Lakhasly

3.Finally, the degradation profile (DP) and recovery profile (RP), which are based on the weighted sum in Munoz and Dunbar (2015) but do not include a time penalty, describe the summed deviation from a linear degradation and recovery, respectively.1: where T0 is the start time of the disruption; T1, T2 and T3 are the end time of the first, second and third phase, respectively; f(t)is the linear degradation function connecting (T0,Q0)and (T1,Qmin); g(t)is the linear recovery function connecting (T2,Qmin)and (T3,Q0); n is the total number of time intervals ti+1 ti, which is equal to the disruption length when taking a step size of one unit; and m and s are the number of equally spaced measurement points in the first and the third phase, respectively.The current DP and RP metrics work well under the assumption of convex/concave behaviour of the performance in degradation/ recovery phases, which is the most commonly observed behaviour in real-life railway disruptions.In the modified BFS algorithm, the start vertex/vertices is/are the disruption location (in case of a timetable point outage) or boundaries (in case of a line blockage), and the end vertices are the second closest decoupling points.To measure both indicators simultaneously, they are combined in a composite performance indicator Q which is calculated as the weighted sum of traffic punctuality (P/R) and traffic intensity (R/S):

3.3.The BFS algorithm starts from a source node, referred to as the start vertex, and visits any adjacent, unvisited vertices until none are left.If no train operations would be possible to and from Apd, this would affect the surrounding lines significantly, and the first decoupling points would be Amf, Dv and Zp. Instead, if limited train operations to and from Apd would be possible, then Apd itself would be the first decoupling point, and Amf, Dv and Zp would be the second decoupling points.The formal procedure of the modified BFS algorithm for a single start vertex is included in Appendix A. For more details on the algorithm, the interested reader may refer to Knoester (2021).Therefore, we also consider the traffic punctuality, which is defined as the proportion of punctual train activities (i.e. with a delay less than three minutes, which is a threshold used at the Dutch railways) relative to the realized train activities in a given time period.Apart from plotting individual resilience curves, one could also identify different shapes of resilience curves and calculate the mean and median resilience curve per disruption type.Disruption management in the Netherlands provides a theoretical foundation for determining which area to consider, based on the concepts of decoupling points and impact areas.3 (a), line Amf-Apd represents a simple, open line, and thus, two start vertices would suffice (being Amf and Apd) which would also be marked as the first decoupling points.Therefore, such system is deemed more resilient, because the gradual degradation gives traffic controllers more time to respond, hence the smaller DP. Similar holds for recovery profile (RP).Methodology A data-driven methodology is developed in this study to ex-post evaluate the resilience of railway networks given a large number of disruptions of varying types.The methodology is generic and can work for different performance indicators, resilience metrics and categorizations of disruption types.Performance measurements at each time instant are stored in a dataframe, which serves as the basis for calculating the resilience metrics.The metrics are stored in a separate dataframe, which serves as the basis for conducting statistical analyses.The resilience curves, which can be drawn directly from the performance mea surements, represent the graphical output of the resilience quantification.We first introduce the boundaries of a disruption, which could be the decoupling points closest to the disruption location, but could also be the timetable points marking a smaller area in which there is no traffic operation, depending on the magnitude of the disruption.The original BFS algorithm, introduced by Moore (1959), is a type of graph search that is used to traverse a graph and find each node or vertex in the graph.The degradation time (DT), response time (RST) and recovery time (RCT) describe the duration of the first, second and third resilience phase, respectively.The seven resilience metrics are defined as follows, using the notation indicated in Fig.It is structured in the resilience quantification framework presented in Fig.The framework has been divided into three parts (i.e. input, processing and output) and works as follows.The resilience metrics and the test statistics represent the numerical output of the resilience quantification.To determine the impact area of each disruption given the disruption location, a modified breadth first search (BFS) algorithm is developed.For example, in case of a line blockage between Amersfoort (Amf) and Apeldoorn (Apd) as in Fig.Calculating performance: In order to calculate performance, it is necessary to specify a performance indicator which measures the level of operations at a given time instant.The statistical analyses are used to identify differences among disruptions of varying types.Determining the impact area: A question arising before calculating the performance during a disruption is which area to consider in this calculation.Since disruption effects can easily spread through the network, it is preferred to study a larger area than just the disrupted line or timetable point.A decoupling point is defined as a timetable point where trains are allowed to start or end their route in case of a disruption.An additional benefit of using impact areas is that it limits possible interactions with disruptions in other parts of the network, both in our quantification approach and in practice.For our railway network context, several new extensions are required on the basic BFS algorithm.Fifth, the algorithm needs to handle more than one start vertex, since a single start vertex only occurs for a timetable point outage.A common approach is to express railway system performance in terms of the traffic level (e.g. Ghaemi et al., 2017), which we refer to as the traffic intensity.Traffic intensity is defined as the proportion of realized train activities relative to the scheduled train activities in a given time period.While such a definition for the traffic intensity indicator accounts for cancellations, it does not account for delays.The maximum impact (MI) describes the vertical distance between target performance (Q0) and minimum performance (Qmin).For all metrics, a higher positive value indicates a stronger disruptive effect, and thus, a less resilient network.Looking at the degradation profile (DP), in case that it degrades gradually and takes a long time to reach minimum performance, the DP value will be smaller than when performance drops quickly and then keeps degrading slowly.The quantification starts with the collection of traffic realization data, disruption log data and network data from a database.Disruption log data refer to information about the disruption, such as the reported start and end time, location, cause, etc.If the studied area is too small, then disruption effects further away from the disruption location could be overlooked, but if the studied area is too large, the impact of the disruption becomes less visible.The first impact area is bounded by the first intercity decoupling points from the disruption location; the second impact area is bounded by the next closest intercity decoupling points from the first ones; and the third impact area is bounded by the next closest intercity decoupling points from the second ones.Since cancellations in the third impact area are in principle not allowed, disruption effects are mostly contained within the first and second impact area.Thus, the first and second impact area are identified as the appropriate area to consider for the resilience quantification.Third, the algorithm must account for the fact that not all vertices may appear in the same path in opposite di rections.Fourth, the algorithm must allow a vertex to have more than one parent (i.e. the previous vertex in the specific path), and to visit this vertex when it has already been visited from another parent.Instead, a line blockage will have two, three or even four start vertices, depending on the network layout.Alternatively, assuming a disrupted station in Apd as in Fig.Calculating the resilience metrics: The shape of the resilience curve is described by seven resilience metrics.Performance loss (PL) describes the area enclosed by target performance and the resilience curve.However, due to a possible complex performance variability during some specific disruptions (e.g. a sudden drop in performance during recovery phase), DP or RP could reach close to zero values.Traffic realization data includes at least the plan time, realization time and location of train activities.Network data specify how each timetable point in the network is connected to neighboring timetable points.Using these data, the evolution of performance over time can be calculated for each disruption in a disruption-specific impact area.First, the algorithm terminates its search along a branch of the network when a second decoupling point is reached in the specific branch.3 (b), one start vertex would suffice (being Apd).Together, the performance indicators account for cancellations as well as delays.The first five metrics can only have nonnegative values, whereas the last two metrics can have positive values as well as negative values.The size of the area depends on the location in the network, but also on the type of impact, as illustrated in Fig.The new extensions are set as follows.The area covered in between represents the impact area.Second, the algorithm cannot follow a path that is not driven by any train.2.3.1.3.3.2.

