Resilience-based protection decision optimization for metro network under operational incidents

LIU Xi-bin, MA Jian, HAO Ru-ru, SONG Qing-song. Design of regenerative braking energy feedback system for ultracapacitor heavy-duty tractor[J]. Journal of Traffic and Transportation Engineering, 2013, 13(2): 60-65. doi: 10.19818/j.cnki.1671-1637.2013.02.009

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LU Qing-chang, LIU Peng, XU Biao, CUI Xin. Resilience-based protection decision optimization for metro network under operational incidents[J]. Journal of Traffic and Transportation Engineering, 2023, 23(3): 209-220. doi: 10.19818/j.cnki.1671-1637.2023.03.016

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Resilience-based protection decision optimization for metro network under operational incidents

doi: 10.19818/j.cnki.1671-1637.2023.03.016

School of Electronics and Control Engineering, Chang'an University, Xi'an 710064, Shaanxi, China

Funds:

National Natural Science Foundation of China 71971029

Fok Ying-Tong Education Foundation for Young Teachers in the Higher Education Institutions of China 171069

Natural Science Basic Research Program of Shaanxi 2021JC-28

More Information

Author Bio:
LU Qing-chang(1984-), male, professor, PhD, qclu@chd.edu.cn
Received Date: 2022-12-28
Available Online: 2023-07-07
Publish Date: 2023-06-25

Abstract

Abstract

The protection decision optimization problem for metro networks was studied to alleviate the negative impacts triggered by operational incidents and improve the capability of metro networks to tackle these incidents. For the network resilience, the variation characteristics of the resilience curve and cumulative loss of the performance in the degradation and recovery of network performance were considered, and a two-layer optimization model for metro network protection decisions was constructed. The upper model was a stochastic integer programming model for identifying the optimal choice of the stations to be protected in the scenarios of uncertain operational incidents. The lower model was a user equilibrium assignment problem, and the variations in the queueing passenger flow and waiting time for recovery at the stations with limited capacity were prioritized to accurately estimate the delay time for passenger travel under operational incidents. The genetic algorithm and Frank-Wolfe algorithm were used to solve the upper and lower models, respectively. The metro network in the central area of Xi'an was taken as an example to verify and analyze the proposed models and algorithms. Analysis results show that the resilience-based protection decision is capable of reducing the loss of network performance by more than 50% by protecting 37.5% of the stations in the research region. It is superior to the vulnerability-based protection decision and the one without considering the substitution role of the bus network. Compared to the situation of the vulnerability-based one, the losses of network performance and passenger flow time reduce by 6.18% and 582 h, respectively, when half of the metro stations in the network are protected by the resilience-based protection decision. The protection priorities of more than two-thirds of stations in the metro network alter due to the substitution role of the bus network. For the same type of stations, their dependence on the substitution role of the bus network enhances with the increase in the passenger flow. The protection priority of a metro station is dependent mainly on the passing passenger flow and transportation capacity. A larger passenger flow is accompanied by a lower transportation capacity and higher protection priority of corresponding stations. The station type is also a factor determining the protection priority, especially for the stations with large passenger flows, and higher protection priorities are required for the non-transfer stations.
- urban rail transit,
- protection decision optimization,
- two-layer optimization model,
- network resilience,
- operational incident,
- genetic algorithm

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0. 引言

1. 网络构建与问题描述

Figure 1. Example of metro network

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Figure 2. Metro network performance evolution curves under operational incidents

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2. 保护决策框架

2.1 地铁运营事件建模方法

$P\left(\eta_{n_{\mathrm{m}}}=\alpha\right)=\frac{\lambda_{n_{\mathrm{m}}}^\alpha}{\alpha !} \mathrm{e}^{-\lambda_{n_{\mathrm{m}}}} \quad \alpha=0, 1, \cdots$

(1)

$\lambda_{n_{\mathrm{m}}}=\frac{\lambda}{j}$

(2)

In the formula:jNumber of subway stations.

Using logarithmic normal distribution to represent failed subway stations under operational eventsn_mProbability characteristics of required recovery time

$P\left(d_{n_{\mathrm{m}}}=\beta\right)=\frac{1}{\beta \sigma_{n_{\mathrm{m}}} \sqrt{2 \pi}} \mathrm{e}^{-\frac{\left[\ln (\beta)-\mu_{n_{\mathrm{m}}}\right]^2}{2 \sigma_{n_{\mathrm{m}}}^2}} \quad \beta>0$

(3)

Based on equation (3), subway stations under operational eventsn_mThe remaining recovery time is determined by a variables_{t_e, n_m}Description, ifs_{t_e, n_m}=0 indicates that the simulated operational event did not occur at the siten_mOtherwise, the subway stationn_mThe recovery times_{t_e, n_m}dod_{n_m}.

