Model and algorithm of stochastic dynamic traffic assignment based on dynamic rainfall intensity

Abstract

Frequent occurrence of urban rainy weather, especially rainstorm weather, affects transportation operation and safety, so it is essential that effective intervention measures to recover disordered traffic be adopted and then analyzed for their influence on the dynamic network. Therefore, models and algorithm to show dynamic traffic flow of traffic network in rainy weather are a fundamental need and have drawn great interest from governments and scholars. In this paper, innovative content contains a travel cost function considering rainfall intensity; considering the travel cost function, a dynamic traffic assignment model based on dynamic rainfall intensity is built. Then a corresponding algorithm is designed. Moreover, this study designs three scenarios under rainfall and analyzes the influence of the rainfall on an example network. The results show that rainfall has a significant effect on traffic flow. The finding proved the proposed models and algorithm can express the development trend of path flow rate on a dynamic network under rainfall.

Keywords

Traffic network rainfall intensity travel cost iterative loading algorithm

1 Introduction

The intensity of rainfall is heavy in the cities of China, especially in the southern coastal cities. In summer, rainstorm weather occurs more frequently. Accumulated ponding in the road caused by rainstorms will reduce the vehicle speed and increase traffic congestion. Simultaneously, the size of urban areas is very large, and different areas of the same city often have different rainfall intensities, which leads to different impacts on roads in the network. Changes in road traffic status lead to new changes in the traffic flow distribution. Therefore, it is necessary to model and analyze traffic assignment problem with dynamic rainfall intensity.

Based on California metropolitan data, Dhaliwal et al. [1] verified that free-flow speed of road reduced by 5.7% under light rain, 11.71% speed under medium rain and 10.22% under heavy rain. Nogal et al. [2] presented an effective probabilistic framework which can be used to evaluate the resilience of a transportation network impacted by extreme weather, and the framework supports the decision-making process. Frei A et al. [3] integrated weather-responsive system into demand models in the network traffic estimation, and the new system was able to analyze the impact of demand strategy. Hou et al. [4] modified the Greenshields under adverse weather and calibrated weather adjustment factors in the model for four urban areas based on the data of United States. The results showed that the weather effects in the calibrated models were realistic. Jia et al. [5] built a rainfall-integrated module in FLOWSIM. FLOWSIM is a microscopic simulation software in the field of traffic. The module could show the impact of rainy weather on urban traffic systems and realize traffic simulation of rainfall scenarios. Ivanović and Jović [6] focused on the rain impact on urban roads in the context of transport supply, and the traffic capacity could be determined as a product of sensitivity coefficients. Ahmed and Ghasemzadeh [7] analyzed driver behavior based on matching trips and naturalistic driving data in normal weather and heavy rain conditions. They discover that the drivers would decrease vehicle speed 5 km/h over the speed-limit, and the probabilities reach 23% in light rain and 29% in heavy rain. Many studies have been performed to explore traffic characteristics in rainy weather. However, most of these studies analyzed the overall traffic characteristic reductions under rainfall, while papers quantifying and formulating the influence of rainfall intensity on road traffic characteristics and road traffic flow are rare.

This study modeled and solved a stochastic dynamic traffic assignment (DTA) problem based on a Logit model with rainfall intensity. The purpose of model and algorithm show that rainfall affects the distribution of traffic flow on the dynamic network. A numerical example containing three scenarios on a test network was designed to validate the impact of rainfall on traffic network. This study is organized as follows. Section 2 briefly reviews the correlational research about traffic characteristics in rainfall and DTA models. This section also show the shortage in existing studies, then proposed the contribution and innovation point of this study. In Section 3, this paper describes a modified travel cost function considering rainfall intensity and establishes the stochastic dynamic traffic assignment model with this travel cost function. In Section 4, a solution algorithm is designed. Section 5 provides and analyzes an example of the road traffic network and scenarios. At the end, the results and suggestions are shown in conclusion.