3. Methodology
   A data-driven methodology is developed in this study to ex-post evaluate the resilience of railway networks given a large number of disruptions of varying types. The methodology is generic and can work for different performance indicators, resilience metrics and categorizations of disruption types. It is structured in the resilience quantification framework presented in Fig. 2. The framework has been divided into three parts (i.e. input, processing and output) and works as follows. The quantification starts with the collection of traffic realization data, disruption log data and network data from a database. Traffic realization data includes at least the plan time, realization time and location of train activities. Disruption log data refer to information about the disruption, such as the reported start and end time, location, cause, etc. Network data specify how each timetable point in the network is connected to neighboring timetable points. Using these data, the evolution of performance over time can be calculated for each disruption in a disruption-specific impact area. Performance measurements at each time instant are stored in a dataframe, which serves as the basis for calculating the resilience metrics. The metrics are stored in a separate dataframe, which serves as the basis for conducting statistical analyses. The statistical analyses are used to identify differences among disruptions of varying types. The resilience metrics and the test statistics represent the numerical output of the resilience quantification. The resilience curves, which can be drawn directly from the performance mea surements, represent the graphical output of the resilience quantification. Apart from plotting individual resilience curves, one could also identify different shapes of resilience curves and calculate the mean and median resilience curve per disruption type.
   3.1. Determining the impact area: A question arising before calculating the performance during a disruption is which area to consider in this calculation. Since disruption effects can easily spread through the network, it is preferred to study a larger area than just the disrupted line or timetable point. If the studied area is too small, then disruption effects further away from the disruption location could be overlooked, but if the studied area is too large, the impact of the disruption becomes less visible. Disruption management in the Netherlands provides a theoretical foundation for determining which area to consider, based on the concepts of decoupling points and impact areas. A decoupling point is defined as a timetable point where trains are allowed to start or end their route in case of a disruption. We first introduce the boundaries of a disruption, which could be the decoupling points closest to the disruption location, but could also be the timetable points marking a smaller area in which there is no traffic operation, depending on the magnitude of the disruption. The first impact area is bounded by the first intercity decoupling points from the disruption location; the second impact area is bounded by the next closest intercity decoupling points from the first ones; and the third impact area is bounded by the next closest intercity decoupling points from the second ones. Since cancellations in the third impact area are in principle not allowed, disruption effects are mostly contained within the first and second impact area. Thus, the first and second impact area are identified as the appropriate area to consider for the resilience quantification. An additional benefit of using impact areas is that it limits possible interactions with disruptions in other parts of the network, both in our quantification approach and in practice. The size of the area depends on the location in the network, but also on the type of impact, as illustrated in Fig. 3. To determine the impact area of each disruption given the disruption location, a modified breadth first search (BFS) algorithm is developed. The original BFS algorithm, introduced by Moore (1959), is a type of graph search that is used to traverse a graph and find each node or vertex in the graph. The BFS algorithm starts from a source node, referred to as the start vertex, and visits any adjacent, unvisited vertices until none are left. For our railway network context, several new extensions are required on the basic BFS algorithm. In the modified BFS algorithm, the start vertex/vertices is/are the disruption location (in case of a timetable point outage) or boundaries (in case of a line blockage), and the end vertices are the second closest decoupling points. The area covered in between represents the impact area. The new extensions are set as follows. First, the algorithm terminates its search along a branch of the network when a second decoupling point is reached in the specific branch. Second, the algorithm cannot follow a path that is not driven by any train. Third, the algorithm must account for the fact that not all vertices may appear in the same path in opposite di rections. Fourth, the algorithm must allow a vertex to have more than one parent (i.e. the previous vertex in the specific path), and to visit this vertex when it has already been visited from another parent. Fifth, the algorithm needs to handle more than one start vertex, since a single start vertex only occurs for a timetable point outage. Instead, a line blockage will have two, three or even four start vertices, depending on the network layout. For example, in case of a line blockage between Amersfoort (Amf) and Apeldoorn (Apd) as in Fig. 3 (a), line Amf-Apd represents a simple, open line, and thus, two start vertices would suffice (being Amf and Apd) which would also be marked as the first decoupling points. Alternatively, assuming a disrupted station in Apd as in Fig. 3 (b), one start vertex would suffice (being Apd). If no train operations would be possible to and from Apd, this would affect the surrounding lines significantly, and the first decoupling points would be Amf, Dv and Zp. Instead, if limited train operations to and from Apd would be possible, then Apd itself would be the first decoupling point, and Amf, Dv and Zp would be the second decoupling points. The formal procedure of the modified BFS algorithm for a single start vertex is included in Appendix A. For more details on the algorithm, the interested reader may refer to Knoester (2021).