2.2 Resilience measurement method for subway network

After an operational event occurs, the efficiency of subway network operation and passenger transportation are important indicators for evaluating network performance. This article constructs a performance evaluation index L (t) based on network operational efficiency, namely

$L(t)=\frac{\sum\limits_n\left(f_{1, t_0, n} z_{1, t_0, n}+f_{2, t_0, n} z_{2, t_0, n}\right)+\sum\limits_a f_{t_0, a} z_{t_0, a}}{\sum\limits_n\left(f_{1, t, n} z_{1, t, n}+f_{2, t, n} z_{2, t, n}\right)+\sum\limits_a f_{t, a} z_{t, a}}$

(4)

In the formula:f_{1, t₀, n}andf_{2, t₀, n}Before the occurrence of operational eventst₀Always pass through any station in the transportation networkn(n∈N）The inbound and outbound passenger flow;f_{t₀, a}Prior to the occurrence of operational eventst₀Always pass through the road sectionaFoot traffic;z_{1, t₀, n}andz_{2, t₀, n}Before the occurrence of operational eventst₀Time StationnImpedance of entry time and exit time;z_{t₀, a}Prior to the occurrence of operational eventst₀Time sectionaTime impedance;f_{1, t, n}andf_{2, t, n}After the operational event, respectivelytTime passes through the sitenThe inbound and outbound passenger flow;f_{t, a}After the operation eventtAlways pass through the road sectionaPassenger flow;z_{1, t, n}andz_{2, t, n}After the operational event, respectivelytTime StationnImpedance of entry time and exit time;z_{t, a}After the operation eventtTime sectionaThe time impedance.

$R=\int_{t_{\mathrm{e}}}^{t_{\mathrm{e}}+D} 1-L(t) \mathrm{d} t \approx \sum\limits_{i \in k}\left[1-L\left(t_i\right)\right] \Delta t$

(5)

In the formula:L(t_i）Fort_iTime network performance.

According to equation (5),RThe smaller the size, the smaller the performance loss during the recovery process, and the higher the resilience of the subway network.

2.3 基于韧性的保护决策优化模型

This article proposes a two-layer optimization model based on resilience to solve the optimization problem of protection decisions in subway networks. The upper level model is an optimization model for subway network protection decisions, with the objective function being the expectationRMinimum, that is, the cumulative performance loss during the entire process of performance degradation and recovery is minimized. The decision variable is the protection status of each subway station, and the objective function isE(R）For

$\min E(R)=E\left\{\sum\limits_i\left[1-L\left(t_i\right)\right] \Delta t\right\}$

(6)

$\left\{\begin{array}{l} t_1=t_{\mathrm{e}} \\ t_k=t_{\mathrm{e}}+D-\Delta t \\ \Delta t=t_{i+1}-t_i \end{array}\right.$

(7)

$L\left(t_{\mathrm{e}}+D\right)=1$

(8)

$\sum\limits_{n_{\mathrm{m}}} y_{n_{\mathrm{m}}} \leqslant B$

(9)

$s_{t_{\mathrm{e}}, n_{\mathrm{m}}} y_{n_{\mathrm{m}}}=0$

(10)

$y_{n_{\mathrm{m}}} \in\{0, 1\}$

(11)

In the formula:y_{n_m}As a 0-1 decision variable, if the subway stationn_mProtected,y_{n_m}=1. Otherwisey_{n_m}0BTo protect the budget, determine the maximum number of sites that can be protected.

Equation (6) minimizesRExpectations; Equation (7) is the discrete-time set calculated by the model; Equation (8) constrains network performance int_e+DAlways restore to the level before the operation event; Equation (9) is a budget constraint, which means that the number of protected subway stations does not exceed the budgetBEquation (10) limits the operation recovery time of any protected road section to 0; Equation (11) is for protecting decision variablesy_{n_m}The binary constraint.

