Reliability Engineering and System Safety 123 (2014) 38–46
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Reliability Engineering and System Safety journal homepage: www.elsevier.com/locate/ress
Comparisons of complex network based models and real train flow model to analyze Chinese railway vulnerability Min Ouyang a,b, Lijing Zhao a,b, Liu Hong a,b,n, Zhezhe Pan a,b a b
School of Automation, Huazhong University of Science and Technology, 1037 Luoyu Road, Wuhan 430074, PR China Key Laboratory for Image Processing and Intelligent Control, Huazhong University of Science and Technology, Wuhan 430074, PR China
art ic l e i nf o
a b s t r a c t
Article history: Received 28 June 2013 Received in revised form 30 September 2013 Accepted 13 October 2013 Available online 18 October 2013
Recently numerous studies have applied complex network based models to study the performance and vulnerability of infrastructure systems under various types of attacks and hazards. But how effective are these models to capture their real performance response is still a question worthy of research. Taking the Chinese railway system as an example, this paper selects three typical complex network based models, including purely topological model (PTM), purely shortest path model (PSPM), and weight (link length) based shortest path model (WBSPM), to analyze railway accessibility and flow-based vulnerability and compare their results with those from the real train flow model (RTFM). The results show that the WBSPM can produce the train routines with 83% stations and 77% railway links identical to the real routines and can approach the RTFM the best for railway vulnerability under both single and multiple component failures. The correlation coefficient for accessibility vulnerability from WBSPM and RTFM under single station failures is 0.96 while it is 0.92 for flow-based vulnerability; under multiple station failures, where each station has the same failure probability fp, the WBSPM can produce almost identical vulnerability results with those from the RTFM under almost all failures scenarios when fp is larger than 0.62 for accessibility vulnerability and 0.86 for flow-based vulnerability. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Railway system Comparison analysis Vulnerability Complex networks Shortest path Train flow
1. Introduction The economy of a nation and the well-being of its citizens depend on the continuous and reliable functioning of infrastructure systems, such as telecommunication systems, electric power systems, gas and oil systems, water supply systems, transportation systems, and so on. However, these systems are subjected to the following issues, such as unavoidability of damage due to natural hazards, cascading failures due to their interdependencies, component aging, demand increase, climatic change, terrorist attacks, which increase their vulnerabilities. Regarding the definitions of vulnerability, they vary by discipline and application [1–5]. For example, Haimes, a scholar in the system and information engineering field, defined vulnerability as the manifestation of the inherent states of the system (e.g., physical, technical, organizational, cultural) that can be exploited to adversely affect (cause harm or damage to) that system [1]. Aven, a professor in the risk management research field, defined vulnerability as the uncertainty about and severity of the consequences of the activity given the occurrence of the initiate event [2]. Considering these available
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definitions in the engineering field and to differentiate with other pertinent terms, such as risk and resilience [1,2], the authors simply define the vulnerability as the performance drop of an infrastructure system under a given disruptive event. Note that the performance can be measured by different metrics, which correspond to various vulnerability values. To better protect infrastructure systems, many scholars recently have applied the complexnetwork based models to describe infrastructure topologies and then study their vulnerabilities from a topological perspective. These models can be simply grouped into two types, depending on whether the “flow” upon infrastructure systems is considered or not. The first type is the purely topological models, which describe infrastructure systems as networks, with system components represented as nodes and component relationships as edges, and then study the performance response of the networks under disruptive events without the consideration of particle transportation. Empirical studies show that some infrastructure topologies have exponential degree distributions and are robust to the failures of both randomly selected nodes and the most connected nodes, such as Chinese bus-transport systems [6], Indian railway system [7], urban street networks [8], North American power grid [9] and southern California power grid [10], water distribution networks in the United Kingdom [11], while some infrastructure topologies have power-law degree distributions and are robust to
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the failures of randomly selected nodes but very vulnerable to the failures of the most connected nodes, such as Indian airline network [12] and USA airline network [13], worldwide cargo ship network [14,15], internet [16], power grid of the western United States [10], eastern interconnected and western system electric transmission networks [17]. Besides the random failures and target attacks, many scholars have studied the vulnerability of infrastructure systems under other hazards by using topologybased approach, such as the seismic vulnerability of interdependent power, gas and water systems in Europe [18] and Shelby County, Tennessee, USA [19,20], the terrorism vulnerability of interdependent power, water, steam supply and natural gas systems in Massachusetts Institute of Technology (MIT) campus [21], the hurricane vulnerability of interdependent power, water and gas systems in Harris County, Texas, USA [22]. The second type is the artificial flow based models, which are based on