2 Literature review

2.1 Traffic characteristics in rainfall

Rainfall can affect the numerical values of speed, free-flow speed, capacity, and other traffic characteristics. Some experts have analyzed the influence of rainfall based on various data. Agarwal et al. [8] classified rain events and analyzed the trends of intensity based on freeway loop data and weather data in the Twin Cities. Their research showed that rain intensity affects speed, headway, and capacity. Billot et al. [9] used the Van Aerde model to fit the measured data of the road and analyzed the velocity–density–flow basic diagram under different rainfall intensities. They found that road free-flow speed and road capacity decreased, road critical speed reduced, and the blocking density would not change, but their research lacked data on heavy rain. Seherman [10] quantified the impact of weather on the queue flow rates at the bottlenecks of freeway and found 27% reduction in heavy rain using the data of California. Alhassan and Ben-Edigbe [11] studied the influence of rain on road traffic flow volume and capacities of highway based data from pneumatic tube detector. It was concluded that speed reduced 3.52% and flow rate reduced 8.64% under rainfall conditions. Liu et al. [12] compared and analyzed the differences in traffic flow characteristics on Beijing expressways under normal weather and snowy weather and described the influence of snow on the traffic flow volume and average speed of expressway in Beijing. Bartlett and Lao [13] analyzed the influence mechanism of severe weather on road traffic rates in freeways. Then the effects of weather factors showed that the volumes decreased when inclement weather occurred comparing with the dry weather volumes. Angel et al. [14] examined driver response to rainfall on freeways under dry versus rainy conditions for each interstate segment. The results indicated that the mean travel speeds were reduced during rainfall events.

Moreover, some studies have analyzed the variation in the relationships of traffic characteristics under rainfall weather. Akin et al. [15] analyzed the average speed, density, and traffic flow rates of expressway separately in rainfall weather and snowy weather. Considering the factors of heavy vehicles and weather, a speed–density logarithm formula was proposed, and the results showed that the average speed reduction rate under rainfall weather is 8–12%. Zhang et al. [16] determined the distribution of microscopic traffic flow parameters at various types of rain with the observed road traffic flow data and measured rainfall data of expressways. William [17] analyzed the effects of rainfall intensities on urban road traffic and found the decrease of road traffic flow parameters, including free-flow speed and road capacity based on the data of urban road section in Hong Kong. Hou et al. [4] calibrated the weather adjustment factors and traffic flow parameters. The results show that the visibility and rain obviously affected road free-flow speed and maximum traffic flow volume. Then, they classified weather conditions and calibrated free-flow speed and maximum flow rate using highway data under each weather condition. Calvert and Snelder [18] showed that rainfall reduced the capacity and traffic demand in highway data from The Netherlands.

Some studies developed systems involving rainfall and traffic, then analyzed the influence of rainfall. Qiu et al. [19] designed a forecasting model to forecast traffic flow during rain events by the methods of data fusion. They verified that the accuracy of model is better under ideal precipitation, but the model is not ideal under heavy rain. Ding et al. [20] developed a weather–traffic index system. The system can be used to verify the regional index about weather–traffic sensitivity on a city area and they found regional characteristics observably affected the region’s weather–traffic index. Guo et al. [21] found that the transportation system suffer various degrees of damage caused by torrential weather. The extent of the influence of torrential weather is affected by the network structure. And torrential weather will not only damage roads in the traffic supply but also decrease the traffic demand.

Existing studies have analyzed the overall reduction in traffic characteristics in rainfall and calibrated the traffic parameters under rainfall. However, papers applying rainfall intensity to the dynamic traffic network are rare.

2.2 DTA models

DTA models are important tools used to analyze problems in traffic networks. They can be classified into analytical model and simulation model [22 –24]. By considering time dependence, traffic assignment models cover static traffic and dynamic traffic. For travel cost perception, models have been developed from certain cost to random cost [25]. Furthermore, some researchers improved the algorithm efficiency of models, which could be more applicable to big data volume and complex networks [26 –30]. Zhao and Huang [31] proposed travel satisfaction and modeled bounded rationality functions of traffic assignment for user equilibrium. Liu et al. [32] proposed a dynamic activity–travel traffic assignment in multistate super networks. This model is formulated discretely to express user equilibrium problem in dynamic network, and a dynamic link function is established to accommodate different characteristics of the links. Hoang et al. [33] built a new mathematical discrete programming model of stochastic DTA considering information and user equilibrium. They proved that linkage between user equilibrium status and system optimal status is underpinned by the first in, first out principle.

