Integrated optimization of train timetables rescheduling and response vehicles on a disrupted metro line

Metro disruption management refers to the field of conventional railway disruption management. In the early ages, disruption management is urgent in daily railway operations, because the railway infrastructures are not complete. Potential influence factors (Jespersen-Groth et al. [2009]) and fault-to-incident propagation (Yan et al. [2019]) are essential for disruption prevention in the railway lines. According to the extent of disruption, railway disruption can be into two types: partial disruption and complete disruption (Cacchiani et al. [2014]). For the partially disrupted railway line, some tracks with the same direction are disrupted. Trains of different directions have to share the same track in the partially disrupted tracks. (Louwerse & Huisman [2014], Luo et al. [2018]). However, the railway line would be divided into two separate parts if it is completely disrupteded (Veelenturf et al. [2016],Nielsen et al. [2012]). This classification is also accepted by metro disruption management. To cope with unexpected disruptions, four strategies are adopted in the metro timetable rescheduling: short-turning, skip-stopping, canceling, and speed adjustment. The short-turning strategy is to make the train circle a partial area of the railway line (Ghaemi et al. [2017],Schettini et al. [2022]). The result of Ghaemi et al. [2018a] have revealed that the optimal short-turning strategy depends on both the start time and the length of the disruption. The skip-stopping strategy is to let the train not stop at some stations, as it can avoid trains loading too many passengers. As this strategy makes tiny changes to the metro timetable, it is usually used together with other strategies (Zhu & Goverde [2020]). Canceling is to cancel some train services in the normal timetable (Zhu & Goverde [2020]). Except for the above three strategies, the normal timetable can also be rescheduled by adjusting train speed. This strategy is from a microscopic aspect, and the train running time of railway tracks would be changed. Speed adjustment is usually adopted to reduce passenger delays (Narayanaswami & Rangaraj [2013]), or to achieve real-time running control of high-speed railway trains (Xu et al. [2017]).

Most literature focus on operator-oriented timetable rescheduling. The objectives of these literature usually includes cost of rescheduling strategies (Schettini et al. [2022]), train delay (Tessitore et al. [2022], Xu et al. [2016]), or energy consumption (Bueno-Cadena et al. [2020]). Few papers focus on integrating timetable rescheduling with other topics.i.g.,Wang et al. [2021] integrates rolling stock circulation and timetable rescheduling. Bešinović et al. [2021] incorporates train timetable rescheduling and passenger flow control. These literature do not conclude the interests of passengers, which is not enough for providing high-quality services. To compensate for this shortcoming, some recent literature focus on passenger-oriented timetable rescheduling. As the disruption information is unknown, passengers have to decide whether to wait until the disruption has ended (Wang et al. [2014], Zhang & Lo [2018]) or board other transport means (Schmidt et al. [2018], Ghaemi et al. [2018b], Xu et al. [2021]). The choice of passengers depends on their general cost, their available incomes (Pnevmatikou et al. [2015]), and so on. To better fit the demand of passengers, passenger-oriented timetable rescheduling is now accepted by scholars. The objective of passenger-oriented timetable rescheduling is to minimize the passenger total traveling time (Zhu & Goverde [2020], Zhu & Goverde [2019]), or the satisfaction of passengers (Binder et al. [2017],Huang et al. [2020]). Usually, these problems are solved by decomposition algorithms (Zhan et al. [2021], Gao et al. [2016]) or intelligent algorithms (Zilko et al. [2016]). However, these literature does not consider the substitute transport, and passengers have to wait during thr overall disruption.