purely topological models and further consider the dynamics of particles of interest over physical infrastructures. Modeling the real particle flow requires modeling the engineering properties of infrastructure systems as well as a huge amount of detailed data on their components, such as generator productions, load levels, line impedances in power grids, which are sometimes difficult to obtain due to security concerns. To overcome this problem, the artificial flow models assume particles move along virtual routes to capture the flow transportation and possible redistribution in real infrastructure systems. Some studies assumed the particles run along the shortest path between a pair of vertices, and then used betweenness as a proxy for the amount of particle passing through a vertex or an edge, where betweenness is computed as the number of shortest paths that pass through every component when connecting vertices. A disruptive event can cause some component failures and alter the infrastructure topology. Depending on the operation mechanisms of the infrastructures under consideration, some studies did not consider the flow or load redistribution, such as the railway systems to be considered in this paper, while some studies assumed that the altered infrastructure topologies further change all components’ betweenness and cause some other components overloaded and failed until all remaining components’ betweennness (load) less than their own capacities. This type of models have been used to study the vulnerability of western U.S. power transmission grid [23], North American power grid [24], Italian power grid [25], trans-European gas networks [26], transportation networks [27], and the seismic and lightning vulnerability of IEEE 118 power grid [28], the hurricane vulnerability of several power grids in Texas, USA [29,30], and so on. For the above two types of models, they both overlooked the engineering properties of infrastructure systems, and then the vulnerability analysis results from these two models could be far from the results from the real flow models. Some scholars have analyzed the differences between the complex-network based models and the real flow models in power grids. For the Italian high-voltage power grid, the critical components identified from the purely topological model do not affect the functioning of the network after their removal when considering the real power flow [31]. However, under some conditions, some studies on power grids showed that the complex network based models can produce almost identical vulnerability results as those from the real flow model [32], which can provide decision makers suggestions to select an efficient model for rapid response to disaster preparation and restoration in some special scenarios. For other types of infrastructure systems, such as railway systems, they have different flow mechanisms, whether similar results can be found as those in power grids and how effective are the complex network based models to analyze railway vulnerability is worthy of research. This paper takes Chinese railway system as an
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example to show how effectively the complex network based models can produce the vulnerability results as those from the real train flow model. The rest of this paper is organized as follows: Section 2 introduces the Chinese railway system and its network-based representation. Section 3 introduces a real train flow model and three typical complex network based railway models, including purely topological model, purely shortest path model and weight based shortest path model for railway vulnerability analysis. Section 4 analyzes and compares the Chinese railway vulnerability results from different models. Section 5 discusses the findings and provides conclusions and directions for future research.
2. Representation of Chinese railway system The Chinese railway system plays a crucial role in the economy of China and the wellbeing of its citizens. In 2012, it transported around 1.89 billion passengers and approximately 3.89 billion metric tons of cargos. This system has approximately 2940 stations in total. This paper picks out important stations in China on a coarse-grained level according to the recent handy book “Chinese Railway Passenger Train Timetable” published in 2010 [33] and combines multiple stations in a city to one station for simplification. Finally, the coarse-grained railway system has N¼399 stations, which are connected together by E¼ 500 railway links. A geographical representation of Chinese railway layout is shown in Fig. 1. Upon the physical layout, on a typical weekday there are 4196 trains running on the railway to transport passengers between different cities. These trains have eight types: high-speed trains, inter-city trains, bullet trains, non-stop or few-stop trains, express trains, fast trains, normal fast trains, normal slow trains, which are denoted, respectively, by type “G”, “C”, “D”, “Z”, “T”, “K”, “P”, and “M”. The number and the average speed of each type of trains are shown in Table 1. The initially departure and finally arrival stations as well as the detailed routines of each train can be also obtained from the handy book. Based on the railway physical layout as well as all train routines, it can construct different railway networks, where nodes represent train stations, but edges can be interpreted differently, depending on the “spaces” under consideration. Kurant and Thiran introduced very clearly three “spaces” [27]: space of stations, where two stations are connected only if they are physically directly connected with no station in between; space of stops, where two stations are connected if they are two consecutive stops on a route of at least one train; space of changes, where two stations are considered to be connected by a link when there is at least one train that stops at both stations. Under different spaces, this paper constructs different Chinese railway networks with their topological properties, including average degree, diameter,
Fig. 1. A geographical representation of Chinese railway system.