Some studies have integrated DTA models with traffic characteristics and applied the models to other traffic fields. Barthélemy and Carletti [34] established a DTA model with strategic agents. They provided travelling agents with a strategy to evaluate in actual time and re-route their travel paths at any road network intersection considering instantaneous changes to the road network. In the research background of a smart city, Yang et al. [35] considered the Internet of Things (IoT) and analyzed dynamic traffic network assignment using a continuous big IoT input database. Zhu et al. [36] integrated DTA with travel behavior model using agent-based microsimulation considering the influence of land development. They calibrated the modeling system with an optimization approach of simulation. Cantelmo and Viti [37] quantified the utility lost caused by traffic congestion. Then they evaluated the impact of activity scheduling and activity time with a DTA model based on user equilibrium. Wang et al. [38] reviewed DTA models applied to sustainable road traffic considering environmental factors, mainly including emission pricing and vehicular emissions.

In the field of emergencies and inclement weather, more studies focus on evacuation, and studies considering DTA are rare. Chen and Xiao [39] modeled the DTA problem based on system optimization considering the shortest evacuation time. Nogal et al. [40] presented a dynamic assignment model with restricted equilibrium. This model could assess traffic network resilience when an emergency event occurs, and obtain the network resilience indices. Gao et al. [41] described day-to-day traffic flow evolution based on a deterministic dynamic assignment model considering degraded link capacity. Then, they analyzed travel time variations using the exponential smoothing filter. He and Peeta [42] analyzed the dynamic resource allocation problem for transportation evacuation with a mixed linear programming model. The model could be applied in evacuation planning and operations.

The studies above promote the research of dynamic traffic flow assignment. However, in the field of inclement weather, the factor of rainfall intensity is rarely considered in the DTA models. Therefore, this study connects dynamic traffic flow assignment with dynamic rainfall to establish the stochastic DTA (SDTA) model considering the free-flow and modified travel cost function with rainfall intensity.

3 DTA model formulation considering rainfall intensity

3.1 Discretize the continuous time

In the DTA network, the time is continuous. For calculating and analyzing dynamic models conveniently, this study discretizes the continuous time. The analysis time period [s₀, s₁] is divided into K intervals. The intervals can be defined as δ = (s₁ - s₀)/K. Then, the analysis time period can be expressed by [k₀, k₁] and k ∈ K. For traffic network, the node set of travel origin is O, and the node set of travel destination is D.

3.2 Modified travel cost function considering rainfall intensity

In urban traffic network, the traffic environment will change with rainfall intensity increasing, especially that the free-flow speed of urban road decreases. William [17] proposed an index function of free-flow speed considering rainfall intensity, shown in Equation (1). The function is appropriate for urban roads, and this study adopted this function to describe the variation in free-flow speed of road. $f_{a} = exp (λ r^{μ} + φ)$ (1)

In the equation above, the free-flow speed of road a with rainfall intensity r is f_a, and λ, μ, and φ are the parameters of the function.

Based the equation of road free-flow speed with rainfall intensity, this study designed a speed function, given in Equation (2). This function states that when the free-flow speed is lower than the vehicle minimum speed, it can be considered that the rain has damaged the road so that it is impossible to pass, and the speed should be zero. The speed is related to the vehicle number on the road and road free-flow speed.

$v_{a} (k) = {\begin{matrix} v_{a}^{min} + (f_{a} - v_{a}^{min}) [1 - (\frac{X_{a} (k)}{L_{a} w_{aj}})^{α}]^{β}, f_{a} > v_{a}^{min} \\ 0, f_{a} \leq v_{a}^{min} \end{matrix}$ (2)

In Equation (2), a is the road corresponding these variables, and k is the discretize time corresponding these variables. v_a (k) is the speed of vehicle on the road; f_a is the free-flow speed of road a; $v_{a}^{min}$ is the vehicle minimal speed on the road; X_a (k) is the vehicle number on the road at time k; w_aj is jamming density of the road; and α, β are parameters of the function.

According to the speed function, it can be seen that rainfall intensity will impact vehicle speed. And vehicle speed will impact travel cost. So combining the above two equations, this paper propose the modified road travel cost function with rainfall intensity shown in Equation (3). $t_{a} (k) = {\begin{matrix} \frac{L_{a}}{v_{a}^{min} + [exp (λ r^{μ} + φ) - v_{a}^{min}] [1 - (\frac{X_{a} (k)}{L_{a} w_{aj}})^{α}]^{β}}, r < \sqrt{(ln v_{a}^{min} - φ) / λ^{μ}} \\ \infty, r \geq {\sqrt{(ln v_{a}^{min} - φ) / λ^{μ}}}_{a} \end{matrix}$ (3)

Where L_a is the road length of a, t_a (k) is the road travel cost of a, and other variables are already described above.