   3.2. Calculating performance:
   In order to calculate performance, it is necessary to specify a performance indicator which measures the level of operations at a given time instant. A common approach is to express railway system performance in terms of the traffic level (e.g. Ghaemi et al., 2017), which we refer to as the traffic intensity. Traffic intensity is defined as the proportion of realized train activities relative to the scheduled train activities in a given time period. While such a definition for the traffic intensity indicator accounts for cancellations, it does not account for delays. Therefore, we also consider the traffic punctuality, which is defined as the proportion of punctual train activities (i.e. with a delay less than three minutes, which is a threshold used at the Dutch railways) relative to the realized train activities in a given time period. Together, the performance indicators account for cancellations as well as delays. To measure both indicators simultaneously, they are combined in a composite performance indicator Q which is calculated as the weighted sum of traffic punctuality (P/R) and traffic intensity (R/S):

3.3. Calculating the resilience metrics:
The shape of the resilience curve is described by seven resilience metrics. The degradation time (DT), response time (RST) and recovery time (RCT) describe the duration of the first, second and third resilience phase, respectively. The maximum impact (MI) describes the vertical distance between target performance (Q0) and minimum performance (Qmin). Performance loss (PL) describes the area enclosed by target performance and the resilience curve. Finally, the degradation profile (DP) and recovery profile (RP), which are based on the weighted sum in Munoz and Dunbar (2015) but do not include a time penalty, describe the summed deviation from a linear degradation and recovery, respectively. The seven resilience metrics are defined as follows, using the notation indicated in Fig. 1:
where T0 is the start time of the disruption; T1, T2 and T3 are the end time of the first, second and third phase, respectively; f(t)is the linear degradation function connecting (T0,Q0)and (T1,Qmin); g(t)is the linear recovery function connecting (T2,Qmin)and (T3,Q0); n is the total number of time intervals ti+1 ti, which is equal to the disruption length when taking a step size of one unit; and m and s are the number of equally spaced measurement points in the first and the third phase, respectively. The first five metrics can only have nonnegative values, whereas the last two metrics can have positive values as well as negative values. For all metrics, a higher positive value indicates a stronger disruptive effect, and thus, a less resilient network. Looking at the degradation profile (DP), in case that it degrades gradually and takes a long time to reach minimum performance, the DP value will be smaller than when performance drops quickly and then keeps degrading slowly. Therefore, such system is deemed more resilient, because the gradual degradation gives traffic controllers more time to respond, hence the smaller DP. Similar holds for recovery profile (RP). The current DP and RP metrics work well under the assumption of convex/concave behaviour of the performance in degradation/ recovery phases, which is the most commonly observed behaviour in real-life railway disruptions. However, due to a possible complex performance variability during some specific disruptions (e.g. a sudden drop in performance during recovery phase), DP or RP could reach close to zero values. Hence, it is recommended to always draw the resilience curve when studying a particular disruption, before interpreting its resilience metric values.” In order to calculate the resilience metrics, it is necessary to identify the timepoints T0,⋯,T3 which are shown in Fig. 1. All timepoints are derived from the resilience curve, because the reported timepoints in the disruption log data may not always be ac curate. Here, the start of disruption T0 is defined as the last moment before the reported start of the disruption when performance is still above target Q0. The end of disruption T3 is defined as the first moment after the recovery is initiated when performance is above target Q 0 again. Target performance Q 0 Transportation Research Part A 179 (2024) 103913 M.J. Knoester et al. is defined as the average networkwide performance during the day, measured over a number of relatively quiet days (i.e. with a limited number of disruptions) throughout the year. Assuming an approximately steady second phase, its start and end time T 