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Table 1 Number and average speed of each type of trains in Chinese railway system. Train type
G
C
D
Z
T
K
P
M
Number of trains Average speed of trains (km/h)
322 240–320
118 240–260
731 130–220
66 90–130
329 70–120
1496 50–90
834 50–70
302 5–40
Table 2 Topological properties of Chinese railway networks and several other railway networks in literature under different spaces. System
Number of stations
Number of trains
Type of space
Number of edges
Average degree
Diameter
Mean distance
Clustering coefficient
Chinese Railway
399
4,196
Indian railway [7] Swiss railway [27]
587 1613
579 6,957
European railway [27]
4853
60,775
Stations Stops Changes Changes Stations Stops Changes Stations Stops Changes
500 989 12,760 705 1,680 1,922 19,827 5,765 8,600 88,329
2.5 4.96 64.0 2.4 2.1 2.4 24.6 2.4 3.5 36.4
39 20 5 5 136 61 8 184 48 8
15.0 5.63 2.13 2.2 46.6 16.3 3.6 50.9 12.6 3.7
0.033 0.39 0.69 0.69 0.0004 0.0949 0.9095 0.0129 0.3401 0.7347
mean distance, clustering coefficient [34], shown in Table 2. Several other railway networks in the literature are also listed in the table for comparison purpose. From the table, it can be found that the Chinese railway system shows similar topological properties as other railway systems. For the station degree, the Chinese railway system has the average degree in the same order with those of other railway systems under different spaces except the Indian railway in space of changes because only a few trains are included for analysis; For the clustering coefficient, its value in the space of changes is very high, which is a direct consequence of a very high density and the existence of many triangles; for the mean distance or average shortest path length, which is computed over the lengths of shortest paths between all pairs of stations, and the diameter is the longest of all shortest path lengths, it can be seen that the diameter and the mean distance of the networks in the space of stations are large, but the networks in the space of changes have very small diameters and mean distance, which are mainly because of their high density number of edges.
3. Railway vulnerability models This paper defines the vulnerability of a railway system as its performance drop under a given disruptive event. Based on this definition, this section will first select two performance metrics and their corresponding vulnerability metrics, and then introduce three typical complex network based railway models as well as a real train flow model for vulnerability quantification and comparisons. As trains have different timetables and the running trains in the network keep changing, then some performance metrics at hourly or smaller scale, such as the fraction of trains in operation at a specific time, are time-dependent and then the corresponding vulnerability measures are also time-dependent. To mainly focus on the model comparisons, this paper does not select timedependent metrics, but only uses two time-independent performance metrics at daily scale for vulnerability assessment. One performance metric is the accessibility between different stations within a typical weekday, which is quantified as the average fraction of reachable stations from each station during the day. Denote Ns as the number of railway stations, and then in a normal day passengers in each station can reach other Ns 1
stations by taking one or several trains. Define nidamg as the number of railway stations which can be reached from the ith railway station within the next day immediately after a disruptive event, and then the accessibility of railway system in the next day immediately after the event is defined as follow: PMAccess;damg ¼
i 1 NS ndamg ∑i ¼ 1 NS NS 1
ð1Þ
The sum part of Eq. (1) is the total value of fraction of reachable stations from each station, which is divided by the number of stations to produce the average fraction, also called accessibility. Note that if in a normal day PMAccess,norm ¼1, and then the railway accessibility vulnerability under the event is calculated as follow: V Access;damg ¼
i PMAccess;norm PMAccess;damg 1 NS ndamg ¼ 1 ∑i ¼ 1 NS PMAccess;norm NS 1
ð2Þ The other performance metric is a flow-based metric, which is quantified by the fraction of trains which can run during a typical weekday. Denote NT by the number of trains which can run in a normal day, i.e., NT ¼4196, and define X idamg i¼1,2,…NT by the status of the ith train within the next day immediately after a disruptive event. If the ith train can still run in the next day, then X idamg ¼1, otherwise, X idamg ¼0. The flow-based performance metric during the next day immediately after the event is defined as follow: PMFlow;damg ¼
1 NT ∑ Xi N T i ¼ 1 damg
ð3Þ
Similarly, if in normal case PMFlow,norm ¼1, and then the railway flow-based vulnerability under that event is calculated as follow: V Flow;damg ¼
PMFlow;norm PMFlow;damg 1 NT ¼ 1 ∑ Xi N T i ¼ 1 damg PMFlow;norm
ð4Þ
If the trains are separately considered according to their types, then the flow-based vulnerability can be computed for each type of trains. Note that for the above two post-event performance metrics in the next day immediately after the event, their values are affected by the duration of the disruptive event (or the repair times of
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damaged railway components) as trains have different timetables. When the event duration is less than one day, the shorter the event duration, the more trains with their departure time after the event would not be interrupted, and then the post-event metrics are non-decreasing function of the event duration; when the event lasts for at least one day, the affected trains in the next day immediately after the event would be fixed and then the postevent metrics are independent of event duration. Hence, to focus on model comparisons, this paper does not consider the effects of event duration on post-event performance and vulnerability measures, and simply assumes all events will last for at least one day (or the repair times of damaged railway components require more than one day). Under this assumption, this paper next selects three typical complex network based models as well as a real train flow model to simulate railway performance