Owing to the blockage of disrupted metro line, some passengers can not successfully reach their destination by metro. Thus, auxiliary transport is needed to serve these passengers. In practical operations, buses have been adopted as their advantage of large capacity (sin [2021]). For example, the first train timetabling (Guo et al. [2019]. Kang et al. [2021]), last train timetabling (Kang & Meng [2017], Yang et al. [2020]), and contract design (Zhang & Lo [2020]). During peak hours, the bus bridging service can also be applied to assist passenger control on a busy metro line (Yang et al. [2017]). While facing with a long-time disruption, the operator has to desgin the bus bridging routes and their frequencies (Kepaptsoglou et al. [2009], Jin et al. [2016]). For a disrupted single metro line, most literature focus on the design of bus bridging to minimize the cost of passengers (Wang et al. [2019], Evelien van der Hurk et al. [2016]). While, for a disrupted metro network, the attention are more paid to the resilience to potential disruptions (Jin et al. [2014]), or to build a integrated rail-bus transit network (Hua & Ong [2018], Tan et al. [2020], Dou et al. [2019]).

However, there are two shortcomings of the current bus bridging research. First, most literature assumes that the auxiliary bus has fixed routes and fixed frequencies. Some researchers have also considered breaking the mindset of bus bridging services, and they have designed a more flexible bus bridging plan in response to metro disruptions (Wang et al. [2016], Gu et al. [2018]). Second, current research on the bus bridging service assumes that the road condition is constant all the time, and the bus can reach each station on time. In daily operation, this assumption is a rather strong hypothesis, as it has not considered the variation of road traffic. For example, bus bunching is a common phenomenon in our daily life, which will lead to an unstable bus delay on the congested road (Daganzo [2009], Daganzo & Pilachowski [2011], Delgado et al. [2012], Petit et al. [2018]). Therefore, to better fit the practical operational demand, the real-time road traffic information should be concluded in our reponse schemes.

To capture the real traffic dynamics, the DTA technologies (Peeta & Ziliaskopoulos [2001], Szeto & Lo [2006]) are considered in our research. Usually, it can be classified into two types: simulation-based approaches (Sbayti et al. [2007],Levin et al. [2015], Shafiei et al. [2018]) and analytical approaches (Carey & Watling [2012], Ke et al. [2013]). The simulation-based models often suffer from a huge computational burden, and the analytical approach also has such a deficiency because the models are often written as path-based formulations. Moreover, the analytical approach has the problem of poor convergence in large-scale networks (Jiang et al. [2016]). Fortunately, a linear programming (LP) framework for system optimal DTA (SO-DTA) problem (Ziliaskopoulos [2000]) was proposed, in which the cell transmission model (CTM) (Daganzo [1994], Daganzo [1995]) is employed to load traffic flow. This framework is widely used in large-scale emergency traffic planning and organization due to its lower computational burden. (Chiu et al. [2007], Yao et al. [2009], Xie et al. [2010], Ben-Tal et al. [2011], Kimms & Maiwald [2018], He & Peeta [2014], Liu et al. [2021] ). These literature has provided the foundation for the design of response vehicles on the road.

As mentioned above, most literature focuses on either train timetable rescheduling or bus bridging service. As far as the author knows, there is no literature integrating the metro timetable rescheduling and bus bridging service. Just like Zhu & Goverde [2020], the metro timetabling rescheduling does not take the substitute transport take into account, they assume that the passengers can wait until the disruption has ended. While in the design of the bus bridging service, the metro timetable is assumed to be fixed (see Jin et al. [2016], etc.). Thus, in this paper, we aim to integrate both the metro timetable rescheduling and response vehicle scheduling.

Besides, the modeling method of timetable rescheduling can be extended to deal with different problems. Zhu & Goverde [2020] has proposed an improved Dynamic Altenative Graph (DAG) formulation to reschedule the train timetable, and the passenger assignment procedure is to assign passengers to the alternative graph. Huang et al. [2020] use big-M and time-indexed formulations to reschedule the metro timetable, and the pasenger accumulation is measured by trains. However, these two important literature can not capture the time-varying accumulation of passengers, and this is required in design of response vehicles. To observe the real-time passenger accumulation in a station, we have introduced instaneous parameters and variables to record the passenger trajectory. Although there are four general reshceduling strategies, the short-turning and canceling strategy are efficient in response to unexpected disruption (Zhu & Goverde [2020]). To simplify the operation of metro operators, our paper only includes these two main strategies.

