Introduction
Roads play a critical role in the development of the economy and society by enabling the mobility of goods and people. Extreme weather, especially heavy precipitation, or natural hazards such as floods, landslides, and earthquakes can cause disruptions to roads and affect normal traffic operations (
Pregnolato et al. 2017;
Hassan et al. 2022). Precipitation could cause adverse effects by influencing road conditions, vehicle performance, and the behavior of drivers (
Donnell and Mason 2004;
Ahmed et al. 2012;
Peterson et al. 2008;
Ahmed et al. 2015). Besides these adverse effects, heavy precipitation could disrupt roads and affect the connectivity and serviceability of the entire road network (
Nowell et al. 2014;
Gori et al. 2020). For instance, the extraordinary heavy precipitation event of Zhengzhou on July 20, 2021, of which the daily precipitation exceeded 600 mm, paralyzed the whole city and traffic. In addition, climate change is expected to increase the frequency and intensity of rainfall worldwide (
Kendon et al. 2012). Climate projections for the rest of the century show continued intensification of daily precipitation extremes (
Donat et al. 2017), and there are expected to be more road disruptions and increased risks in the coming decades (
Pregnolato et al. 2017;
Koetse and Rietveld 2009).
Road network disruption has been a focus of many studies for the past years. Some studies investigate the impact analysis on the road network after disruptions such as travel time and travel cost (
Rosyida et al. 2019), network connectivity and network efficiency (
Zhou et al. 2021), etc. Resilience assessment of road networks when disruption occurs has been widely studied. This research mainly focuses on three aspects. One is the road network vulnerability analysis under disruption conditions (
Jenelius and Mattsson 2012;
Dehghani et al. 2014;
Abdulla and Birgisson 2020;
Starita and Scaparra 2020;
Redzuan et al. 2022). Vulnerability is the system’s susceptibility to disruption (
Jenelius et al. 2006). Dehghani et al. (
2014) assumed particular road status that disruption probabilities of roads were randomly generated from a uniform distribution. The vulnerability of the network was then assessed by computing the change in performance measures: network efficiency and vehicle miles traveled, before and after the disruption. The system’s vulnerability shows a nonlinear increasing trend with the disruption probability. Papilloud and Keiler (
2021) evaluated the vulnerability patterns of road network and identified the most vulnerable traffic zones to 150 flood scenarios based on accessibility measures which include flood-affected populations, opportunities, and average shortest travel time. The second is to assess the robustness of road network (
Sohouenou et al. 2021;
Jin et al. 2022;
Casali and Heinimann 2020;
Gauthier et al. 2018). Robustness indicates the system’s ability to absorb disturbance and remain functional (
Bruneau et al. 2003). Sohouenou et al. (
2021) simulated the localized, targeted, random disruptive events and then evaluated the robustness of road network using robustness indicators: travel time relative change index and unsatisfied demand indicator. The critical index which combines the robustness indicators was also employed to identify the most critical links whose disruptions would cause the most severe disturbance. Jin et al. (
2022) developed a bilevel mixed-integer nonlinear problem (MINLP) model for multiple disruption scenarios to identify the critical combination of roads which play a vital role in ensuring the robustness and function of road network. The last is devoted to the evaluation of road network reliability when disruption occurs (
Duan et al. 2023;
Dong et al. 2023;
Jiang and Huang 2022). Reliability of the traffic system is interpreted as the ability to serve a specific number of travelers in a given time interval or under other certain conditions (
Iida 1999). Connectivity reliability and travel time reliability are two common measures of road network reliability (
Bell and Lida 1997). The former describes the probability of at least one path remaining connected when disruptions occur, and the latter refers to probability that a trip can be completed within an acceptable travel time tolerance (
Iida 1999). For example, Jiang and Huang (
2022) used a Bayesian network (BN) model to evaluate the connectivity reliability and identify critical units for the highway network of G213 and S9 in the Wenchuan Earthquake. The connectivity probability was calculated using empirical prediction method. Fujita and Wakabayashi (
2021) calculated the link reliability of road network in urban areas under the condition of heavy rainfall using a probability density function of depth in flood hazard map, and the probability of exceeding a reference value of depth can be considered as link unreliability.
