Risk control of the cascading failure of multimodal transport network considering uncertain disturbance factors 1

Abstract

The multimodal transport network in the region with complex environment and being easily affected by disturbance factors is used as the research object in our work. The characteristics of the cascading failure of such multimodal transport network were analyzed. From the perspective of network load redistribution, the risk control methods for the cascading failure of the multimodal transport network were investigated. This research aims to solve the problem that traditional load redistribution methods usually ignore the original-destination (OD) constraint and uncertain risks. The conditional value-at-risk (CVaR) was improved based on the Bureau of Public Roads (BPR) road impedance function to quantify the uncertainty of the disturbance factors. A nonlinear programming model was established with the generalized travel time as the objective function. A parallelly-running cellular ant colony algorithm was designed to solve the model. Empirical analysis was conducted on the multimodal transport network in Sichuan-Tibet region of China. The results of the empirical analysis verified the applicability of the proposed load redistribution method to such kind of regions and the effectiveness of the algorithm. This research provides theoretical basis and practical reference for the risk control of the cascading failure of multimodal transport networks in some regions.

Keywords

Uncertain disturbance multimodal transport network risk control load redistribution cellular ant colony algorithm

1 Introduction

Multimodal transport refers to the transport process which is completed by two or more modes of transport connecting and transferring each other. Accordingly, the transport network composed of two or more modes of transport that can be connected and transferred to each other is called multimodal transport network. The construction of a multimodal transport network is an important means to promote regional development, and it has attracted the attention of many countries in the world. For example, the Chinese government has put forward a series of plans to promote the deployment of multimodal transport network since 2018, so as to accelerate the development of multimodal transport in China. Based on the characteristics of regions, the highlight of multimodal transport network optimization also varies. This research focuses on the regions that are located in complex environments and are susceptible to disturbance factors, such as dangerous mountainous regions, geologically active regions, and plateau regions. The multimodal transport network in such regions often plays an important role in promoting economic growth and social development. However, the influence of disturbance factors in the network will cause it to locally fail and be at risk of cascading failure, which will affect the normal transportation of the loads in the network [1]. In fact, cascading failures in multimodal transport networks are very common. For example, in Sichuan, which is a transport hub in Southwest China, its multimodal transport network has a large load, which is very important for economic and social development. However, earthquakes occur frequently here, once a part of the network fails, the failure risk will spread rapidly as the load flows in the network, making a part of network or even the whole multimodal transport network stagnation, causing great economic loss and negative social impact. Therefore, the risk control of multimodal transport networks under the influence of uncertain disturbance factors is the key of network optimization in such regions.

Previous researchers have carried out the risk control of network cascading failure from the perspectives of changing the topology [2 –4], improving the performance of nodes and edges [5 –7], and conducting load redistribution [8 –10]. However, in multimodal transport networks, the strategies of changing network topology or node attributes are time-consuming, costly, and poorly viable. Especially for regions in complex environments, such strategies have low applicability. By contrast, the risk control of cascading failure is mainly to carry out research on the load redistribution after a network is disturbed.

At present, researchers have mainly adopted three redistribution methods when studying the issue of load redistribution in network cascading failures. 1) Performing the even distribution [11, 12]. The load on the failed node is evenly distributed to the neighbor nodes. 2) Performing the distribution according to the node attributes [13 –15]. The distribution weights are calculated according to the attributes of neighbor nodes such as node degree and node betweenness, and the load of the failed node is weighted to the neighbor nodes. 3) Performing the distribution according to the node capacity [16 –19]. The distribution weights of the residual value of the neighbor nodes are calculated, and the load of the failed node is weighted and distributed to the neighbor nodes. However, these types of redistribution methods are idealized operations, and they have certain drawbacks when applied in transport networks. The loads in a transport network are rational, and there is “destination", that is, there is a clear original-destination (OD). That means, even if the original path cannot be reached, the purpose of transportation will not be ignored to blindly adjust the path. The “idealization” of the existing load redistribution methods is that only the flow of loads is considered, and the “purpose” of transportation is ignored, where the load distribution is limited to the local structure. In fact, when a network is subjected to disturbances that fail some networks, the loads are forced to redistribute. At this time, in order to ensure the reliability of the transportation process, the failure risk is evaluated during load redistribution, and the risk of different load avoidance will produce different distribution results, which is in line with reality and is also ignored in the existing research. In summary, the existing research ignores the influence of OD constraints and uncertain disturbance factors when researching the load redistribution.