$\begin{array}{l} \min Z_{t_i}(f)=\sum\limits_a z_{t_i, a}+\sum\limits_{n_{\mathrm{m}}} \int_0^{f_{1, t_i, n_{\mathrm{m}}}} z_{1, t_i, n_{\mathrm{m}}}(x) \mathrm{d} x+ \\ \sum\limits_{n_{\mathrm{b}}} \int_0^{f_{1, t_i, n_{\mathrm{b}}}} z_{1, t_i, n_{\mathrm{b}}}(x) \mathrm{d} x+\sum\limits_{n_{\mathrm{m}}} z_{2, t_i, n_{\mathrm{m}}}+ \\ \sum\limits_{n_{\mathrm{b}}} z_{2, t_i, n_{\mathrm{b}}} \end{array}$

(12)

$\sum\limits_u f_{t_i, u, w}=q_w$

(13)

$f_{t_i, a}=\sum\limits_w \sum\limits_u f_{t_i, u, w} \delta_{t_i, a, u, w}$

(14)

$f_{1, t_i, n_{\mathrm{m}}}=\sum\limits_{w w} \sum\limits_u f_{t_i, u, w} \delta_{1, t_i, n_{\mathrm{m}}, u, w}$

(15)

$f_{1, t_i, n_{\mathrm{b}}}=\sum\limits_w \sum\limits_u f_{t_i, u, w} \delta_{1, t_i, n_{\mathrm{b}}, u, w}$

(16)

$f_{2, t_i, n_{\mathrm{m}}}=\sum\limits_w \sum\limits_u f_{t_i, u, w} \delta_{2, t_i, n_{\mathrm{m}}, u, w}$

(17)

$f_{2, t_i, n_{\mathrm{b}}}=\sum\limits_w \sum\limits_u f_{t_i, u, w} \delta_{2, t_i, n_{\mathrm{b}}, u, w}$

(18)

$f_{t_i, u, w} \geqslant 0$

(19)

$z_{1, t_i, n_{\mathrm{m}}}(x)=\frac{H_{n_{\mathrm{m}}}}{2}+\left\lceil\frac{F_{t_{i-1}, n_{\mathrm{m}}}+x}{C_{n_{\mathrm{m}}}}\right\rceil H_{n_{\mathrm{m}}}+s_{t_i, n_{\mathrm{m}}}$

(20)

$z_{2, t_i, n_{\mathrm{m}}}=\frac{z_{\mathrm{d}, n_{\mathrm{m}}}}{\zeta_{t_i, n_{\mathrm{m}}}}$

(21)

$z_{1, t_i, n_{\mathrm{b}}}(x)=\frac{H_{n_{\mathrm{b}}}}{2}+\left\lceil\frac{F_{t_{i-1}, n_{\mathrm{b}}}+x}{C_{n_{\mathrm{b}}}}\right\rceil H_{n_{\mathrm{b}}}$

(22)

$z_{2, t_i, n_{\mathrm{b}}}=z_{\mathrm{d}, n_{\mathrm{b}}}$

(23)

$z_{t_i, a}=z_{\mathrm{d}, a} \quad a \in\left\{A_{\mathrm{x}}, A_{\mathrm{m}_1}, A_{\mathrm{m}_2}, A_{\mathrm{b}}\right\}$

(24)

$F_{t_i, n_{\mathrm{m}}}=F_{t_{i-1}, n_{\mathrm{m}}}+f_{1, t_i, n_{\mathrm{m}}}-\left\lfloor\frac{\Delta t}{H_{n_{\mathrm{m}}}}\right\rfloor C_{n_{\mathrm{m}}} \zeta_{t_i, n_{\mathrm{m}}}$

(25)

$F_{t_i, n_{\mathrm{b}}}=F_{t_{i-1}, n_{\mathrm{b}}}+f_{1, t_i, n_{\mathrm{b}}}-\left\lfloor\frac{\Delta t}{H_{n_{\mathrm{b}}}}\right\rfloor C_{n_{\mathrm{b}}}$

(26)

$s_{t_i, n_{\mathrm{m}}}=s_{t_{\mathrm{e}}, n_{\mathrm{m}}}-(i-1) \Delta t$

(27)

$\zeta_{t_i, n_{\mathrm{m}}}= \begin{cases}1 & s_{t_i, n_{\mathrm{m}}}=0 \\ 0 & s_{t_i, n_{\mathrm{m}}}>0\end{cases}$

(28)

$F_{t_i, n_{\mathrm{m}}}, F_{t_i, n_{\mathrm{b}}} \geqslant 0$

(29)

