# Assessing Protection Strategies for Urban Rail Transit Systems: A Case-Study on the Central London Underground

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

#### 2.1. Single-Asset Metrics for Fortification

**Connectivity-driven metrics**

**Path length-driven metrics**

**Flow-driven metrics**

**Combined metrics**

#### 2.2. Systemic Approach to Fortification

## 3. Methodology

#### 3.1. Sequential Approach

#### 3.2. Integrated Approach

- N is the set of network nodes, indexed by i, s or d,
- A is the set of network arcs,
- $P\left(sd\right)$ is the set of paths connecting nodes s and d, indexed by p,
- $N\left(p\right)\subseteq N$, is the set of nodes in path p,
- ${l}_{p}$ is the length of path p,
- $L{P}_{sd}$ is the length of the longest path connecting nodes s and d,
- ${f}_{sd}$ is the amount of passenger flow travelling from node s to node d,
- D is the maximum number of nodes that can be disrupted simultaneously,
- B is the amount of budget available for protection,
- ${\varphi}_{sd}$ is a penalty cost incurred when nodes s and d are disconnected,
- ${\eta}^{p},{\eta}^{f}$ are constants used to normalise path and flow objectives, respectively,
- ${\alpha}^{C},{\alpha}^{P},{\alpha}^{F}\in \left[0,1\right]$ are real coefficients used to weight connectivity, path and flow objectives, respectively, whose sum is equal to 1.

- ${W}_{i}$ is equal to 1 if node i is protected, 0 otherwise,
- ${X}_{i}$ is equal to 1 if node i is disrupted, 0 otherwise,
- ${Y}_{sd}$ is the length of the shortest non-disrupted path from node s to node d,
- ${Z}_{sd}$ is equal to 1 if there is no connection between nodes s and d, 0 otherwise.

**The Railway Fortification Problem (RFP)**

## 4. Central London Underground: A Case Study

## 5. Discussion

#### 5.1. Sequential Approach Analysis

#### 5.2. Integrated Approach Analysis

- connectivity-based model (${\alpha}^{C}$ = 1, ${\alpha}^{P}$ = 0, and ${\alpha}^{F}$ = 0), referred to as ${\mathrm{RFP}}^{c}$;
- path-based model (${\alpha}^{C}$ = 0, ${\alpha}^{P}$ = 1, and ${\alpha}^{F}$ = 0), referred to as ${\mathrm{RFP}}^{p}$;
- flow-based model (${\alpha}^{C}$ = 0, ${\alpha}^{P}$ = 0, and ${\alpha}^{F}$ = 1), referred to as ${\mathrm{RFP}}^{f}$; and
- multi-criteria model with equal weights model (${\alpha}^{C}$ = 0.33, ${\alpha}^{P}$ = 0.33, and ${\alpha}^{F}$ = 0.33), referred to as ${\mathrm{RFP}}^{m}$.

#### 5.3. Comparing Sequential and Integrated Approaches

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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ND | HC | NV | NB | PF |

King’s Cross | OxfordCircus | GreenPark | GreenPark | Bank |

Bank | GreenPark | OxfordCircus | OxfordCircus | King’s Cross |

BakerStreet | Bank | Bank | LeicesterSquare | Waterloo |

Waterloo | BakerStreet | BakerStreet | PiccadillyCircus | Victoria |

OxfordCircus | Embankment | King’s Cross | Embankment | LiverpoolStreet |

Moorgate | King’s Cross | Embankment | TottenhamCourtRoad | BakerStreet |

GreenPark | LeicesterSquare | Victoria | CharingCross | Moorgate |

Embankment | Holborn | BondStreet | King’s Cross | LondonBridge |

LiverpoolStreet | TottenhamCourtRoad | Waterloo | Westminster | Paddington |

Aldgate | Waterloo | Holborn | WarrenStreet | Farringdon |

ST | IM | SV | WA | WI |

Bank | GreenPark | Bank | Bank | King’s Cross St Pancras |

King’s Cross | OxfordCircus | King’s Cross | King’s Cross | Bank |

Waterloo | LeicesterSquare | Waterloo | Waterloo | Waterloo |

BakerStreet | PiccadillyCircus | Victoria | Victoria | BakerStreet |

Moorgate | Embankment | BakerStreet | BakerStreet | GreenPark |

LiverpoolStreet | TottenhamCourtRoad | Moorgate | LiverpoolStreet | OxfordCircus |