Then, based on the formula of road travel cost, the real path travel cost can be given by Equation (4). $C_{p} (k) = \sum_{a \in p} t_{a} (k), \forall k$ (4)

In an actual transportation network, traveler chooses travel path by perceiving the cost of the potential paths between origin and destination. The perceived path travel cost of path p is τ_p (k) and it is an estimated value taking into account real path travel cost C_p (k). A difference value between perceived travel path cost and real travel path cost is defined as random variable ɛ_p (k). Based on this random variable, this paper can formulate the perceived path travel cost as shown in Equation (5). (Gao, 2005) $τ_{p} (k) = C_{p} (k) - ɛ_{p} (k) / θ_{r}, \forall p \in p^{od}, \forall k$ (5)

Then this paper supposes that random variable ɛ_p (k) follows the Gumbel distribution, the minimum expected travel cost between the OD pair is described as $C^{od} (k)$ , function is shown in Equation (6): $C^{od} (k) = - \frac{1}{θ_{r}} ln \sum_{p} exp [- θ_{r} C_{p}^{od} (k)], \forall k \subseteq [k_{0}, k_{1}]$ (6)

In the formulas above, $C^{od} (k)$ is minimum expected value of travel cost between OD at k time; $C_{p}^{od} (k)$ is path travel cost using path p between OD at k time; and θ_r is error measure of path perceived cost.

3.3 Departure time choice and path choice model

In a transportation network, considering both departure time and path, the network will be defined as equilibrium status, only under the condition that no traveler could reduce their perceived path travel cost by unilaterally transforming their path and departure time.

Considering perceived path cost at different departure time, travelers make the choice of travel path and departure time. Supposing the perceived random error follows the Gumbel distribution, the Logit model can be gotten to describe the preference of travel path choice and departure time choice [43]. Then based on the Logit model, the departure time choice and travel path choice in DTA model can be expressed as Equations (7) and (8). $f^{od} (k) = T^{od} \frac{exp [- θ_{t} C^{od} (k)]}{\sum_{k = 1}^{k} exp [- θ_{t} C^{od} (k)]}, \forall k$ (7) $f_{p}^{od} (k) = f^{od} (k) \times \frac{exp [- θ_{r} \times C_{p}^{od} (k)]}{\sum_{p \in P^{od}} exp [- θ_{r} \times C_{p}^{od} (k)]}, \forall k$ (8)

In Equations (7) and (8), the departure flow rate for OD pair at k time is $f^{od} (k)$ , which represents thedeparture time choice for travelers. $f_{p}^{od} (k)$ is defined as the path flow rate using path p for OD pair at k time, which describes the path choice for travelers. T^od is the traffic generated for OD pair. The minimum expected value of travel cost of OD pair $C^{od} (k)$ can be obtained from Equation (6). The path travel cost using path p between OD pair $C_{p}^{od} (k)$ can be obtained from Equations (3) and (4). θ_t is the error measure of the perceived departure time; and θ_r is the error measure of perceived path cost.

3.4 Road state equations

In dynamic traffic assignment model, this paper uses traffic loading to describe road state. The traffic loading is the instantaneous vehicle number on the road, which is a spatiotemporal indicator. As this paper discretizes the time in DTA model, road state formula can be expressed as follows [44]:

$\begin{matrix} X_{ap}^{od} (k) = B_{ap}^{od} (k) - E_{ap}^{od} (k), \forall (o, d), \\ \forall p \in P^{od}, \forall a \in A, \forall k \end{matrix}$ (9)

Where $X_{ap}^{od} (k)$ is the instantaneous vehicle number on road a at k time following path p for OD pair. The accumulative vehicle amount flowing into road a at k time following path p for OD pair is defined as $B_{ap}^{od} (k)$ . And $E_{ap}^{od} (k)$ is the accumulative vehicle amount exiting from road a at time k following path p for OD pair.