response, and also introduces how to compute the accessibility and flow-based vulnerability under that event. In addition, note that in practice there may exist network congestion after a disruptive event, whose modeling requires more detailed information, such as the event occurrence time and location, each train location at the occurrence of the event, and so on. Due to lack of sufficient data, this paper simply assumes all interrupted trains can find some nearby stations or emergency railway segments to stop so that they would not cause the congestion and affect other trains. The first model is a purely topological model (PTM). A disruptive event initially causes the direct failures of some railway components (stations or railway links or both), which change the railway topology and lead to the disconnection of some railway stations from others. During the next day immediately after the disruptive event, the number of stations nidamg which can be reached from the ith station is the number of nodes minus one in the post-event connected cluster including the station i, while the status X idamg of the ith train is set as 1 if its initially departure and finally arrival stations are still in the same connected cluster, 0 otherwise, and then the accessibility and flow-based vulnerability can be respectively computed according to Eqs. (2)–(4). The second model is a purely shortest path model (PSPM). It assumes each railway link have the same weight 1.0 and each train run along the shortest path between its initially departure and finally arrival stations and stop at each station in its routine. A disruptive event initially causes some component failures, and during the next day immediately after the event, if one of these failed components is a part of the shortest path routine for a train i, and then the ith train is assumed to be interrupted and X idamg ¼ 0 in that day. The post-event railway physical topology together with not affected trains can construct a new network under space of changes, the number of nodes minus one in the post-event connected sub-network (of the new network) including the ith station is the value of nidamg , which together with the value of X idamg can then give the accessibility and flow-based vulnerability. The third model is the weight based shortest path model (WBSPM), which takes the real railway link length as the edge weight. Each train runs along the shortest path with the consideration of the edge weight and stops at every station in its routine. The forth model is the real train flow model (RTFM), where each train routine is extracted from the handy book “Chinese Railway Passenger Train Timetable”. Based on these train routines, then similar to the PSPM, the values of X idamg and nidamg in each of these two models can be computed and then provide the accessibility and flow-based vulnerability under the event. From the above introduction of the four models, it can be seen that the last three models (PSPM, WBSPM and RTFM) are different mainly due to their different train routines. This paper will next analyze the differences of these routines, as the results shown in Table 3. First, train routines produced by PSPM and WBSPM can construct different networks under space of changes, and the
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Table 3 Properties of networks produced by PSPM and WBSPM under space of changes and comparison of the routines from PSPM or WBSPM and RTFM. Parameters
PSPM
WBSPM 12203 61.2 5 2.13 0.65
Network Properties
Number of edges Average degree Diameter Mean distance Clustering coefficient
10028 50.3 5 2.24 0.66
Routine Differences
Components All G C D Z T K P M Load level
Stations 0.74 0.67 1.0 0.81 0.59 0.68 0.69 0.69 0.98 0.57
Links 0.65 0.48 1.0 0.74 0.48 0.61 0.60 0.60 0.96 0.37
Stations 0.83 0.94 1.0 0.92 0.67 0.76 0.77 0.80 0.99 0.92
Links 0.77 0.86 1.0 0.88 0.60 0.71 0.70 0.71 0.98 0.81
topological properties of these networks are computed to compare with those from the RTFM. From the Tables 2 and 3 together, the network from RTFM has more edges than those from the other two models because the routine of a train in practice should not only minimize the distance between its initially departure station and its finally arrival station, but also need to pass through large population cities as many as possible. The tradeoff between these two objectives may increase the number of stations in the routines, and then bring more edges in the network under the space of changes and larger clustering coefficient. However, the diameter and the shortest path length do not change a lot compared with those from the WBSPM, because the number of edges does not increase too much. Second, for each train, the ratio of the number of stations (or railway links) included in both routines from PSPM (or WBSPM) and RTFM to the total number of stations (or railway links) in the real routine can be computed. The average value of these ratios over all trains is further calculated to reflect the routine differences. The results are also shown in Table 3. From the table, for trains with the type “C”, their routines produced by PSPM and WBSPM are totally the same with the real routines, which is because all these trains travel between two neighboring stations. When taking all types of trains into consideration, there are 83% stations and 77% railway links included in both routines from WBSPM and RTFM, compared with 74% stations and 65% railway links for PSPM and RTFM. These results indicate the WBSPM can approach the real train flow routine better, which also holds true when the trains are grouped by different types. These conclusions are also supported by the Pearson's correlation coefficient of the load level vectors from PSPM or WBSPM and RTFM, where the load level vector consists of the number of trains passing through each railway station (or railway link). The load levels from WBSPM and RTFM have a strong correlation coefficient 0.92 for stations and 0.81 for railway links, while for PSPM and RTFM, it has mild correlation with 0.57 for stations and 0.37 for railway links. Despite the WBSPM can provide good approximation for the train flow routines, whether it can also perform well on the real vulnerability assessment will be discussed in next section.