Victoria | CharingCross | GreenPark | Moorgate | Moorgate |

GreenPark | King’s Cross | OxfordCircus | OxfordCircus | Victoria |

OxfordCircus | Westminster | LiverpoolStreet | GreenPark | Embankment |

Paddington | WarrenStreet | LondonBridge | LondonBridge | LiverpoolStreet |

ND | HC | NV | NB | PF |

GoodgeStreet | Barbican | Southwark | EdgwareRoad(Bak) | LeicesterSquare |

EdgwareRoad(Bak) | LambethNorth | Angel | Aldgate | Queensway |

CoventGarden | EdgwareRoad(Cir) | LambethNorth | TowerHill | RussellSquare |

ChanceryLane | SloaneSquare | Borough | LambethNorth | LambethNorth |

CannonStreet | Bayswater | TowerHill | Borough | Borough |

Blackfriars | Borough | EdgwareRoad(Bak) | Elephant-Castle | GoodgeStreet |

Bayswater | Pimlico | Elephant-Castle | Pimlico | RegentsPark |

Angel | EarlsCourt | Vauxhall | EarlsCourt | CoventGarden |

Vauxhall | Vauxhall | EarlsCourt | Vauxhall | Bayswater |

AldgateEast | AldgateEast | AldgateEast | AldgateEast | EdgwareRoad(Bak) |

ST | IM | SV | WA | WI |

Queensway | EdgwareRoad(Bak) | GoodgeStreet | HydeParkCorner | Southwark |

AldgateEast | Aldgate | Queensway | RussellSquare | TowerHill |

RussellSquare | TowerHill | AldgateEast | GoodgeStreet | EdgwareRoad(Bak) |

LambethNorth | LambethNorth | RussellSquare | Queensway | Pimlico |

Borough | Borough | CoventGarden | LambethNorth | Elephant-Castle |

GoodgeStreet | Elephant-Castle | RegentsPark | RegentsPark | LambethNorth |

RegentsPark | Pimlico | LambethNorth | CoventGarden | Borough |

CoventGarden | EarlsCourt | Borough | Borough | EarlsCourt |

Bayswater | Vauxhall | Bayswater | Bayswater | Vauxhall |

EdgwareRoad(Bak) | AldgateEast | EdgwareRoad(Bak) | EdgwareRoad(Bak) | AldgateEast |

Model | Connectivity | Path | Flow | Multi | AVG | Max |
---|---|---|---|---|---|---|

${\mathrm{RFP}}^{c}$ | 0% | 1% | 41% | 6% | 12% | 41% |

${\mathrm{RFP}}^{p}$ | 6% | 0% | 52% | 10% | 17% | 52% |

${\mathrm{RFP}}^{f}$ | 22% | 7% | 0% | 9% | 10% | 22% |

${\mathrm{RFP}}^{m}$ | 6% | 2% | 20% | 0% | 7% | 20% |

**Table 4.**Metrics vs. ${\mathrm{RFP}}^{c}$, ${\mathrm{RFP}}^{p}$ and ${\mathrm{RFP}}^{f}$: Relative objective increment.

Connectivity | Path | Flow | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Metric | Metric | Metric | |||||||||||||||

D | q | IM | ND | ST | WI | HC | IM | NB | NV | SV | WA | WI | PF | ST | SV | WA | WI |

1 | 5% | 18% | 18% | 18% | 18% | 1% | 1% | 1% | 1% | 1% | 1% | 1% | 0% | 0% | 0% | 0% | 0% |

10% | 30% | 10% | 10% | 10% | 4% | 4% | 4% | 4% | 3% | 3% | 4% | 0% | 12% | 0% | 0% | 12% | |

15% | 54% | 31% | 31% | 31% | 7% | 7% | 7% | 2% | 2% | 6% | 7% | 0% | 27% | 13% | 0% | 27% | |

20% | 71% | 46% | 20% | 20% | 9% | 9% | 9% | 2% | 3% | 3% | 3% | 0% | 13% | 13% | 13% | 68% | |

25% | 55% | 7% | 28% | 28% | 11% | 11% | 11% | 3% | 3% | 5% | 5% | 0% | 23% | 13% | 9% | 23% | |