3.5 Flow propagation function

On the dynamic traffic network, vehicles are flowing and traffic flow is propagating between origin and destination. So a flow propagation function is given in Equations (10), (11) and (12). $\begin{matrix} E_{ap}^{od} (k) = \sum_{k_{0}}^{J_{k}} b_{ap}^{od} (j) δ, \forall (o, d), \forall p \in P^{od}, \forall a \in A, \forall k \\ J_{k} = max j, j \in {j δ + t_{a} (j) \leq k δ} \end{matrix}$ (10) $b_{ap}^{od} (k) = {\begin{matrix} f_{p}^{od} (k), & the first road of path p \\ e_{a^{'} p}^{od}, & a^{'} is the road after a \end{matrix}, \forall (o, d), \forall p \in P^{od}$ (11) $e_{ap}^{od} (k) = \frac{E_{ap}^{od} (k) - E_{ap}^{od} (k - 1)}{δ}, \forall (o, d), \forall p \in P^{od}$ (12)

In the above equations, $b_{ap}^{od} (k)$ is the traffic rate flowing into road a at k time following path p for OD pair, which describes the instantaneous vehicle number flowing into road a per unit time. $e_{ap}^{od} (k)$ is the traffic flow rate exiting from link a at time k following path p for OD pair, which describes the instantaneous vehicle number exiting from road a per unit time. And the path flow rate at k time using path p for OD pair $f_{p}^{od} (k)$ can be obtained from Equation (8).

3.6 Other constraints

In addition to the above formulas, this paper defines other general constraints for DTA model. The constraints mainly contain initial variable constraints, traffic flow conservation constraints, and nonnegative constraints in Equations (13) to (17). $B_{ap}^{od} (0) = 0, E_{ap}^{od} (0) = 0, V_{ap}^{od} (0) = 0$ (13) $B_{ap}^{od} (k) = \sum_{j = 0}^{k} b_{ap}^{od} (j) δ$ (14) $\sum_{k} f^{od} (k) = T^{od}, \forall k$ (15) $\sum_{p \in P_{od}} f_{p} (k) = f^{od} (k), \forall k$ (16) $\begin{matrix} B_{ap}^{od} (k) \geq 0, b_{ap}^{od} (k) \geq 0, E_{ap}^{od} (k) \geq 0, e_{ap}^{od} (k) \geq 0 \\ X_{ap}^{od} (k) \geq 0, \forall (o, d) \forall p \in P^{od}, \forall a \in A, \forall k \end{matrix}$ (17)

4 Solution algorithm

Considering the models in Section 3 and the discrete dynamic network, this paper designs an iterative loading algorithm to solve the stochastic dynamic traffic assignment model with rainfall intensity. The detailed steps are as follows.

Step 1: Initialization

Set original state of traffic network. The network is empty. $\begin{matrix} B_{ap}^{od} (0) = 0, E_{ap}^{od} (0) = 0, X_{ap}^{od} (0) = 0, \\ \forall (o, d), \forall p \in P^{od}, \forall a \in A \end{matrix}$

Set discrete time interval δ, rainfall intensity r, and other parameters.

Set the traffic start time k = 1 and the start iteration number of algorithm n = 1.

Effective path set P^od containing all effective paths between OD pair is obtained by the traversing graph method.

Step 2: Travel cost

Calculate the road travel cost $t_{a} (k)$ with the designated rainfall intensity and number of vehicles on road a.

Based on the road travel cost, obtain the set of path travel cost by calculating path cost $C_{p}^{od} (k)$ between OD pair. Then obtain minimum expected travel cost C^od (k) for each OD.

Step 3: Traffic flow assignment

Calculate the departure flow rate f^od (k) between OD pair.

Calculate the travel path flow $f_{p}^{od} (k)$ between OD pair.

Step 4: Dynamic iterative network loading

Obtain $E_{ap}^{od} (k)$ , $e_{ap}^{od} (k)$ , $B_{ap}^{od} (k)$ , and $b_{ap}^{od} (k)$ using their corresponding formulas.

Based on the results, figure out the vehicle number on road a between OD pair.

Step 5: Modify the travel path flow. $\begin{matrix} f_{p}^{(n + 1)} = f_{p}^{(n)} + ψ^{(n)} (f_{p}^{(n)} - f_{p}^{(n)}) \\ ψ^{(n)} = \frac{n^{d}}{1^{d} + 2^{d} + 3^{d} + \dots + n^{d}}, d = 2 \end{matrix}$

Step 6: Convergence judgment

If $\frac{\sum_{od} \sum_{p \in P^{od}} \sum_{k = 1}^{K} | f_{p}^{od (n)} (k) - f_{p}^{od (n - 1)} (k) |}{\sum_{od} \sum_{p \in P^{od}} \sum_{k = 1}^{K} | f_{p}^{od (n)} (k) |} \leq η$ , then stop; otherwise, n = n + 1, k = 1 and return to Step 2.