To deal with the shortcomings of the existing research, this research proposes a load redistribution method considering uncertain disturbance factors and OD constraints. The Bureau of Public Roads (BPR) road impedance function was used to improve the conditional value-at-risk (CVaR), and the travel time was used to construct a loss function to quantify the uncertainty of the impact of disturbance factors on the road segments in the network. This is the first innovation of this research. Further, the load redistribution problem was transformed into the path selection problem with potential disturbances of the road segments in the failure scenario. Using the generalized travel time as the objective function, a nonlinear programming model was constructed, and a cellular ant colony algorithm was designed to solve the model. This is the second innovation of this research. Our research has significant values in both theoretical innovations and practical applications, and can provide a theoretical basis for the risk control of multimodal transport networks in vulnerable regions.

The rest of this paper is organized as follows. Section 2 analyzes the characteristics of a multimodal transport network, mainly focusing on the structure of the network and the characteristics of cascading failures. In Section 3, the quantitative analysis of uncertain disturbance factors is conducted. Section 4 presents the risk control modeling and algorithm design, which is the theoretical research on the modeling and algorithms for load redistribution methods. Section 5 performs the empirical analysis to verify the feasibility of the proposed method and algorithm. Conclusions are made in Section 6.

2 Analysis of the characteristics of a multimodal transport network

2.1 Network structure

The multimodal transport network G (N, E) in this research is composed of three modes of transportation: rail, road, and aviation, which is essentially a multi-layer network, as shown in Fig. 1. In the figure, N ={ 1, 2, ⋯ , n } is the set of nodes in the network, and E ={ (i, j) |i, j ∈ n } is the set of road segments in the network. The nodes and road segments are classified according to their attributes: $N = N_{rail} \cup N_{road} \cup N_{aviation}$ (1) $E = E_{rail} \cup E_{road} \cup E_{aviation} \cup E_{transfer}$ (2)

Fig. 1

Multi-layer network diagram.

The linking edge between different modes of transportation is E_transfer, as shown in Fig. 2. All the E_transfer are defined as non-main roads, and the transfer process is to connect the corresponding nodes through the non-main roads.

Fig. 2

Network links diagram.

2.2 Analysis of network cascading failure

Multimodal transport network is a multi-state system (MSS), that is, the network not only has two extreme states of “normal operation” and “complete failure", but also has many states between the two states [20]. Usually, the abnormal state of a network is called failure state, and the failure degree is characterized by the change of travel time in the network. The polymorphism of a multimodal transport network results from the state of each road segment in the network. When the network is affected by disturbance factors and some nodes and road segments are interrupted, the original load on the network will flow to other nodes and road segments according to certain rules, causing fluctuations in the loads on the road segment. When these fluctuations exceed a certain threshold, the road segment will enter a failure state, and the newly-added load on the road segment will flow in other directions, causing network cascading failure. The greater the degree of network failure, the worse the network reliability.

3 Quantitative analysis of uncertain disturbance factors

3.1 Analysis of the influence of disturbance factors

The multimodal transport networks in complex environments are affected by uncertain disturbance factors, which will increase the network’s travel time, thereby affecting the network’s functions to varying degrees. In this research, the influences of different types of disturbance factors on network travel time are quantified. In fact, uncertain disturbance factors mainly come from the attack of natural disasters outside the network and the influence of load flow inside the network.

It is generally considered that the attack from external factors has the characteristics of strong destructiveness and long recovery time, such as debris flow, earthquake, tsunami, etc., which is characterized by the non-temporary interruption of some nodes and road segments, as well as the stopped flow of loads on nodes and road segments. The influence of internal factors of the network mainly manifests in the temporary interruption of nodes and road segments, which can be restored after a certain time, such as traffic control, traffic accidents, weather factors, etc. Both of these two kinds of disturbances will cause load stagnation on nodes and road segments, resulting in greater road impedance on the road segments. This research extends the road impedance function to calculate the change in network travel time.

The BPR road impedance function calculates the road impedance that varies with the load in different modes of transportation [21] according to: $t_{ij} (l_{ij}) = t_{0 ij} (1 + ɛ (\frac{l_{ij}}{c_{ij}})^{γ})$ (3) where t_ij (l_ij) represents the average travel time when load l_ij flows on the road segment E_ij, t_0ij represents the average travel time when the load on the road segment flows freely, $t_{0 ij} = \frac{d_{ij}}{v_{ij}}$ , d_ij is the length of the road segment E_ij, v_ij is the average traveling speed on the road segment E_ij, represents the load capacity c_ij on the road segment E_ij, ɛ and γ are the parameters. Here, we consider the influence of the characteristics of the disturbance factors on the travel time in the network. Let t_ij be the travel time of the road segment E_ij under the influence of the disturbance factors, and t_ij (break) be the interruption time of the road segment E_ij. The total road impedance of the road segment E_ij can be calculated by: $t_{ij} = t_{ij} (l_{ij}) + t_{ij} (break)$ (4)

Compared with the disturbance of internal factors, external factors will cause orders of magnitude change.