4. Vulnerability comparisons This section will compare three complex network based models with the real train flow model from the vulnerability perspective.
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The vulnerability comparison method will be first introduced and then followed with the simulation results. 4.1. Comparison method To compare the complex network based models with the real train flow model, this paper will use them to respectively compute the accessibility and the flow-based vulnerability of the Chinese railway under both single and multiple component failures and then analyze the relationships of different vulnerability results. Considering the vulnerability under single component failures is mainly because single component failures accounts for large fraction of disruptive events in practice, and it is also usually used to quantify the importance of the component. When the ith component is initially failed, the four models can respectively generate the accessibility and flow-based vulnerability. The vulnerability results for all components make up the corresponding vulnerability vectors. The relationships of these vulnerability vectors from different models are then analyzed from the following two aspects for model comparisons. One is to compute the Pearson product-moment correlation coefficient CC for two vulnerability vectors; the other is to compute the fraction of strictly identical or almost identical results FIR for vulnerability vectors, here “almost” means the absolute difference of the vulnerability result from one of the complex network based models and the real train flow model is within err. Note that if err ¼ 0, it is strictly identical. Considering the vulnerability under multiple component failures is mainly because it is usually used to analyze the performance response of railway system under natural hazards or some other extreme hazards, which can cause many components failed simultaneously. To model multiple component failures, this paper assumes each component have an identical failure probability, denoted by fp A(0, 1]. Given a fp, a failure scenario is produced according to the following rule: for each component, a uniformly distributed random number ε within [0, 1] is generated, and compared with fp. If ε ofp, then the component is assumed failed, otherwise, it is still normal. By this way, it can generate a lot of failure scenarios, and then for each failure scenario, different models can produce different accessibility and flow-based vulnerability. The results under all failure scenarios together make up the vulnerability vectors. With these vulnerability vectors, similar to the comparison analysis on single component vulnerabilities, the correlation coefficients CC and the fraction of strictly or almost identical results FIR for each of the complex network based modes and the RTFM can be computed for vulnerability comparisons under multiple component failures. 4.2. Vulnerability comparison under single component failures Railway component failures can be station failures or link failures or both. This subsection will mainly analyze the station failures for illustrative purpose, and the link failures will be further discussed in the section of “Conclusions and Discussions”. The CC and the FIR for the accessibility and flow-based vulnerability results under single station failures from each complex network based model and the RTFM are shown in Table 4. When studying the flow-based vulnerability, the results are also grouped according to the train types. When using the accessibility vulnerability metric, the value of CC is 0.83 for PTM and RTFM, 0.92 for PSPM and RTFM, and 0.96 for WBSPM and RTFM, which indicates the WBSPM can approach the RTFM the best for single station vulnerability assessment. If checking the FIR with err ¼0, similar results are found: the value of FIR is 0.48 for PTM and RTFM, 0.73 for PSPM and RTFM, and 0.83 for WBSPM and RTFM. Note that for the PTM, two stations are
Table 4 The correlation coefficients and the fractions of identical results for the accessibility and flow-based vulnerability results under single station failures from each of the complex network based models and the real train flow model. Model
PTM
PSPM
WBSPM
Parameter
CC
FIR err ¼ 0
CC
FIR err ¼0
CC
FIR err ¼ 0
Accessibility Flow-based All G C D Z T K P M
0.83 0.58 0.57 1.0 0.79 0.51 0.46 0.41 0.49 0.91
0.48 0.26 0.96 1.0 0.78 0.77 0.52 0.31 0.35 0.82
0.92 0.67 0.70 1.0 0.78 0.69 0.71 0.66 0.63 0.98
0.73 0.26 0.97 1.0 0.73 0.73 0.54 0.30 0.33 0.96
0.96 0.92 0.96 1.0 0.96 0.80 0.76 0.89 0.88 0.99
0.83 0.28 0.99 1.0 0.78 0.77 0.55 0.36 0.38 0.99