In the field of metro disruption management, DTA is a new technique in response to the disruption. Compared with traditional optimization method (i.g., Kepaptsoglou et al. [2009], Jin et al. [2016]), the biggest advantage of DTA is that it can capture real-time information of vehicles on the road.

Therefore, this paper contributes to the literature through three aspects:

$\bullet$A new two-stage optimization model is built to coordinate the metro timetable rescheduling and response vehicle scheduling. In particular, the passenger accumulation is measured by our proposed model, which provides a new approach to passenger-oriented timetabling compared with current research.

$\bullet$Most literature take bus bridging service as a substitute for trains, we explore to provide more flexible service by scheduling response vehicles. This provides a new perspective to the response of metro disruption.

$\bullet$The application of SO-DTA can capture the real-time road traffic, which is closer to reality compared with current metro disruption research.

The remainder of this paper is organized as follows. In section 2, the problem statements of our research are described. Section 3 presents a MILP model to reschedule the metro timetable. Furthermore, the real-time routes of response vehicles are optimized based on the SO-DTA model in section 4. Numerical cases of Beijing Metro Line 9 are conducted in section 5. Finally, the conclusions and possible topics for future research have been presented in section 6.

The metro timetable includes the train route as well as station arrival and departure times. The normal timetable is generated using daily passenger demand data. However, if an unexpected disruption occurs on the metro line, the normal timetable should be adjusted to ensure train safety. That is, in the disruption scenario, the train should not pass through the spatio-temporal disruption area and should turn around at the disruption area’s terminal station. If the train service in the normal timetable conflicts with the spatio-temporal disruption area, the train trajectory should be revised. The following constraints depict the activation of normal train services:

where binary parameter $\Theta_{{\rm{u}}}$ with value 1 indicating that train service u $\in\mathbb{U}$ conflicts with the spatio-temporal disruption area, and 0 otherwise. Constraint (2a)-(2b) means that the rescheduled arrival/departure times of a train service remain unchanged if it is activated, and does not conflict with the spatio-temporal disruption area. Constraint (3a)-(3b) mean that the arrival/departure time of an activated positive train service u $\in\mathbb{U}$ at station ${\rm{r}\in\mathbb{R}_{O}^{1}}\setminus\mathbb{R}_{\rm{T}}^{1}$ remains unchangesd if it conflicts with spatio-temporal disruption area. Constraint (4a) means that the arrival times of an actived positive train service u $\in\mathbb{U}$ at terminal station ${\rm{r}\in\mathbb{R}_{\rm{T}}^{1}}$ remain unchanged if it conflicts with the spatio-temporal disruption area. Constraint (4b) means that the departure time of an activated positive train service u $\in\mathbb{U}$ at terminal station ${\rm{r}\in\mathbb{R}_{T}^{1}}$ equals -1 if it conflicts with the spatio-temporal disruption area. Constraint (5a)-(5b) mean that the arrival/departure time of an activated positive train service u $\in\mathbb{U}$ at station ${\rm{r}\in\mathbb{R}\setminus\mathbb{R}_{\rm{O}}^{1}}$ equals -1 if it conflicts with the spatio-temporal disruption area.

For an activated negative train service, constraints (6a)-(6b) mean that the arrival/departure time of an activated train service u $\in\mathbb{U}$ at station ${\rm{r}\in\mathbb{R}_{O}^{2}}\setminus\mathbb{R}_{\rm{T}}^{2}$ remains unchanged if it conflicts with the spatio-temporal disruption area. Constraint (7a) means that the arrival time of an actived positive train service u $\in\mathbb{U}$ at terminal station ${\rm{r}\in\mathbb{R}_{\rm{T}}^{2}}$ remains unchanged if it conflicts with the spatio-temporal disruption area. Constraint (7b) mean that the departure time of an activated positive train service u $\in\mathbb{U}$ at terminal station ${\rm{r}\in\mathbb{R}_{T}^{2}}$ equals -1 if it conflicts with the spatio-temporal disruption area. Constraints (8a)-(8b) mean that the arrival/departure time of an activated negative train service u $\in\mathbb{U}$ at station ${\rm{r}\in\mathbb{R}\setminus\mathbb{R}_{\rm{O}}^{2}}$ equals -1 if it conflicts with the spatio-temporal disruption area. Note that, the arrival/departure time of a train service equals -1 if it is not activated (constraints (9a)-(9b)). Constraint (10) means that train service u $\in\mathbb{U}$ should be activated if it does not conflict with the spatio-temporal disruption area.