3.2 Risk measurement of uncertainties

The risk measurement of a multimodal transport network is used to calculate the degree of damage to the network caused by the potential failure risks. Here, CVaR is introduced for the calculation of any road segment in the network. CVaR is a method for assessing uncertain risks and is often used in investment decision-making in the financial field [22 –25]. Later, with the development of related theories, CVaR is gradually applied in the resource allocation [26, 27], network design [28 –30], system optimization [31 –33], etc. in the fields of power, transportation, and logistics, etc. The multimodal transport network is a transportation service system, and its failure risk is ultimately reflected in the travel time of the loads in the network. In the network G (N, E), t_ij is the travel time of the road segment E_ij under the influence of disturbance factors, and $t_{ij}^{0}$ is the travel time of the road segment E_ij under normal conditions. The variation $t_{ij}^{'}$ in weighted travel time is modified by: $t_{ij}^{'} = t_{ij} - t_{ij}^{0}$ (5)

Let f_ij (x, t′) be the loss function, where X = (x₁, x₂, ⋯ x_m) ^T is the weight of the different disturbance factors of the road segment E_ij, and α is the minimum threshold of the loss function. For a certain confidence level β ∈ (0, 1), the value-at-risk (VaR) can be obtained as: $VaR (t_{ij}^{'}) = min {α \in R : ψ (x, α) > β}$ (6)

Then the CVaR at the confidence level β is given by:

$\begin{matrix} CVaR (t_{ij}^{'}) & = E [f_{ij} (x, t^{'}) | f_{ij} (x, t^{'}) > VaR (t_{ij}^{'})] \\ = {(1 - β)}^{- 1} \int_{f_{ij} (x, t^{'}) > VaR (t_{ij}^{'})} f_{ij} (x, t^{'}) p_{ij} (t^{'}) {dt}^{'} \end{matrix}$ (7)

In order to facilitate the calculation, we draw on the research of ROCKAFELLAR [34] and simplify $CVaR (t_{ij}^{'})$ by:

$\begin{matrix} CVaR (t_{ij}^{'}) \\ = α + \frac{1}{m (1 - β)} \sum_{k = 1}^{m} [max (g (x_{k}, t_{ijk}^{'}) - α, 0)] \end{matrix}$ (8)

Equation (9) is used to calculate the CVaR of any road segment. Considering of the decision process of the road segment E_ij, a decision variable χ_ij ={ 0, 1 } is introduced. By adjusting the value of χ_ij, the CVaR of the corresponding path can be obtained as:

$\begin{matrix} F = CVaR (\sum_{i \in n} \sum_{j \in n} χ_{ij} t_{ij}^{'}) = \sum_{i \in n} \sum_{j \in n} χ_{ij} \\ [α + \frac{1}{m (1 - β)} \sum_{k = 1}^{m} [max (g (x_{k}, t_{ijk}^{'}) - α, 0)]] \end{matrix}$ (9)

4 Risk control modeling and algorithm design

The disturbing factors in a multimodal transport network are generated by the objective environment, which is unavoidable. The influences of the disturbance factors on the network is manifested in the failure state and the failure degree. This research considers that although the disturbance factors cannot be changed, we can reduce the network’s failure scope and degree to a certain extent, that is, control the cascading failure of the network, and the key is the load redistribution on the road segment.

4.1 Problem description

The multimodal transport network reaches equilibrium in the initial state. When disturbance factors in the network occur, the equilibrium is broken, and the attributes of some road segments and nodes change. The loads of the network need to be redistributed. In order to facilitate mathematical analysis and consider both the global and local factors, we convert the load redistribution problem into a path optimization problem that considers uncertain risks. The original network is updated to a new network after being disturbed, and the original OD of the load changes accordingly. The new OD of the load starts from the position in the equilibrium state, and the original destination does not change. Minimizing the generalized travel time is used as the optimization goal to establish a mathematical model, and the optimization result is the result of load redistribution.