Equations (12) to (19) are standard user equilibrium allocation problems, where equation (12) is used to calculatet_iThe equilibrium flow of the network at time, equation (13) represents that all travel demands can be met, and equations (14) to (18) are used to calculatet_iAlways pass through the road sectionaAnd the siten_mandn_bThe passenger flow is limited by equation (19)f_{t_i, u, w}Is a non negative variable; Equations (20) to (23) are road resistance calculation functions, and equation (20) ist_iTime subway stationn_mImpedance of station entry time, including the time passengers wait for the subway to arrive at the stationH_{n_m}/2^[22]Recovery waiting times_{t_i, n_m}Waiting time in line $\left\lceil\frac{F_{t_{i-1}, n_{\mathrm{m}}}+x}{C_{n_{\mathrm{m}}}}\right\rceil H_{n_{\mathrm{m}}}$ Equation (21) ist_iTime subway stationn_mThe impedance of departure time; Equation (22) ist_iTime Bus Stopn_bImpedance of station entry time, including the time passengers wait for the bus to arrive at the bus stopH_{n_b}/2. Waiting time in line $\left\lceil\frac{F_{t_{i-1}, n_{\mathrm{b}}}+x}{C_{n_{\mathrm{b}}}}\right\rceil H_{n_{\mathrm{b}}}$ Equation (23) ist_iTime Bus Stopn_bThe impedance of departure time; Equation (24) represents the driving section inside the subway cara_m₁∈A_m₁Bus routes within the busa_b∈A_bTransfer sections between subway linesa_m₂∈A_m₂Transfer section between bus and subwaya_x∈A_xFixed time impedance; Formula (25) calculationt_iTime subway stationn_mAccumulated passenger flow, byt_i－1Accumulated passenger flow upon arrival at the station at any given timeF_{t_i－1, n_m}、t_iThe passenger flow arriving at the momentf_{1, t_i, n_m}andt_iThe passenger flow leaving by subway at all times $\left\lfloor\frac{\Delta t}{H_{n_{\mathrm{m}}}}\right\rfloor C_{n_{\mathrm{m}}} \zeta_{t_i, n_{\mathrm{m}}}$ Decision, and only for subway stationsn_mNormal operation, i.eζ_{t_i, n_m}=Passengers can only leave the station at 1 o'clock; Formula (26) calculationt_iAlways at the bus stopn_bAccumulated passenger flow at the station, byt_i－1Accumulated passenger flow over timeF_{t_i－1, n_b}、t_iThe passenger flow arriving at the momentf_{1, t_i, n_b}andt_iPassenger flow leaving by bus at all times $\left\lfloor\frac{\Delta t}{H_{n_{\mathrm{b}}}}\right\rfloor C_{n_{\mathrm{b}}}$ decision; Formula (27) calculationt_iTime subway stationn_mRemaining operational recovery time; Equation (28) is used to determinet_iThe operational status of each subway station at all times; Equation (29) isF_{t_i, n_m}andF_{t_i, n_b}Non negative constraints.

3. Algorithm design

Figure 3. Flow of algorithm

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4. 案例分析

4.1 试验设计与参数设置

Figure 4. Metro network in Xi'an central area

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Table 1. Metro station parameters

编号	站点名称	车头时距/min	地铁列车容量/(人次·veh^-1)	正常运营路段客流/(人次·h^-1)
1	西北工业大学(5号线)	4	560	2 232
2	边家村	4	460	2 127
3	省人民医院·黄雁村	4	460	2 505
4	南稍门(5号线)	4	380	2 949
5	文艺路	4	440	1 428
6	建筑科技大学·李家村(5号线)	4	540	2 547
7	太乙路	4	560	1 242
8	雁翔路北口	4	560	1 482
9	青龙寺(5号线)	4	480	1 389
10	青龙寺(3号线)	3	480	1 563
11	北池头	3	480	1 299
12	大雁塔(3号线)	3	320	3 852
13	小寨(3号线)	3	380	3 624
14	吉祥村	3	480	2 568
15	太白南路	3	480	2 103
16	科技路(3号线)	3	560	2 985
17	科技路(6号线)	4	560	2 307
18	西北工业大学(6号线)	4	560	2 307
19	南稍门(2号线)	2	360	2 565
20	建筑科技大学·李家村(4号线)	4	400	2 343
21	体育场	2	400	1 629
22	西安科技大学	4	400	1 341
23	小寨(2号线)	2	340	2 820
24	大雁塔(4号线)	4	280	2 682

| Show Table

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4.2 模型效果

In order to verify the necessity and effectiveness of considering cumulative performance loss and bus network substitution in the optimization problem of subway protection decisions, this paper analyzes the proposed optimal protection decision P, vulnerability based protection decision P1, and protection decision P2 that only considers the subway network. Protect decision P1 based on the degree of network performance degradation at the time of the operational eventL(t_e）As the objective function, the cumulative performance loss during the performance degradation process was not taken into account. The protection decision P2 is only based on the analysis of the subway network, taking into account the cumulative performance loss, but does not consider the substitution effect of the public transportation network under operational events.