assumed reachable if they are included in the same connected sub-network after a disruptive event, and then the FIR value for PTM and RTFM is the fraction of stations whose individually failure or removal can make any two stations in the same post-event connected sub-network still reachable no matter the real train flow is considered or not. This metric may be taken by the decision makers to quantify the robustness of the train routines. When using the flow-based vulnerability metric, as the railway operator usually does not change train routines under disruptive events and then there is no train flow redistribution in railway system. The flow-based vulnerability under a station failure is just the load level of this station normalized by the total number of trains, and the values of CC for the PSPM and RTFM and the WBSPM and RTFM are just the station load correlations. Despite there is high CC ¼0.92 for the flow-based vulnerability results from WBSPM and RTFM, the FIR value is very low, only 0.28. In addition, the CC and FIR for “C” trains are both 1.0 for each pair of models, because different models produce the same routines for these trains. For trains with each of the other types, the CC and FIR for WBSPM and RTFM are both larger than the values from other two pairs of models, which indicates the WBSPM can approach the RTFM the best for the flow-based vulnerability assessment, even considering the train types. 4.3. Accessibility vulnerability comparison under multiple component failures This subsection will analyze and compare the CC and FIR of the accessibility vulnerability under multiple station failures from each of the three complex network based models and the real train flow model. The results under multiple link failures will be further discussed in the section of “Conclusions and Discussions”. For different random failure probabilities fp, the accessibility vulnerability Vaccess, multiple(fp) from different models can be computed, with the results shown in Fig. 2. From the figure, the vulnerability results from different models become close to 1.0 when the fp is larger than 0.5, which is mainly because larger fp makes the network separate into many small connected clusters, and then any two stations have a probability close to zero to be still connected. Also, the vulnerability curve for PTM is always below that for RTFM, this is because in PTM, any two stations in the same post-event connected sub-network is assumed reachable, but in RTFM, it needs to further consider whether there are trains between them, which indicates that the conditions in RTFM to make two stations reachable are more strictly than those in PTM. Hence, the accessibility vulnerability curve from the PTM can provide a lower bound for the real vulnerability curve. In addition, the curves from PSPM, WBSPM
M. Ouyang et al. / Reliability Engineering and System Safety 123 (2014) 38–46
and RTFM are very close to each other, which indicate that the two artificial flow models can both provide good approximation for the expected vulnerability assessment under random failures. But to further clearly show which artificial flow model is better, the inset figure displays the values of the vulnerability from RTFM minus the vulnerability from PSPM or WBBM at different fp. It shows that the differences between WBSPM and RTFM are always smaller than those for PSPM and RTFM, which indicates the WBSPM can better approach the RTFM for the accessibility vulnerability analysis under multiple station failures. Although the two artificial flow models can provide good approximation for the expected accessibility vulnerability, whether they can also perform well under individual failure scenarios will be discussed next. As introduced in Section 4.1, for random failures, this paper generates 2,000 failure scenarios and computes the accessibility vulnerability from different models. The values of CC and FIR as a function of fp are shown in the Fig. 3. The FIR curve with err¼0.001 is also plotted and shown in the inset figure of Fig. 3. From the figure, for the PSPM and RTFM and the WBSPM and RTFM, their values of CC both first decrease and then increase to 1.0, but keep above 0.83. These indicates the two artificial flow models both can performance well for accessibility vulnerability under individual failure scenarios, but the WBSPM can perform better as its CC curve is almost always above that for PSPM, which is also supported by the FIR curves in
Fig. 2. The accessibility vulnerability under different random failure probabilities fp, the inset figure is the vulnerability from PSPM and WBSPM minus the vulnerability from RTFM under different fp.