In order to facilitate the research, the following assumptions are made:

The total load in the network is constant, and it will not increase or decrease;

All loads in the network are completely rational, and can fully obey the scheduling during the process of load redistribution;

The failure effect of the node is equivalent to the failure of all edges connected to the node.

The connections between different transportation modes in the network all pass through non-main roads.

The meanings of the parameters as displayed in Table 1.

Table 1
Parameter description

Parameter Meaning Parameter Meaning

$χ_{ij}^{h}$ Decision variable $g (x_{k}, t_{ijk}^{'})$ Loss function, the estimated loss of disturbance factors acting on the road segment E_ij

t _0ij The travel time for free flow through road segment E_ij L Total number of loads

t_ij (break) The interruption time of road segment E_ij, only for the disturbed road segments ɛ, γ Parameter

α The threshold of loss function, when the system is not disturbed, α = 0 β Confidence level, which represents the degree of trust to the loss function by the load

Parameter	Meaning	Parameter	Meaning
$χ_{ij}^{h}$	Decision variable	$g (x_{k}, t_{ijk}^{'})$	Loss function, the estimated loss of disturbance factors acting on the road segment E_ij
t _0ij	The travel time for free flow through road segment E_ij	L	Total number of loads
t_ij (break)	The interruption time of road segment E_ij, only for the disturbed road segments	ɛ, γ	Parameter
α	The threshold of loss function, when the system is not disturbed, α = 0	β	Confidence level, which represents the degree of trust to the loss function by the load

4.2 Model establishment

Minimizing the generalized travel time is used as the objective function to establish a mathematical model. The generalized travel time includes the ideal travel time, the travel time increment caused by road impedance, and the travel time increment based on CVaR at a certain confidence level. The travel time increment caused by road impedance is shown in Equation (10), and the constraints are shown in Equations (11)–(18).

$\begin{matrix} min Z = min (z_{o} + Δ z_{path} + Δ z_{CVaR}) \\ \begin{matrix} \end{matrix} = min {\begin{matrix} \sum_{h} \sum_{i = 1}^{n} \sum_{j = 2}^{n} χ_{ij}^{h} t_{0 ij} + \sum_{h} \sum_{i = 1}^{n} \sum_{j = 2}^{n} χ_{ij}^{h} [t_{0 ij} (1 + ɛ {(\frac{\sum_{h} χ_{ij}^{h}}{c_{ij}})}^{γ}) + t_{ij} (break) - t_{0 ij}] \\ + \sum_{h} \sum_{i = 1}^{n} \sum_{j = 2}^{n} χ_{ij}^{h} [α + \frac{1}{m (1 - β)} \sum_{k = 1}^{m} [max (g (x_{k}, t_{ijk}^{'}) - α, 0)]] \end{matrix}} \\ \begin{matrix} \end{matrix} = min {\sum_{h} \sum_{i = 1}^{n} \sum_{j = 2}^{n} χ_{ij}^{h} [\begin{matrix} (t_{0 ij} (1 + ɛ {(\frac{\sum_{h} χ_{ij}^{h}}{c_{ij}})}^{γ}) + t_{ij} (break)) \\ + [α + \frac{1}{m (1 - β)} \sum_{k = 1}^{m} [max (g (x_{k}, t_{ijk}^{'}) - α, 0)]] \end{matrix}]} \end{matrix}$ (10)

$s . t . g (x_{k}, t_{ijk}^{'}) = {\begin{matrix} 0, & The disturbance does not affect the travel time of E_{ij} \\ > 0 & The disturbance can affect the travel time of E_{ij} \end{matrix}$ (11) $χ_{ij}^{h} = {\begin{matrix} \begin{matrix} 0, & T he load h does not select E_{ij} \end{matrix} \\ \begin{matrix} 1, & T he load h can select E_{ij} \end{matrix} \end{matrix}$ (12) $t_{ij} (break) {\begin{matrix} 0, & E_{ij} is undisturbed \\ > 0 & E_{ij} is disturbed \end{matrix}$ (13) $\sum_{h} χ_{ij}^{h} = L$ (14) $t_{0 ij} = \frac{d_{ij}}{v_{ij}}$ (15) $α ⩾ 0$ (16) $β \in (0, 1)$ (17) $i < j$ (18)

In the model, Equations (11)–(13) are the description of the functions and parameter values, Equation (14) guarantees the conservation of loads, and Equation (15) is the way to calculate t_0ij. Equations (16)-(17) define the range of parameter values, and Equation (18) ensures that the road segments of the network will not be repeatedly calculated.

