3.1. Vulnerability Assessment System Establishment
In order to assess the vulnerability of highway transport system based on cascading failure phenomenon, we must identify the status of vulnerable sections. If the traffic volume is less than or equal to the maximum highway capacity, the section is in a normal status; On the contrary, if the traffic flow is more than the maximum highway capacity, the section is in a failed status.
In the latter status, traffic flow is reallocated, and an alternative route is chosen by users who have selected the vulnerable section of highway transport system. If the traffic flow is more than the maximum highway capacity after flow redistribution, the other sections would convert from a normal status to a failed status, which causes more serious congestion. As a result, another reallocation of traffic flow is necessary. The cascading failure would not end out until the traffic flow is less than the maximum highway capacity.
Traffic flow reallocation is a repeated iteration step which is critical to vulnerability assessment of highway transport system. User equilibrium assignment model (UE model) and system optimal allocation model (SUE model) are widely used in traffic flow reallocation. When the vulnerable section is exposed, users who have selected the vulnerable section need to choose an alternative route. Note that the users cannot fully understand real-time traffic information and other travelers’ choice before making decisions, and they seek to select the optimal path to minimize their transportation cost. Therefore, the UE model can be used to evaluate methods of vulnerability.
The equivalent mathematical formulation is
where,
is the number of trips between the O-D pair (
i,
j) that uses path
,
is the cost of flow
using link
,
is the flow in link
, and
equal to one if part of path
between
and
belongs to link
and zero otherwise.
Some measure indices are frequently used in complex network assessment. They include point degree, degree distribution, average distance, shortest distance, effectiveness, clustering coefficients, betweenness and so on.
As static indices of complex systems, they are often used to compare the effectiveness of different networks. However, the vulnerability excitation of highway network system is a dynamic, concatenate and time-delay process, the static indices cannot accurately measure the vulnerability excitation impact to users in the system. Therefore, we consider the congestion-caused failure process and attempt to define vulnerability measure indices of highway network system.
Cascading failure of highway network system caused by vulnerable sections may bring impact to users in two ways. If users at vulnerable sections cannot find an alternative route to reach the destination, their travel demands could not be met, which will result in delays in travel. For another, if users at vulnerable sections reach the destination through a detour on a trip, they would spend more travel time and travel costs to meet their travel demand.
In case of vulnerable section failure, if the demand of travelling to destination could not be met, the users have to wait until the vulnerable section becomes normal, or exit the highway and enter other low grade road. Then the traffic supply of the section drops to 0 due to the vulnerable section failure. In this issue, assessment formula proposed by Jenelius is used for failure section [
19].
If the users have to detour to meet their travel demands, the economic costs of travel can be measured by the total cost of travel, see Equation (3). And the travel time cost can be measured by total travel time in highway network, see Equation (4).
where,
is the total travel cost after traffic flow allocation due to section failure,
is the Length of section
,
is the toll station costs per kilometer,
is the fuel costs per kilometer. According to the highway toll station fees standard of Heilongjiang Province,
is set to be 0.5 Yuan.
is set to be 0.6 Yuan due to the average fuel consumption of vehicles.
where,
is the total travel time after traffic flow allocation due to section failure,
represents the scenario that section
is in failure status,
is the traffic flow allocated for section
before iterations,
is the traffic flow allocated for section
after iterations,
is the section travel time, and we use BPR function
, where we set 0.15 for
, and 4 for
.
The total cost of travel
is
where,
is time value per hour, and we set 20 for it here.
Considering these two factors, we defined vulnerability value
as follow,
where,
is the total cost of travel without UE equilibrium process.
After dimensionless, the vulnerability assessment indices of highway network system are represented as a value between 0 and 1, the formula as follows:
where,
is the dimensionless vulnerability value,
is the original vulnerability value,
is the maximal vulnerability value and
is the minimal vulnerability value for each highway section.
The levels of vulnerability are graded as shown in
Table 2.