30% | 58% | 9% | 31% | 31% | 12% | 12% | 12% | 4% | 4% | 6% | 6% | 0% | 0% | 34% | 18% | 52% | |

AVG | 48% | 20% | 23% | 23% | 7% | 7% | 7% | 3% | 3% | 4% | 4% | 0% | 13% | 12% | 7% | 30% | |

2 | 5% | 27% | 6% | 6% | 6% | 7% | 7% | 7% | 7% | 3% | 3% | 3% | 4% | 4% | 4% | 4% | 4% |

10% | 52% | 27% | 27% | 27% | 1% | 11% | 11% | 1% | 8% | 8% | 8% | 0% | 2% | 0% | 0% | 2% | |

15% | 65% | 8% | 19% | 8% | 6% | 18% | 18% | 4% | 10% | 14% | 6% | 24% | 7% | 3% | 24% | 27% | |

20% | 46% | 20% | 12% | 12% | 9% | 13% | 13% | 1% | 6% | 6% | 6% | 59% | 0% | 0% | 0% | 28% | |

25% | 72% | 26% | 32% | 32% | 12% | 17% | 17% | 4% | 5% | 11% | 11% | 52% | 15% | 15% | 15% | 14% | |

30% | 52% | 36% | 42% | 42% | 17% | 17% | 17% | 8% | 10% | 15% | 15% | 26% | 26% | 26% | 26% | 24% | |

AVG | 52% | 21% | 23% | 21% | 8% | 14% | 14% | 4% | 7% | 9% | 8% | 27% | 9% | 8% | 11% | 17% | |

3 | 5% | 16% | 1% | 1% | 1% | 5% | 5% | 5% | 5% | 2% | 2% | 2% | 5% | 5% | 5% | 5% | 5% |

10% | 32% | 16% | 16% | 16% | 2% | 11% | 11% | 2% | 8% | 8% | 9% | 14% | 14% | 14% | 14% | 14% | |

15% | 48% | 4% | 26% | 4% | 3% | 16% | 16% | 1% | 11% | 13% | 3% | 30% | 5% | 5% | 30% | 5% | |

20% | 36% | 11% | 9% | 9% | 7% | 16% | 16% | 5% | 6% | 6% | 6% | 40% | 0% | 0% | 0% | 9% | |

25% | 42% | 23% | 23% | 26% | 13% | 16% | 16% | 8% | 10% | 10% | 11% | 61% | 20% | 17% | 17% | 16% | |

30% | 46% | 36% | 40% | 43% | 17% | 17% | 17% | 12% | 15% | 15% | 16% | 21% | 21% | 27% | 23% | 27% | |

AVG | 37% | 15% | 19% | 17% | 8% | 13% | 13% | 5% | 9% | 9% | 8% | 29% | 11% | 11% | 15% | 13% | |

AVG | 46% | 19% | 22% | 20% | 8% | 11% | 11% | 4% | 6% | 8% | 7% | 19% | 11% | 11% | 11% | 20% |

Metric | q | ||||||
---|---|---|---|---|---|---|---|

D | 5% | 10% | 15% | 20% | 25% | 30% | |

WI | 1 | 0% | 8% | 21% | 12% | 8% | 9% |

2 | 3% | 15% | 2% | 2% | 11% | 20% | |

3 | 0% | 13% | 3% | 2% | 15% | 29% | |

AVG | 1% | 12% | 8% | 5% | 11% | 19% |

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## Share and Cite

**MDPI and ACS Style**

Esposito Amideo, A.; Starita, S.; Scaparra, M.P.
Assessing Protection Strategies for Urban Rail Transit Systems: A Case-Study on the Central London Underground. *Sustainability* **2019**, *11*, 6322.
https://doi.org/10.3390/su11226322

**AMA Style**

Esposito Amideo A, Starita S, Scaparra MP.
Assessing Protection Strategies for Urban Rail Transit Systems: A Case-Study on the Central London Underground. *Sustainability*. 2019; 11(22):6322.
https://doi.org/10.3390/su11226322

**Chicago/Turabian Style**

Esposito Amideo, Annunziata, Stefano Starita, and Maria Paola Scaparra.
2019. "Assessing Protection Strategies for Urban Rail Transit Systems: A Case-Study on the Central London Underground" *Sustainability* 11, no. 22: 6322.
https://doi.org/10.3390/su11226322