4.3 Algorithm design

In order to solve the model, this research combines the idea of cellular automaton (CA) with ant colony optimization (ACO) to propose the cellular automaton ant colony optimization (CAACO) algorithm with efficient solving performance. Ant colony algorithm is a heuristic algorithm that simulates ant foraging behavior. Ants keep a kind of biopheromone in the process of foraging. In the process of foraging, a single ant will tend to search along the path with the largest concentration of pheromone. Ants will affect the surrounding environment by releasing pheromone. At the same time, they will also be affected by the pheromone in the environment. After a period of searching, the optimal path will be repeated by a large number of ants. The pheromone of this path will be much more than other paths, and the opti-mal result will be determined finally. ACO algorithm has good performance in path optimization, but has the disadvantage of slow convergence and easily falling into local optimization, especially when we consider the path CVaR and disturbance response. This algorithm enriches the attributes of the path and increases the complexity of the calculation, making the search method in the algorithm particularly important. CA is a grid dynamic model with discrete time, space, and state. The spatial interaction and temporal causality are local. It has the ability to simulate the spatiotemporal evolution of complex systems and it can parallelize the evolution. The advantage of its combining with ACO algorithm is that it can put the process of ant colony optimization in the cell neighborhood structure of the cell, which reduces the blindness of the search and improves the algorithm performance. In this research, we design the algorithm from the perspectives of cell information setting, heuristic function design, pheromone update and CAACO algorithm flow.

4.3.1 Cell information setting

Each node in the network is defined as a cell. The cell information includes cell state, cell space, cell neighborhood, and cell rules, which is denoted as C = (C_S, C_W, C_N, C_R).

Cell state C_S: each cell has only two states, occupied or idle, which are labeled as 1 and 0, respectively, and the occupied cell can still be occupied again;

Cell space C_W: On the basis of considering the 2D geographic coordinates of the cell, we record the effect of the disturbance factor on the cell as the third coordinate, that is, the interruption time is taken as the third coordinate to fully reflect the cell attributes, i.e. W = 3. The cell space diagram is shown in Figs. 3 and 4. The height of the cell in Fig. 3 indicates the strength of the disturbance factor. The higher the cell, the stronger the disturbance. The number in each grid in Fig. 4 represents the interruption time of the corresponding cell.

Cell neighborhood C_N: The model we are solving is a single-target model. According to the conclusion of J. Kennedy [35], the closer the cells in the neighborhood are to each other, the more conducive to the optimization of the single-target problem. The extended Moore-type neighborhood structure can be applied to this problem.

Cell rule C_R: The evolution of cells is the process of updating the pheromone and finding the optimal value by continuously comparing the target function values of the central cell and its neighboring cells. When the ants are transferred from cell a to cell b, cell a is placed on the taboo list. The probability of the neighboring cell being selected is given by:

Fig. 3

3D diagram of cell space.

Fig. 4

2D diagram of cell space.

$P_{an} = {\begin{matrix} \frac{{(τ_{ab})}^{ς} \times {(η_{ab})}^{κ}}{\sum_{w \notin tabu} [{(τ_{aw})}^{ς} \times {(η_{aw})}^{κ}]}, & j \notin tabu \\ 0, & others \end{matrix}$ (19) where τ_ab and τ_aw represent the pheromone concentration between cells, η_ab and η_aw represent the heuristic function between cells, ς and κ are the pheromone heuristic factor and expected heuristic factor, respectively, tabu is the taboo list in the ant colony algorithm.

4.3.2 Heuristic function design

The heuristic function reflects the expected value of ants in selecting any cell. We convert the objective function in Equation (11) as the heuristic function:

$\begin{matrix} η_{ab} = [max (t_{a} (break), t_{b} (break)) \\ {+ [α + \frac{1}{m (1 - β)} \sum_{k = 1}^{m} [max (g (x_{k}, t_{abk}^{'}) - α, 0)]]]}^{- 1} \end{matrix}$ (20)

4.3.3 Pheromone update

After ants find a feasible path, the need to update the pheromone of all cells in the cell space to find the optimal path. Pheromone can be updated by: $τ_{ab} (t + 1) = (1 - ρ) τ_{ab} (t) + Δ τ_{ab} (t)$ (21) where ρ ∈ [0, 1) represents the pheromone volatilization factor, Δτ_ab (t) is the global update factor, A is the total number of ants, Q is the pheromone in a cycle, usually being a constant, Z_k is the objective function value of a feasible path in a cycle, S^* is the set of cells in the selected path. Δτ_ab (t) and $Δ τ_{ab}^{k} (t)$ can be calculated by: $Δ τ_{ab} (t) = \sum_{k = 1}^{A} Δ τ_{ab}^{k} (t)$ (22) $Δ τ_{ab}^{k} (t) = {\begin{matrix} \frac{Q}{Z_{k}}, & a, b \in S^{*} \\ 0, & others \end{matrix}$ (23)