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Fig. 3(b). However, for PTM and RTFM, its CC curve almost keeps decreasing until fp¼1, while its FIR curve starts to increase from fp¼0.86, and especially at fp¼ 0.98, CC¼ 0.5 but FIR¼ 0.87, which indicates CC alone cannot well reflect the vulnerability relationship of two models in some conditions. In addition, for the FIR curves, it can be found that when fp is larger than a certain value, the FIR is almost close to 1.0, which means the two models can produce identical results under almost all failure scenarios. As these results are based on Monte Carlo simulation, to make a conclusion for exact FIR ¼1.0 is impossible, hence, this paper computes the lower bound for the FIR under 95% confidence interval at each fp, whose value larger than 0.995 will be regarded as “almost all failure scenarios”. From the figure, the PSPM and WBSPM both can produce the strictly identical results with those from RTFM under almost all failure scenarios when fp is larger than 0.94, but it is 1.0 for PTM. However, if relaxing the strictly identical (err¼ 0) condition, at almost identical (err ¼0.001) case, the critical fp is 0.78 for PTM, 0.64 for PSPM and 0.62 for WBSPM. These results indicate under some extreme events, a simple topology based model is enough to provide good approximation for the railway accessibility vulnerability. 4.4. Flow-based vulnerability comparison under multiple component failures When taking all types of trains into consideration, at different failure probabilities fp, the flow-based vulnerability VFlow, Multiple(fp) can be computed, with the results shown in Fig. 4. Similar to the accessibility vulnerability curves, the flow-based vulnerability curves from the two artificial flow models are both close to the curve from RTFM, but the WBSPM can better approach the RTFM, as clearly shown in the inset figure for the vulnerability from the RTFM minus the vulnerability from PSPM or WBSPM. Also, the PTM can provide the lower bound for the real flow-based vulnerability curve, which is because a train with its initially departure and finally arrival stations in the same post-event connected subnetwork is assumed in operation for PTM, but in RTFM, it needs to check whether all the stations and railway links in its routines are still in the same connected cluster, hence, RTFM requires more strictly conditions to ensure a train in operation, and then the PTM can also produce the lower bound for the real flow-based vulnerability curve. When considering the flow-based vulnerability for each type of trains, Fig. 5 shows the results. From the figure, it can be seen that the trains with the type “C” are the least vulnerable, while the trains with the type “Z” are the most vulnerable, and the second to
Fig. 3. (a) Correlation coefficients for the accessibility vulnerability results from each of the complex network based models and the RTFM under different failure probabilities fp; (b) Fraction of strictly identical (err ¼ 0) results for the vulnerability results from each of the complex network based models and the RTFM under different fp. The inset figure is for almost identical case with err ¼0.001.
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seventh vulnerable trains are the type “T”, “K”, “P”, “G”, “D”, “M”, respectively. This is because a train passing through larger number of stations (routine length) in its routine has relatively higher probability to be interrupted and trains grouped by some type are
Fig. 4. The flow-based vulnerability under different failure probabilities fp. The inset figure is the vulnerability from RTFM minus the vulnerability from PSPM or WBSPM.
Flow-based Vulnerability
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Failure Probability fp Fig. 5. Flow-based vulnerability for each type of trains under different failure probability fp.
more vulnerable if they have relatively longer routine length on average. The routine length is 14.3, 13.9, 11.9, 10.3, 6.6, 5.7, 3.1 stations for “Z”, “T”, “K”, “P”, “G”, “D”, “M”, “C” trains, respectively. The ranking of these routine lengths corresponds to the vulnerable ranking for different types of trains. In addition, the flow-based vulnerability curves (not shown) for each type of trains show similar results displayed in Fig. 4: the PTM can provide the lower bound vulnerability curve, while the WBSPM can approach the RTFM the best except the “C” trains, where all four railway models produce the same vulnerability curve. Similar to the comparison analysis on accessibility vulnerability, this paper next discusses whether the complex network based models can also perform well for flow-based vulnerability under individual failure scenarios. When taking all types of trains into consideration, the CC and FIR for the flow-based vulnerability results from each of the complex network based models and the train flow model are shown in Fig. 6. From the Fig. 6(a), the CC curves for each of the complex network based models and the RTFM keeps increases for fp 40, and the CC curve for WBSPM and RTFM is always above those from other two pairs of models and has the minimum value larger than 0.95, which indicates the WBSPM can approach the RTFM the best for flow-based vulnerability under individual failure scenarios. Also, the CC values for each pair of models reach close to 1.0 after fp4 0.8, but when checking the FIR curves, all three complex network based models can produce strictly identical flow-based vulnerability results under almost all failure scenarios after fp 40.98. If relaxing the strictly identical condition and set err ¼0.001, the critical fp becomes 0.88 for both PTM and PSPM, and 0.86 for WBSPM. These larger critical values of fp than those for accessibility vulnerability indicate that complex network based models are less capable to estimate flow-based vulnerability. When considering the flow-based vulnerability for each type of trains, similar results are found. For “C” trains, the CC and FIR curves keep at 1.0 for any fp; for each of other types of trains, the CC and FIR curves for WBSPM and RTFM are almost always above other two curves; there also exists a critical value of fp, above which, all three complex network based models can produce identical flow-based vulnerability results under almost all failure scenarios. The critical fp to make WBSPM and RTFM produce strictly identical flow-based vulnerability is 0.68 for “G” trains, 0.84 for “D” trains, 0.52 for “Z” trains, 0.74 for “T” trains, 0.92 for “K” trains, 0.98 for “P” trains, and 0.94 for “M” trains. Despite the “Z” trains are the most flow-based vulnerable (shown in Fig. 5), the condition of a complex network model to exactly approach the RTFM for flow-based vulnerability analysis is the easiest.