Most of the preceding research concentrates on road network disruptions caused by floods, earthquakes, landslides or accidents. While road network disruption caused by precipitation has been less investigated. Existing studies related to the influence of precipitation on road focus on the reduction of traffic speed (
Chung 2012;
Lam et al. 2013;
Hooper et al. 2014;
Ahmed and Ghasemzadeh 2018), or damage loss to precipitation (
Espinet et al. 2018;
Zhu et al. 2022). Due to insufficient recorded data, relatively few studies have been performed to construct disruption probability model of road to precipitation, which is a prerequisite for road network reliability analysis. Considering the previous challenges, this study focuses on the empirical evaluation of road network vulnerability and reliability due to disruptions caused by precipitation using the historical recorded data.
In this study, we propose an approach for assessing the reliability of road network to precipitation based on the disruption vulnerability model constructed using the historical recorded data. The road network in Fujian Province, China was selected as a case to validate the proposed framework. The vulnerability model and the framework of reliability assessment would contribute new knowledge to the literature and could be used for reliable performance assessment of road networks in response to precipitation events. The rest of this paper is organized as follows. The Data and Methods section describes the recorded road disruption data, precipitation data, and overall framework of this study, as well as the methods used for developing road disruption vulnerability model and performing road reliability assessment. In the Results section, the developed disruption vulnerability model is presented, and the reliability assessment of road network for two precipitation events is elaborated. The contributions and limitations of this study are illustrated in the Discussion section. Finally, a summary of this study is included in the Conclusion section.
Methods
We propose a framework for assessing the reliability of road network to precipitation based on recorded disruptions. The overall framework is shown in Fig.
3. The framework consists of three main components. First, based on recorded disruptions, the vulnerability model of precipitation-disruption probability and precipitation-amount of disruptions were constructed. Then, a Monte Carlo approach was applied to simulate the disruption scenarios of the whole road network for one precipitation event based on the disruption vulnerability model. At last, the travel time reliability was assessed based on the simulated status of road network.
Disruption Vulnerability Model
In this study, the models linking precipitation and the disruption vulnerability of roads were developed to evaluate the impact of precipitation on road disruptions. Based on the 536 recorded disruptions, the vulnerability models of precipitation-disruption probability and precipitation-amount of disruptions were constructed.
Disruption Probability
We first divided the road segments into grid cells. We assigned a value of 1 to grid cells if there is at least one disruption point; otherwise, a value of 0 was assigned. For each disruption event, the cumulative precipitation of three days before disruption for all grid cells was further collected, as roads are often damaged by cumulative precipitation. In this way, the data set of cumulative precipitation of three days and corresponding disruption state of all grid cells was got.
The disruption probability is calculated based on Eq. (
1)
where
= road disruption probability at the given cumulative precipitation interval (CPI). For a given cumulative precipitation interval,
and
= number of disrupted grid cells and total number of grid cells, respectively.
Model Fitting and Uncertainty
The disruption vulnerability model was developed using a data set of precipitation and disruption information. The least squares (LS) method, a mathematical method often used to numerically estimate a relationship between variables using sample data, was utilized for model fitting (
Ezell and Land 2005;
Griva et al. 2009).
To evaluate the uncertainty of the fitted model, the 99% confidence interval was selected to express the uncertainty. For the fitted model, the 99% confidence interval was calculated based on Eqs. (
2)–(4)
where
= confidence interval limit,
= lower limit of the 99% confidence interval, and
= upper limit of the 99% confidence interval.
for the 99% confidence interval,
= standard deviation,
= average value of the sample, and
= amount of the sample.
Monte Carlo Simulation
Monte Carlo simulation is commonly used for disaster simulation and analysis (
Buslenko et al. 1966;
Crowley and Bommer 2006). To evaluate the performance of road network under the condition of precipitation, Monte Carlo simulation is employed here to simulate the probable disruption scenarios. The procedure to implement the disruption is shown in Fig.
3.
The detailed algorithm of Monte Carlo simulation is as follows:
Step 1: The road network is represented by G = (N, L), where N (N = 1, 2,…,n) is the set of nodes, L (L = 1, 2,…,l) is the set of links, and then the links are divided into 0.1 degree × 0.1 degree grid cell links .