5 Scenario design and numerical example

To validate the models and algorithms, this paper designed a numerical example and rain scenarios. In the example network, 9 nodes and 24 directed links were shown in Fig. 1. The length of each link was given in Table 1. Then this paper chose one origin and one destination to test, such as the OD pair is (1, 9).

Fig. 1

The network structure of the example.

Table 1

The lengths of links in the network (m)

Node ID	Node ID	Length (m)
1	2	4000
1	4	5000
2	3	5000
2	5	3600
3	6	5000
4	5	3500
4	7	5000
5	6	4000
5	8	4000
6	9	4000
7	8	4000
8	9	5000

The parameter setting of the example is as follows: The traffic generations are T¹⁹ = 1500 vehicle/h. The test duration is 3600 s, and the time interval δ is 10 s. The minimal velocity is $v_{a}^{min} = 5 km / h$ . Other parameters are listed:

θ_r = 0.01, θ_t = 0.005, α = 1.4, β = 3, λ = -0.047, μ = 2.739, and φ = 4.19.

Considering that different districts in cities often have different rainfall intensities because of the large size of cities, two scenarios of rainfall were set up to analyze the influence of rainfall, and one scenario of normal weather was set as the control. In the scenarios, some links were set under different rainfall intensities and other links were under normal weather. Rain intensity settings: In Scenario 1, the weather was normal weather. In Scenario 2, the rain intensity was 60 mm/h over link 4–5, and the other links were under normal weather. In Scenario 3, the rain intensity was 60 km/h over link 4–5 and 120 mm/h over link 7–8, and the other links were under normal weather.

During the test, the normal weather of Scenario 1, rainstorm of Scenario 2, and rainstorm of Scenario 3 were designed and analyzed. Different rainfall intensities were taken into account, resulting in different changes in free-flow speed, then the equilibrium traffic flow distribution was changed. Substituting the relevant parameter values into the program, the program reached the convergence condition in 3.5 seconds, after about 20 iterations, and then computed the results.

Table 2 shows the nodes of all paths in OD pair (1,9). Figure 2 reveals the relationship between the flow rates of all travel paths and time intervals under normal weather in Scenario 1. Each curve represents the flow rate of each path over time. The overall tendency for the flow rate of every path is to gradually decline from high to equilibrium status and then remain constant. We find that initial path flow rates are high. The reason is that empty initial network makes path travel costs low and attracts high traffic demand. As vehicles flow on the network, the vehicle number on the road increases, then road and path travel costs rise. As a result, the path flow rates reduce gradually and reach network equilibrium status.

Table 2

The nodes of paths in OD pair (1,9)

Path 1	1	2	3	6	9
Path 2	1	2	3	6	5	4	7	8	9
Path 3	1	2	3	6	5	8	9
Path 4	1	2	5	4	7	8	9
Path 5	1	2	5	6	9
Path 6	1	2	5	8	9
Path 7	1	4	5	2	3	6	9
Path 8	1	4	5	6	9
Path 9	1	4	5	8	9
Path 10	1	4	7	8	9
Path 11	1	4	7	8	5	2	3	6	9
Path 12	1	4	7	8	5	6	9

Fig. 2

The path flow rates of each path on each time in normal weather.

The path flow rates in the rainy weather of Scenario 2 are as shown in Fig. 3. Figire 4 describes the most significantly affected paths in Scenario 2 and the normal comparison curves. In Fig. 3, rainfall have a little influence on some paths which have high travel cost, because of high travel costs without rain resulting in few travel choices. For instance, the path flow rates of Path 7 and Path 11 are approximately zero all the time. Furthermore, from Figs. 3 and 4, we can see that the rain has a significant influence on the flow rates of Path 8 and Path 9. The two curves decrease overall, and the flow rates of the two paths in Scenario 2 are lower than those under normal weather. In the road network, Path 8 and Path 9 contain the rainy link 4-5. The initial path travel costs of the two paths are lower, so rainfall causes a decrease in free-flow speed and increases the path travel cost. The path flow rates decline obviously as the path travel cost increases.

Fig. 3

The path flow rates of each path in Scenario 2.

Fig. 4

The path flow rates of the most significantly affected paths in Scenario 2 compared with normal weather.