4.3.4 CAACO algorithm flow

The flow of CAACO algorithm is as follows:

Step 1: Determine the cell information C = (C_S, C_W, C_N, C_R) and construct a cell map;

Step 2: Initialize the parameters and record the number of iterations as Iter = 0;

Step 3: Initialize the pheromone and heuristic functions of all cells to obtain the initial pheromone matrix and heuristic function matrix;

Step 4: Set A ants in each ant colony, a total of L ant colonies are placed at their corresponding cells, perform parallel path search, and record the number of iterations Iter = Iter + 1;

Step 5: Obtain A feasible sub-paths in each ant colony, and calculate the generalized travel time Z^* under the most feasible L sub-paths;

Step 6: Update the heuristic function and pheromone, update the feasible path, compare the generalized travel time before and after the update, and assign a smaller value to Z^*;

Step 7: Determine whether the maximum number of iterations is reached, if yes, go to Step 8, otherwise, go to Step 4;

Step 8: Output the optimal path and end.

5 Empirical analysis

Empirical analysis was conducted in the Sichuan-Tibet region in China. The Sichuan-Tibet region is a typical region in China with a complex environment. The occurrence of natural disasters in this region shows a seasonal change pattern. The disasters are frequent in winter and summer. Due to its special trade channel location, the medium load flow of the network is inevitable Therefore, under the influence of uncertain failure factors, it is of great significance to research the network risk control from the perspective of load redistribution in Sichuan-Tibet region. The multimodal transport network in Sichuan-Tibet region is composed of three modes of transportation: rail, road and aviation. The network G (N, E) consists of 90 nodes and 177 edges, as shown in Fig. 5. Through the surveys conducted by the transportation departments of Sichuan Province and the Tibet Autonomous Region, we obtained the capacity on each road segment of the network, and determined the initial load on the road segment to be 2/3 of the capacity.

Fig. 5

Schematic diagram of the multimodal transport network in Sichuan-Tibet region.

5.1 Setting of related parameter

According to the BPR function given by the Bureau of Public Roads of the United States, in general, the value of the road resistance function is ɛ = 0.15, γ = 4, which is recognized by scholars in related fields and widely used. In fact, the road impedance varies for different modes of transportation. Rail is limited by infrastructure. When the load is less than the capacity, the increase in load has little effect on the speed of the rail. When the load is greater than the capacity, the impact is very large. The laws for road, aviation and transfer are reversed. According to the research of Berche et al. [36] and Mattsson et al. [37], the road resistances of different modes of transport are expressed by parameters adjustment, which is more suitable for the actual situation. The transfer edges were set as non-main roads. The parameter values of different transportation modes are shown in Table 2.

Table 2
Parameters of BPR road impedance function

Transportation mode ɛ γ v_ij (km/h)

Rail 0.15 8 90

Road 0.15 4 60

Aviation 0.15 4 800

Transfer 0.15 4 30

Transportation mode	ɛ	γ	v_ij (km/h)
Rail	0.15	8	90
Road	0.15	4	60
Aviation	0.15	4	800
Transfer	0.15	4	30

Through the investigation and survey at Meteorological Service and Department of Transportation of Sichuan Province and Tibet Autonomous Region as well as the Chinese Meteorological Disasters, we classified the disturbance factors and their effects on the network, as shown in Table 3. It’s worth noting that we can’t accurately describe the specific time of each disturbance, which is represented by the average interruption time. For the short-term unrecoverable disturbance, the value of 100 is to distinguish it from the short-term recoverable disturbance on the order of magnitude. For the short-term recoverable disturbance, this data is obtained from the experience of the staff of the Department of transportation. In fact, it is consistent with our cognition Of. The weights of disturbance factors on each road segment are different.