Step 2: For a precipitation event, the cumulative three-day precipitation of each grid cell are obtained. The vulnerability models of precipitation-disruption probability and precipitation-amount of disruptions are used to calculate the probability of disruption for each grid cell link in the network and the number range of disruptions that may occur in the whole network due to precipitation.
Step 3: For each grid cell link in the network, a Monte Carlo approach is used. The random number is compared with the disruption probability to determine whether the grid cell link is disrupted, with disruptions being recorded as 1 and nondisruptions being recorded as 0.
Step 4: Count the number of disruptions in the road network . If the number of disruptions falls within the range evaluated by Step 2, then the network status is recorded, otherwise, go to Step 3.
Step 5: Step 3 and Step 4 are repeated until 1,000 network status have been recorded and the results of the 1,000 simulations are taken as the possible status of the road network under the precipitation event.
Step 6: End.
Travel Time Reliability
Road disruptions always lead to a significant increase in travel time and reduce the probability of arriving at destination on schedule. Travel time reliability is measured as the probability that the origin-destination connection can be completed in a time less than threshold value (
Bell et al. 1999;
Jamous and Balijepalli 2018). In this study, we assessed the travel time reliability between cities in Fujian Province which were considered as the origin-destination (O-D) pairs in road network. The travel time reliability is calculated according to Eq. (
5)
where
is the travel time reliability,
is the travel time on abnormal network under precipitation,
is the travel time on normal network,
is the acceptable upper limit.
and
are calculated by Eq. (
6). Under abnormal conditions, the shortest path between origin (
) and destination (
) will change.
The travel time from an origin point to a destination point
is expressed as:
where
is the length of the shortest path from origin (
) to destination (
) and
is the travel speed. Moreover, the shortest path length
is calculated using the Dijkstra algorithm. The Dijkstra algorithm is a typical method to solve the shortest path from one point to other points in a network, and it was proposed by Dijkstra, a computer scientist from Holland, in 1959 (
Dijkstra 1959).
Discussion
Heavy precipitation can seriously affect roads, resulting in negative consequences for road services. Under the context of climate change, the road disruption risk is expected to increase due to the continued intensification of precipitation extremes. As an important part of risk and resilience, reliability assessment of the road network to precipitation is indispensable for road authorities to prepare for possible adverse impacts. A framework for assessing the reliability of the road network to precipitation based on historical recorded disruption data was therefore proposed. The framework consists of three main components, the vulnerability model, a Monte Carlo approach, and travel time reliability. The disruption vulnerability model describing the relationship between precipitation intensity and the disruption vulnerability of roads, was first constructed using the recorded data. The validity of the vulnerability model was verified using R squares, 99% confidence interval, and two precipitation events, which showed a good estimation. Based on the disruption vulnerability models, a Monte Carlo approach was used to simulate the probable disruption scenarios of road network for precipitation events. At last, the travel time reliability was assessed based on the simulated status of road network.
There are still some limitations as follows. First, the data set of road disruptions was collected from 2016 to 2020. The function built based on the data set would be robust if more disruption data were included. Moreover, it should be updated when more data sets of recent years are collected. Second, the impact of precipitation on speed reduction of vehicles was not considered in this study since we focus on the disruption of the road network, the dual effect of these two factors should be considered in further studies. Third, the basic data set in this study was collected for Fujian, China. Therefore, the applicability of the estimated model may be limited by the region. More studies can be performed for different regions to develop locally specific functions.
Conclusion
Road network reliability assessment is essential for the management of road networks when considering service failure in network. Our study proposes a framework for assessing the reliability of road network to precipitation based on the disruption records. The vulnerability models of precipitation-disruption probability and precipitation-amount of disruptions show a nonlinear increasing trend with precipitation. Based on the disruption vulnerability models, the reliability of road network of Fujian is assessed by using the Monte Carlo approach. For an event with an average precipitation of 73.32 mm on May 21, 2021, the travel time reliability of 11.12% city O-D pairs in Fujian Province decreased to below 0.85.
To the best of our knowledge, the developed disruption vulnerability models in this study are one of the first findings on road disruption vulnerability to precipitation and thus contribute new knowledge to the literature. More importantly, the proposed framework for assessing the reliability of road network to precipitation based on historical recorded disruptions in this study can serve as a basis for reliable performance and risk assessment of road networks in response to precipitation events.