In the traffic network, the variation in path flow rates caused by different rainfall intensities in different regions is more complicated. In Scenario 3, two links of the road network were designated different rainfalls. Figure 5 shows the flow rates of paths in Scenario 3. Figure 6 shows the most significantly affected paths in Scenario 3 in comparison with Scenario 1 and Scenario 2.

Fig. 5

The path flow rates in Scenario 3.

Fig. 6

The path flow rates of the most significantly affected paths compared across the three scenarios.

For convenience in analyzing the results of Scenario 3, the most significantly affected paths in Fig. 6 are classified into three categories: the paths that do not include link 4-5 or link 5-6 are shown in Fig. 7, the path including link 5-6 is shown in Fig. 8, and the path including link 4-5 and link 5-6 is shown in Fig. 9.

Fig. 7

The flow rate of paths not including link 4-5 or link 5-6 compared across the three scenarios.

Fig. 8

The flow rates of Path 5 (including link 5-6) compared across the three scenario.

Fig. 9

The flow rates of Path 8 (including link 4-5 and link 5-6) compared across the three scenarios.

From Figs. 6 and 7, it can be seen that the path flow rate of Path 1 in Scenario 3 is higher than the flow rates of Path 1 in Scenario 1 and Scenario 2. Further, the flow rate of Path 1 in Scenario 2 is higher than the flow rate of Path 1 in Scenario 1. The trends of Path 5 and Path 10 in the three scenarios are consistent with that of Path 1. Each link in these paths which do not include link 4-5 or link 5-6 is always in normal weather. Other path flow rates affected by rainfall are partially transferred to paths that are not affected by rainfall, resulting in the flow rates of Path 1, Path 6, and Path 10 increasing with increasing rainfall intensity and rainfall range. Therefore, the flow rate in Scenario 3 is higher than that in Scenario 2, and the flow rate in Scenario 2 is higher than that in Scenario 1.

As can be seen from Figs. 6 and 8, the flow rate of Path 5 in Scenario 3 is lower than that in Scenario 1, and that in Scenario 2 is higher than that in Scenario 1. Path 5 includes link 5–6, and link 5–6 is under normal weather in Scenario 1 and Scenario 2. In Scenario 3, link 5-6 is affected by rainfall, so the flow rate in Scenario 3 is lower than those in Scenario 1 and Scenario 2. In Scenario 2, although Path 5 does not cover rainfall, other paths affected by rainfall result in an increase in the flow rate of Path 5. Therefore, the path flow rate in Scenario 2 is higher than that in Scenario 1.

In Figs. 6 and 9, the flow of Path 8 is highest in Scenario 1, followed by Scenario 2, and lowest in Scenario 3. The rainy links 5-6 and 4-5 are included in Path 8. In Scenario 2, link 4-5 is affected by the rain, so the path flow rate is reduced. In Scenario 3, link 4-5 and link 5-6 are simultaneously affected by rain, resulting in a further increase of the path cost; thus, the path flow rate is further reduced.

Based on the above analysis, the results of the example in three scenarios conform to the dynamic tendency of traffic network and proves the availability of models and algorithm under rainfall.

6 Conclusions

In an urban road network, rainy weather, especially rainstorm weather, affects normal vehicle driving, and the free-flow speed declines obviously under rainstorm weather. Existing studies focused on the overall reduction in traffic characteristics in rainfall and calibrated the traffic parameters under rainfall. This study extended existing research to dynamic traffic network. This study included the proposal of a travel cost function considering the variation in free-flow speed with rainfall intensity. Combing the modified travel cost function, a stochastic dynamic traffic assignment model based on a Logit model with rainfall intensity was established. Then, a numerical example containing three scenarios on a test network was designed to validate the models and algorithms.

The results of the three scenarios from the numerical experiment show that rainfall affects the distribution of traffic flow on the dynamic network. The path flow rates of travel paths affected by rainfall will decrease, and the path flow rates of travel paths not affected by rainfall will increase, which is consistent with the traffic flow being transferred from rainy paths to normal paths in the actual network. The iterative convergence speed and analysis results prove the validity of model and the algorithm.

This research can provide a method to analyzing dynamic traffic assignment problem of networks under rainfall. Based on the model in this study, policy-makers can apply and analyze traffic intervention countermeasures. In further research, the effects of some countermeasures to the network will be quantitatively analyzed.

Footnotes

Acknowledgments

This work was supported by National Natural Science Foundation of China (No. 51678044, 71621001, 71210001).

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