Recall that turnaround train services are saved in the set $\mathbb{\tilde{U}}$ and are activated by an activated normal train service if they conflict with the spatio-temporal disruption area. The following constraints depict the activation of turnaround train services:

where binary parameter $\delta_{{\rm{u,v}}}^{{\rm{r}}}$ with value 1 indicating that train service ${\rm{v}\in\mathbb{\tilde{U}}}$ is activated by the turnaround operations of train service ${\rm{u}\in\mathbb{U}}$, and 0 otherwise. Constraints (11a)-(11b) mean that an activated normal train service ${\rm{u}\in\mathbb{U}}$ turns around if it conflicts with the spatio-temporal disruption area, and then its turnaround train service ${\rm{v}\in\mathbb{\tilde{U}}}$ is activated. Constraints (12a)-(12b) mean that the arrival/departure times of an activated train service ${\rm{v}\in\mathbb{\tilde{U}}}$ at station ${\rm{r}\in\mathbb{R}}$ remain unchanged if it is activated. On the contrary, constraints (13a)-(13b)) mean that the arrival/departure times of an activated train service ${\rm{v}\in\mathbb{\tilde{U}}}$ at station ${\rm{r}\in\mathbb{R}}$ euqal -1 if it is not activated. Remark that the station arrival/departure time of train service ${\rm{v}\in\mathbb{\tilde{U}}}$ is calculated in advance, which is an input to our model. Constraint (14) means that a train service ${\rm{u}\in\mathbb{U}}$ does not turn around if it is not activated, and also a train service ${\rm{u}\in\mathbb{U}}$ does not turn around if it does not conflict with the spatio-temporal area. The turnaround train service ${\rm{v}\in\mathbb{\tilde{U}}}$ of a train service ${\rm{u}\in\mathbb{U}}$ would not be activated in both two mentioned conditions.

Figure 8 depicts the entire Beijing Metro Line 9, which has thirteen metro stations from Guo Gong Zhuang Station (GGZ) to National Library Station (NL). An unexpected disruption is taking place between Feng Tai South Road Station (FTSR) and Military Museum Station (MM), with the disruption lasting from 8:00 to 9:00. The metro transportation system is divided into two areas: in MS-1, the station set concludes four stations: GGZ station, FTSP station, KYR station, and FTSR station. We denote this set as $\mathbb{R}_{{\rm{O}}}^{1}=\{1,2,3,4\}$; In the MS-2, it also concludes four stations: MM station, BDZ station, BSBS station, NL station, and we denote this set as $\mathbb{R}_{{\rm{O}}}^{2}=\{10,11,12,13\}$. Furthermore, the station set of disruption area can be denotes as $\mathbb{R}_{{\rm{D}}}=\{4,5,6,7,8,9,10\}$. The terminal station sets of disruption area can be denoted as: $\mathbb{R}_{{\rm{T}}}^{1}=\{4\}$, $\mathbb{R}_{{\rm{T}}}^{2}=\{10\}$.

The partial timetable of Beijing Metro Line 9 is shown in Figure 9. In the original normal timetable generated by the Yizhuang metro line’s workday timetable, there are 118 train services from 7:20 to 9:30. There are 58 positive-direction train services and 60 negative-direction train services in this normal timetable. The volume of train service is set as $\rm{\bar{C}_{{\rm{u}}}}=1000$.