Table 3

Classification of disturbance factors

Disturbance factor	Type	Interruption time (h)
Severe natural disasters (earthquakes, debris flows, landslides, etc.)	Unrecoverable in a short time	100
Adverse weather conditions (torrential rain, blizzard, gusty wind, etc.)
Factors of the road segment (maintenance, upgrade, etc.)
Major traffic accidents
Traffic control	Recoverable in a short time	4
General traffic accidents		5
Other emergencies		3

In CVaR, the road segment is generally considered as not being affected by the disturbance factor, that is α = 0. The disturbance to the road segment is “extra". The m in X = (x₁, x₂, ⋯ x_m) ^T was all set as m = 8, where x₁ ∼ x₇ is the weight of 7 disturbance factors. In order to ensure the computability of α = 0 in the calculation, we added “zero disturbance” as one of the disturbance factors, with the corresponding interruption time as 0 and x₈ as the corresponding weight.

The setting of the parameters in the CAACO algorithm is shown in Table 4, these parameters are used to represent the behavior of ants in the algorithm. Although each parameter can have different values, in fact, this does not have a subversive change to the solution results. In the research of related scholars, the values in a certain range are reasonable [38, 39]:

Table 4

Setting of the parameters in the CAACO algorithm

Parameter	Meaning	Value	Parameter	Meaning	Value
A	The number of ants in each ant colony	50	ρ	Pheromone volatilization factor	0.3
Q	Pheromones in a cycle	500	ς	Pheromone heuristic factor	3
Iter (max)	The maximum number of iterations	200	κ	Expected heuristic factor	2

5.2 Numerical analysis

Factors such as the confidence level of the load in the network and the number of disturbed road segments all have an impact on the effect of load redistribution. In order to fully measure the effectiveness of the network’s load redistribution method, we randomly selected the road segments in the initial multimodal network to add disturbance factors. In consideration of the actual situation, the proportion of the disturbed road segment was set in the range of 0% -20%. The ratio of unrecoverable disturbances in short time to recoverable disturbances in short time was set as 1:2. The ratio of recoverable disturbance factor was set as 1: 1: 1, and the confidence level as 0.5, thereby changing the state of the network. The confidence level was set to vary between 0 and 1, and the proportion of the disturbed road segments was 10%. Comparison was conducted between the proposed method of load redistribution and the method without load redistribution. The comparison results are presented in Figs. 6 and 7.

Fig. 6

Analysis of load redistribution effect.

Fig. 7

Effect of the network without load redistribution.

By comparing Figs. 6 and 7, it can be found that the proposed load redistribution method shows a good optimization effect when the network is disturbed, which weakens the role of the disturbance factor, reduces the total travel time of the network, and improves the performance of multimodal transport services. From the perspective of the proportion of disturbed road segments, with the gradual increase of the proportion, the difference between the results of the two methods become increasingly larger, the growth rate of the generalized travel time gradually slows down, the optimization effect gradually becomes obvious. The difference at 20% is the largest, indicating that the greater the disturbance to the network, the more reasonable measures should be taken to control the risk of cascading failure of the network, otherwise the network will easily collapse. The impact of fluctuations in the confidence level on the network shows a periodic change. The operation without load redistribution is not affected by the confidence level. For the operation of load redistribution, when the confidence level is between 0 and 0.733, the generalized travel time of the network and load redistribution results remain unchanged, indicating that when β ∈ (0, 0.733], the load exhibits a neutral attitude to risk. When β ∈ [0.733, 1), the generalized travel time begins to increase, and the load redistribution results also change, indicating that when β ∈ [0.733, 1), the load shows a disgusted attitude towards risk, and as the level of disgust increases, the generalized travel time increases increasingly faster. At this time, the load chooses to sacrifice travel time to meet the risk avoidance, which is in line with the real situation. The load redistribution based on the confidence level of uncertain disturbance factors can provide a reasonable load distribution basis for decision makers.

5.3 Analysis of the proposed algorithm’s effectiveness

The proposed CAACO algorithm was compared with ACO algorithm, Cellular Automaton Genetic Algorithm (CAGA). The maximum number of iterations was set to 200 when the confidence level was 0.5 and the proportion of disturbed road segments was 10%. The relevant parameters of the ant colony algorithm were consistent with those of the CAACO algorithm. The CAGA adopted real number coding, and its relevant parameters are shown in Table 5. The convergence effects of different algorithms were obtained, as shown in Fig. 8.

Table 5
Setting of the parameters of CAGA

Parameter Value Parameter Value

Population length 10 Number of iterations 200

Population width 10 Crossover probability 0.8

Population size 100 Mutation probability 0.35

Parameter	Value	Parameter	Value
Population length	10	Number of iterations	200
Population width	10	Crossover probability	0.8
Population size	100	Mutation probability	0.35

Fig. 8

Comparison of different algorithms.

