=0\). Edge Capacity: The road capacity refers to the maximum number of vehicles that can reasonably pass through a lane or road within a given unit of time. The actual capacity of a road is dynamic and varies based on the specific characteristics of time and space. The base capacity, on the other hand, denotes the capacity of a road under ideal road and traffic conditions. Given the challenges in observing the actual road capacity during real-world operations, the base capacity is used as the maximum capacity of the road. The capacity drop phenomenon is caused by various factors, including road infrastructure, traffic flow, weather conditions, and management measures. To simulate the cascading failure process under different capacities, we can introduce a capacity reduction factor \(\gamma\) that adjusts the effective capacity of each road section based on a specific parameter. Specifically, \(\gamma\) is a coefficient ranging from 0 to 1 that comprehensively accounts for the impacts of traffic accidents, adverse weather conditions, and abnormal traffic flow on road capacity. This approach enables us to examine how different levels of reduced capacity affect the network's performance and the propagation of failures. The effective capacity \(C_{a}\) of a road section can be defined as a function of the base capacity \(C_{max}\) and the capacity reduction factor \(\gamma\). The capacity reduction factor \(\gamma\) can be defined as a function of the base capacity \(C_{max}\) and the reduction factor \(\gamma\). The formula for the effective capacity \(C_{a}\) of a road section \(a\) is given by  
\[C_{a}=\gamma\times C_{max}^{a} \tag{2}\]  
where \(C_{max}^{a}\) is the maximum capacity of the road section \(a\). When \(\gamma=0\), it means the road has completely lost its capacity, for example, due to severe accidents or complete  
Figure 2: Cascading progress in a simplified network. closure. When \(\gamma=1\), the road operates at full capacity. When \(0<\gamma<1\), it indicates that the road capacity is partially restricted, lying between completely unobstructed and fully blocked, reflecting the degree of capacity reduction caused by real-world factors. By adjusting the value of \(\gamma\), the model can flexibly simulate the impact of different levels of capacity reduction on network performance and the cascading failure propagation process. Edge State: During the cascading process, the state of the edges in the network is a critical factor in determining the overall resilience and stability of the network. We need to clearly define the state of the edges to accurately model and analyze the network. In this section, edge states are defined as either normal or failed.