Figure 5. Fitness convergence processes of genetic algorithm and exhaustive method when B=9

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Figure 6. Network performance loss reduction rates and passenger flow time losses based on different calculation methods under different protection budget levels

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In order to visually describe the differences in protection priority levels among different stations, this article is based on the subway stationsB=3，B=Until 6B=At 21:00, the order of protection in the protection decision is divided into 8 protection priorities for the 24 subway stations in the study area, with 3 stations as one protection priority. of whichB=The protection priority of the protected site at 3 o'clock is the highest, with a protection priority of 1. The protection priority of the site that does not appear in any budget protection decisions is the lowest, with a protection priority of 8.Figure 7The change in protection priority from decision P2 to decision P for each subway station shows significant differences in the protection priority of 16 stations in 2 decisions, indicating that the ground bus network, as an important alternative to the subway network, will have an impact on subway protection decisions. Specifically, if the change in protection priority of a subway station is negative, then the station has a higher protection priority in the protection decision P2 that does not consider the substitution effect of buses, indicating that it relies more on the substitution effect of buses; Among the 14 transfer stations, only Dayan Pagoda Station (Line 3) and Xiaozhai Station (Line 3) have higher protection priority in protection decision P2; Among the 10 non transfer stations, Jixiang Village Station, Bianjia Village Station, Taibai South Road Station, Yanxiang Road North Exit Station, and Wenyi Road Station have higher protection priority in decision P2. The other 5 non transfer stations have the same protection priority in both decisions, and no station has a higher protection priority in decision P. This result indicates that overall, non transfer stations rely more on the substitution role of the public transportation network. Due to the lack of redundant subway lines at non transfer stations during operational events, they require alternative transportation options provided by other modes to ensure the topological connectivity of these stations. combineFigure 7andTable 1It can be seen that for the same type of station, the larger the passenger flow, the more dependent it is on the substitution effect of the public transportation network, such as a flow greater than 2500 people per hour^-1The Huangyan Village Station and Jixiang Village Station of the Provincial People's Hospital rely most on the substitution effect of public transportation, with a flow of less than 1500 people per hour^-1The Taiyi Road Station, Yanxiang Road North Exit Station, Sports Stadium Station, and Xi'an University of Science and Technology Station have the same protection priority in the two decisions. This is because the number of passengers who transfer to the bus network after these stations fail is relatively higher. Without the bus network as an alternative travel option, more passengers will be affected.

Figure 7. Changes in station protection priorities from decision P2 to decision P

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4.3 不同预算下的最优保护决策

Table 2. Optimal protection decisions under different protection budget levels

预算/个	韧性	保护决策站点选择
0	2.72	无
3	2.09	12、14、15
6	1.61	4、12、13、14、15、16
9	1.25	2、3、4、12、13、14、15、16、21
12	0.91	2、3、4、6、12、13、14、15、16、19、21、24
15	0.62	2、3、4、6、12、13、14、15、16、17、18、19、21、22、24
18	0.38	2、3、4、6、8、10、12、13、14、15、16、17、18、19、21、22、23、24
21	0.17	1、2、3、4、6、7、8、10、12、13、14、15、16、17、18、19、20、21、22、23、24

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4.4 不同运营恢复时间对最优保护决策的影响

Figure 8. Resilience losses of metro network under different operation recovery times

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5. 结语

(1) Resilience based protection decisions are superior to vulnerability based protection decisions and protection decisions that do not consider the substitution effect of public transportation. When protecting a small number of sites in the network, vulnerability based protection decisions can provide similar protection effects as resilience based protection decisions, with a difference of less than 0.50% between the two; When there are many sites involved in the network, the difference in protection effectiveness between the two decisions can be significant, up to 6.18%. Therefore, resilience based protection decisions are suitable for major events that cause widespread network failure.

(4) The failure of the subway network under operational events will affect the normal operation of the public transportation network. Analyzing the resilience of the multi-mode transportation system coupled with the bus subway and carrying out protection decision optimization of the coupled transportation system will be the future research content.

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Resilience-based protection decision optimization for metro network under operational incidents

doi: 10.19818/j.cnki.1671-1637.2023.03.016