Fig. 6. (a) Correlation coefficients for the flow-based vulnerability results from each of the complex network based models and the RTFM under different failure probabilities fp; (b) Fraction of identical results for the flow-based vulnerability results from each of the complex network based models and the RTFM under different fp. The inset figure is for almost identical case with err ¼ 0.001.
M. Ouyang et al. / Reliability Engineering and System Safety 123 (2014) 38–46
5. Discussions and conclusions This paper selects three complex network based models, including purely topological model, purely shortest path model and weight based shortest path model, to simulate the railway vulnerability under both single and multiple station failures, and compare the accessibility and flow-based vulnerability results with those from the real train flow model. The results show that the PTM can provide the lower bound of the real vulnerability curves, and the two artificial models can both approach the RTFM well, but the WBSPM is better. The correlation coefficient for accessibility vulnerability results from WBSPM and RTFM under single station failures is 0.96 and it is 0.92 for flow-based vulnerability. Also, when considering the multiple station failures, the WBSPM can produce almost identical vulnerability results with those from the RTFM under almost all failure scenarios when the fp is larger than 0.62 for accessibility vulnerability and 0.86 for flow-based vulnerability. Note that the accessibility and flowbased metrics quantify the vulnerability from different perspectives, but how correlated are they? This paper further computes the Pearson product-moment correlation coefficients CC of the accessibility and flow-based vulnerability results from RTFM under single and multiple station failures. The results show that under single station failures, CC¼ 0.068; under multiple station failures, the value of CC fluctuates between 0.4 and 0.6 for any 0o fpo1, which indicates a mild correlation between the two vulnerability metrics. The above paper only compares the models under station failures. When considering the railway link failures, similar results are also found and the WBSPM can still approach the RTFM the best. Under single railway link failures, the correlation coefficient (fraction of strictly identical results) for the accessibility vulnerability from WBSPM and RTFM is 0.95 (0.84) and it is 0.81 (0.22) for flow-based vulnerability; under multiple link failures, the correlation coefficient for WBSPM and RTFM keeps above 0.8 for accessibility vulnerability and 0.9 for flow-based vulnerability; the critical fp to make WBSPM and RTFM produce almost identical vulnerability results under almost all failure scenarios is 0.68 for accessibility vulnerability and 0.96 for flow-based vulnerability. In addition, the two vulnerability metrics have a weak correlation with CC ¼ 0.08 under single railway link failures, and have the value of CC fluctuate between 0.25 and 0.45 under multiple railway link failures for any 0o fpo1. According to the above results for Chinese railway system under station and link failures and comparing them with the findings from power grids [32], some common results can be concluded for both two types of infrastructure systems: the purely topological model can be used to estimate the lower bound value of real infrastructure vulnerability, and under some extreme events which can cause almost 60–70% of components fail, complex network based models can provide well approximation on the real topology-based (for power grid) or accessibility-based (for railway system) vulnerability, but to produce almost identical real flow-based vulnerability from complex network based models, it requires almost all components failed. Also, the flow-based vulnerability and topology-based or accessibility-based vulnerability have mild correlation. Hence, for vulnerability bound estimation or for topology-based or accessibility based vulnerability assessment under extreme events, the complex network based models can work well, but in other case, they should be carefully used. However, this paper only takes the Chinese railway system under single component failures and randomly multiple component failures for vulnerability comparisons, whether similar results can be also found in other railway systems and under other failure types, such as intentionally multiple component
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failures, are still questions for future research. In addition, there are many limitations and assumptions in this paper for the railway vulnerability assessment, such as time-independent performance metrics at daily scale, long event duration, no consideration of network congestion, relaxing these assumptions to analyze timedependent vulnerability is also an interesting topic.
Acknowledgments This material is based upon work supported by the National Science Foundation of China under Grants 51208223, 90924301 and 91024032, and the Independent Innovation Foundation of Huazhong University of Science and Technology under Grant 2012QN088. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.
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