It can be seen that the results of the three algorithms are the same when they are stable, and the generalized travel time is 674 h, so the pros and cons of the algorithms need to be compared for the convergence speed. In terms of convergence speed, the convergence speed of the CAACO algorithm is significantly higher than that of the other two algorithms. Due to the similarity of the solution mechanism, the trends of CAACO and ACO algorithms during the iteration process are similar, and this is because both of them use the pheromone concentration of ants to find the target path. With the constant update of pheromone concentration, the search of the target path will be faster and faster. The CAACO algorithm can obtain the optimal solution when the number of iterations is about 75, and the optimal solution of the ACO algorithm can be obtained only when the number of iterations is close to 160, indicating that the cellular automaton’s neighborhood mechanism can well solve the problem of low efficiency in the ACO algorithm, and the key to improve the solving speed of ACO is to use the evolution rules of cells to carry out parallel computing and improve the global sharing ability of pheromones; By contrast, for the same neighborhood mechanism in CAGA, the optimal solution is obtained when the number of iterations is around 130. This is due to a defect in the search mechanism of the genetic algorithm. Genetic algorithm uses a random search when solving, which may result in repeated calculations multiple times, reducing the efficiency of the solution. At the same time, random search will lead to a large uncertainty in the optimization process of the solution, causing the quality of the solution to fluctuate. Therefore, the comparison results have verified the effectiveness of the CAACO algorithm in this research.

6 Conclusion

The risk control in the multimodal transport network is investigated from the perspective of network load redistribution. By analyzing the characteristics of the multimodal transport network, the uncertain disturbance factors in the network are quantified, a mathematical model is constructed and an improved solution is designed on this basis. The conclusions are summarized as follows:

This research considers the impact of the characteristics of disturbance factors on the travel time in the network, and uses the BPR road impedance function to improve the CVaR to measure the uncertain risk factors in the multimodal transport network. These factors are used as an important indicator for selecting the load distribution path, which is of applicability to the research of load re-distribution in multimodal transport networks.

This research transforms the load redistribution problem in the multimodal transport network into a path optimization problem that considers uncertain risk factors, and builds a mathematical model of path optimization with the smallest travel time by including ideal travel time, travel time increment based on road impedance, and CVaR-based travel time increment at a confidence level. This makes up for the shortcomings of the traditional load redistribution strategies that do not consider OD constraints and uncertainties of network risk.

Aiming at the model, the CAACO algorithm with efficient solving ability is proposed. The algorithm combines the advantages of cellular automaton and ant colony algorithm. In the empirical analysis, the comparison of the proposed CAACO algorithm, ant colony algorithm and CAGA verifies that the CAACO algorithm has good convergence in solving this kind of problem.

The empirical analysis takes the intermodal transport network of Sichuan-Tibet region in China as the research object. The region is highly affected by the complex environment and the probability of network interruption is high. The path optimization model based on CVaR and the CAACO algorithm proposed in this paper are applied to dealing with the problem of load redistribution in the multimodal transportation network in the region. The effect of network load redistribution indicates that the model and algorithm also have good applicability.

The risk control method of multimodal transport network is not limited to the load redistribution in the network. We will conduct in-depth research on network risk control from different perspectives, and improve related theories and methods.

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Risk control of the cascading failure of multimodal transport network considering uncertain disturbance factors 1

Abstract

Keywords

1 Introduction

2 Analysis of the characteristics of a multimodal transport network

2.1 Network structure

3 Quantitative analysis of uncertain disturbance factors

3.1 Analysis of the influence of disturbance factors

4.1 Problem description

4.3.1 Cell information setting

5 Empirical analysis

Table 2 Parameters of BPR road impedance function Transportation mode ɛ γ v ij (km/h) Rail 0.15 8 90 Road 0.15 4 60 Aviation 0.15 4 800 Transfer 0.15 4 30

Table 5 Setting of the parameters of CAGA Parameter Value Parameter Value Population length 10 Number of iterations 200 Population width 10 Crossover probability 0.8 Population size 100 Mutation probability 0.35

References

Table 2
Parameters of BPR road impedance function

Transportation mode ɛ γ v_ij (km/h)

Rail 0.15 8 90

Road 0.15 4 60

Aviation 0.15 4 800

Transfer 0.15 4 30

Table 5
Setting of the parameters of CAGA

Parameter Value Parameter Value

Population length 10 Number of iterations 200

Population width 10 Crossover probability 0.8

Population size 100 Mutation probability 0.35