The practical road condition should be considered when scheduling response vehicles in the disruption area. Figure D.1 of D depicts the topological structure of the road network along the disrupted metro line and its cell discretion grapha. The cell length is equal to the free-flow speed multiplied by the 20-second time interval. As a result, the cell length is 400 m, the maximum capacity is 11 vehicles per time step, and the maximum occupancy per cell is 33 vehicles. Because the stations from FTES to BJW are closed for safety, response vehicles evaluate twelve OD pairs: from the origin station FTSR to the destination station FTES, QLZ, LLQ, LLQE, BJW, and MM; and from the origin station MM to the destination stations BJW, LLQE, LLQ, QLZ, FTNA, and FTSR. The response vehicle parameters are consistent, as shown in Table 1. The maximum flow velocity of the bus is 1992 vph for each lane, according to the triangle fundamental diagram. Table 2 shows the signal intersections and their fixed time plans.

As shown in Figure 9, an unexpected disruption occurs during the normal timetable from 8:00 to 9:00, and the disruption area spans from FSTR station to MM station. The spatio-temporal disruption area is a rectangle area in the new rescheduled timetable (as shown in Figure 10), and trains cannot run through it. To ensure the safety of two successive trains traveling in the same direction, we have set all minimum headways (including AA, AD, DA, and DD headways) to one minute. 13 train services before the disruption are canceled in the rescheduled timetable to meet headway constraints with other train services. That is, if these canceled train services were activated, the headway constraints would be violated. As a result, the unexpected disruption would result in train service cancellations before it occurred.

To avoid the conflict between train services and the spatio-temporal disruption area, train services have to be rescheduled to turn around at terminal stations of the disruption area. As shown by Figure 10, In the positive direction, 31 (of 58) train services have been rescheduled. In the meanwhile, 29 (of 60) negative direction train services have been rescheduled. These rescheduled train services are truncated as the blockage of disruption. For the positive direction train services, they have to turn around at FTSR station; while, for the negative direction train services, they turn around at MM station. Furthermore, it can be seen that no train services turn around after the disruption, and these turnaround operations are executed only before the disruption has ended. Therefore, the unexpected disruption would result in turnaround operations before the disruption has ended.

The unexpected disruption would result in a short recovery period after it has ended. During this period, the transport capacity is imbalanced. For the positive direction train services, the recovery time is 17 minutes, whereas the recovery time for negative direction train services is 22.5 minutes. The duration of the recovery period depends on the first train service after the disruption.

Statistical analysis shows that 61.9% of train services in the normal timetable have been rescheduled or canceled, while the disruption time occupancy is 46.2%. More effect on the normal timetable might be caused while the disruption area lies in the middle position of the normal timetable, this observation can be proved by the sensitive analysis part of this paper. As the directional difference of train services, even although the number of affected train services is similar in both two directions of train services, more disturbances would be caused for the positive direction train services as they have fewer train services. Therefore, the direction with fewer train services would occur more effect while facing with an unexpected disruption.

The normal metro timetable has been rescheduled to ensure the safety of trains under the disruption condition, and also the passengers are assigned to this new rescheduled timetable. Note that, passengers boarding these rescheduled train services would accumulate at the terminal stations of the disruption area, if their trip crosses the disruption area, this would bring a high safety risk to FTSR station and MM station. Therefore, response vehicles should be scheduled timely to transport these accumulated passengers.

Passenger accumulation of each station is an important index in metro disruption management since it reflects the real-time variation of passengers in a station. By observing the real-time passenger accumulation in a station, we can assess the passenger loading pressure of each station. There are 13 metro stations in our basic experiment, and the observation time is set from 8:00 to 9:00 (disruption period). Note that, the accumulated arrival/departure/accumulation flow at a station are calculated by the accumulated form of passenger flow,i.e., variables $\sum\limits_{{\rm{p}}\in\mathbb{P}}A_{{\rm{p}}}^{{\rm{r,t}}}/\sum\limits_{{\rm{p}}\in\mathbb{P}}D_{{\rm{p}}}^{{\rm{r,t}}}/\sum\limits_{{\rm{p}}\in\mathbb{P}}G_{{\rm{p}}}^{{\rm{r,t}}}$. The result of passenger accumulation at each station is shown in Figure 11.