1 and T 2 are determined by a modified steady state detection algorithm based on the approach of Dalheim and Steen (2020). A steady state detection algorithm is generally used to identify the steady parts of time series data. The approach of Dalheim and Steen (2020) involves performing regression analysis on consecutive, overlapping time windows. Their approach is modified to account for the dynamic nature of railway system performance, which results from the interplay between rescheduling the timetable, rolling stock and crew, and also for example from the up- or downscaling of a contingency plan and from deviations from the plan. Our modified steady state detection algorithm works by per forming regression analysis on consecutive, overlapping time windows in the lower part of the resilience curve. Modifications to the approach of Dalheim and Steen (2020) include the following. First, the length of the time windows is made dependent on the total number of measurement points by taking the square root of this number and multiplying it by a scaling factor δ. Taking the square root allows short windows for relatively short disruptions and longer windows for longer disruptions, while it prevents too long windows for very long disruptions. The scaling factor δ is set to 1.8 based on initial testing of the algorithm, after establishing that lower values (e.g. δ = 1.5) can yield too short windows, and higher values (e.g. δ = 2.0) can yield too long windows occasionally. Second, the identification of a steady state for a particular window is only allowed in case the first performance measurement in the window is below a certain threshold, to avoid a steady state from being detected in the first or third resilience phase. Third, evaluation by the sliding window from Dalheim and Steen (2020) is dropped, since the algorithm only has to produce the first and last steady point in the lower part of the resilience curve. The formal procedure of the modified steady state detection algorithm is included in Appendix B. For more details on the algorithm, the interested reader may refer to Knoester (2021).
3.4. Conducting statistical analyses :
Statistical analysis is a means to investigate patterns and relationships in quantitative data. Group comparisons are a class of statistical analysis, which we use to identify differences in the resilience metrics among disruptions of varying types. Knowing where and how large these differences are could help improve resilience in specific types of disruptions and in a specific phase or phases of a disruption. First, it is necessary to categorize the disruptions into groups. The disruption cause is taken as the variable that defines group membership, since for example, a line blockage due to a train defect might be inherently different than a line blockage due to an overhead line failure. The standard parametric option for group comparisons of a single dependent variable is one-way analysis of variance (ANOVA), while the nonparametric alternative is the Kruskal-Wallis test. In both cases, a number of assumptions must be satisfied in order to draw justified conclusions from the test results. Since the distributions of the resilience metrics violate the assumptions of both one- way ANOVA and the Kruskal-Wallis test, we use Welch’s ANOVA for the group comparisons. This method is similar to one-way ANOVA, but applies weights to adjust the grand mean (i.e. the mean of the total sample) based on the group means. Since ANOVA is robust against violation of the normality assumption (Mertens et al., 2017), Welch’s ANOVA is useful for analyzing data that are nonnormally distributed and that have unequal variances among groups. The F-statistic, which describes the part of the variation in the dependent variable that is explained by group membership, is defined as follows: where SS is the weighted sum of squares of the residual error, g is the number of groups and Λ is a factor based on the weights and group sizes. A high F-statistic indicates that differences among groups likely exist. ANOVA is an omnibus test, which is two-sided by defi nition. This means it reveals whether a difference exists among groups, but not where exactly the difference lies or how large it is (Mertens et al., 2017). A post hoc test is required to explore the results in more detail. The common post hoc test for Welch’s ANOVA is the Games-Howell test, which applies a series of pairwise comparisons among the groups while controlling for the family error rate. This test is similar to Tukey’s Honest Significant Difference, but does not require equal variances among groups.