As shown in Figure 11 (a), the accumulated arrival curve approximates a linear relationship, while the accumulated departure curve climbs along the accumulated departure curve. The passengers originating from GGZ station can be boarded in time, as adequate train services are departing from GGZ station. Meanwhile, it can be seen that the passengers standed on the platform of GGZ station can be controlled under 500 all the time. In Figure 11 (b), the accumulated arrival curve approximates a linear relationship, and the accumulated departure curve climbs along the accumulated departure curve. Different from GGZ station, the arriving rate of passengers at FTSP station increase suddenly after 8:38, and it can also be met by our train services. The number of passengers standed on the platform of GGZ station and FTSP station can be controlled under 500 all the time, which is acceptable by metro managers.

The FTSR station, being the first terminal station of the disruption area in the positive direction, has a different accumulation tendency, as shown in Figure 11 (d). There are no positive direction train services departing from FTSR station; all positive direction train services finish their tasks at FTSR station, while a new negative direction train service departs from FTSR station. Thus, the newly activated turnaround train service will serve passenger demand in the negative direction; however, passengers originating from the FTSR will still have to wait in this station. Furthermore, transfer passengers wishing to transfer from the FTSR station to other stations in the positive direction must wait at the FTSR station. These two sorts of stranded passengers have caused the FTSR station’s passenger accumulation to significantly expand. However, it is not worried that the response vehicles would serve the stranded passengers in time.

Figure 11 (e) depicts the passenger accumulation at closed stations, where the accumulation of passenger flows equals zero because these stations are closed and no passengers or trains arrive.

The passenger accumulation at MS-1 stations has been depicted in Figure 11 (a)-(e). Similarly, the passenger accumulation at MS-2 stations follows a similar pattern as that of MS-1 stations. The accumulated arrival curve approximates a linear relationship, as shown by the passenger accumulation at TJS station in Figure 11 (f), while the accumulated departure curve climbs along the accumulated arrival curve. Passengers traveling in the negative direction tend to gather at station MM. There are two sorts of passengers accumulating at MM station: passengers who originate at MM station and transfer passengers who travel from MM station to other stations. It should be noted that the response vehicles would serve the transfer passenger flows.

In Figure 11 (g)-(i), the passenger accumulation at stations of MS-2 (except for the terminal station of the disruption area) has been demonstrated. The accumulated arrival curve of these two stations approximates a linear relationship, while the accumulated departure curve climbs along the arrival curve. It is obvious that the passengers departing from these three stations will be picked up in time. Passenger accumulation at each station can be controlled within an acceptable range.

The average waiting time and passenger arrival are depicted in Figure 12. The bar chart depicts the number of passengers arriving at each stop. It can be seen that the FTSR and MM stations have the highest average number of arriving passengers. These two stations’ platforms must service more than 300 people per minute. There is also no passenger demand from FTES to BJW because these five stations are blocked during the disruption period. Furthermore, the average number of people coming in MS-1 is larger than in MS-2, indicating that MS-1 is under greater passenger pressure. The line chart depicts the average time passengers spend waiting at each station. Passengers’ average waiting time is greatest at FTSR and MM stations, with values greater than 30 minutes. This is because some passengers departing from these two stations must wait until the disruption is over if no response vehicles arrive. Furthermore, the average wait time at other open stations is less than 3 minutes, which is consistent with train service departure frequency. This results show that the new rescheduled timetable can provide passengers with an efficient mode of transportation, and the passengers will be picked up in time. As a result, passengers stranded at FTSR and MM stations would have a shorter wait if we offered timely response vehicle service, and other passengers would only have to wait 2 minutes before boarding a train. The impact of disruption can be mitigated by coordinating the optimization of metro and response vehicles.

Fig 13 (a) and Fig 13 (b) depict the cumulative arrival curves in two directions (i.e., positive direction and negative direction). The cumulative curves show a step-by-step change because the response vehicles are scheduled at fixed time intervals of 5 minutes. We can obtain the departure sequence and fleet size of response vehicles, which are combined with a space-time diagram to provide real-time guidance.

The network clearance time (NCT) is made up of two components: path holding time and path travel time. For example, the larger the NCT without considering dynamic traffic, the farther the station is from the origin station. However, the NCT of S4-S7 and S4-S8 is quite similar because S8 receives more response vehicles than S7. The NCT gap between S10-S5 and S10-S4 is 1 minute in the direction of S4, whereas the corresponding fleet size gap is 84 vehicles. The slope of the cumulative arrival curves at station S4 is greater becausethe shorter routes are prioritized for the fleets reaching station S4.

The results of fleet size and network clearance time for various OD pairs are summarized in Table 3. The maximum network clearance time provides a lower time limit for temporary exclusive lanes. The exclusive lanes are maintained in this case for at least 63.67 minutes, which can be used as the time threshold for designating exclusive lanes for response vehicles.

To compare the transport efficiency of response vehicles, the total travel time of the response vehicle is compared to the shortest path scheme. The shortest path scheme sets fixed routes with the shortest distance between two stations. In this case, the shortest route is (1, 5, 9, 10, 14, 19, 20, 23, 24, 29, 30, 32, 36, 37), and vehicles are unable to change routes. The results of the response vehicles and shortest path schemes are shown in Figure 14, where the TTT using the response vehicles scheme is 7% lower than the fixed shortest path scheme. However, as illustrated in Fig 15 (a)-(b), the NCT of different OD pairs may be greater than the fixed shortest path. Because the goal of scheduling response vehicles is to reduce total travel time for all vehicles and the NCT is determined by the capacity entering the destination station, some vehicles may stay at their original stations to avoid congestion and increase system costs.

To investigate the phenomenon throughout different disruption periods, the disruption area is set from the FTSR station to the MM station, with a one-hour duration. Figure 16 depicts the sensitive analysis of the disruption period on the normal timetable rescheduling. Apart from the disruption period of 6:00–7:00, it can be seen that the number of rescheduled train services is greater than the number of normal train services. Early train services would be less affected by the disruption since they would have a higher chance of avoiding the spatio-temporal disruption area. Furthermore, we can see that during other disruption periods, the disruption may not only affect normal train services during this disruption period, but also train services later in the period. This observation reflects the spread of disruptions to normal train services.

Furthermore, we have chosen two typical disruption periods as our focused disruption periods: peak hour (8:00–9:00 a.m.) and off-peak hour (10:00–11:00 a.m.). The sensitive analysis of the disruption period on the normal timetable rescheduling has been depicted in Figure 17. The number of rescheduled train services grows as the number of disrupted stations increases. Moreover, during the peak hour disruption period (see Figure 17 (a)), more train services are rescheduled in the metro line’s central area. For example, if the number of stations in the disruption area is fixed at three, the number of rescheduled train services in the disruption area from QLZ to LLQ has the greatest value, and this value decreases when the disruption area is closed to the metro line’s terminal stations. During the off-peak hour disruption period (see Figure refsensitive area2 (b), more train services are rescheduled in terminal stations.

Apart from the normal metro timetable, the sensitive analysis of the disruption on passenger accumulation is also crucial in metro disruption management. Due to the time-varying passenger demands at different stations, we have set the disruption area from FTSR station to MM station, and the disruption period is one hour. As a result, the sensitive analysis of the disruption period on passenger accumulation is depicted in Figure 18.

Figure 18 shows that the total number of accumulated passengers is higher before 9:00. This is due to the fact that passenger demand is usually high during the normal timetable, causing the passenger accumulation to be significantly higher during the disruption period. When an unexpected disruption occurs before 9:00 a.m., the metro manager should pay more attention. Furthermore, the average waiting time at both the FTSR station and MM station is around 30 minutes, whereas at other stations it is less than 3 minutes. This is due to the fact that passengers stranded at these two terminals will not be able to board any transportation means until the disruption is over. As a result, the response vehicles must arrive in time, or passengers may be left waiting for a long period of time on the platforms of the FTSR and MM stations.