# Resilience-Based Recovery Assessments of Networked Infrastructure Systems under Localized Attacks

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Recovery Strategies against Localized Attacks

#### 2.1. Localized Attacks

#### 2.2. Recovery Strategies

- Periphery recovery (PR). Recovery priorities are given to the most populated isolated node at the boundary [4]. In Figure 2(C1), the blue edges with arrowheads are the damaged edges adjacent to the functional components of the network. The red node n1 is the most populated boundary node of the functional network. According to this recovery rule, either edges m1 or m2 would be repaired first randomly. In this case, m1 is selected to be restored first and colored green. After all the isolated nodes are connected, m2 is repaired and colored yellow. At the next step, the node n2, in Figure 2(C2), is the most populated boundary node of the functional network, and either edge m3 or edge m4 is supposed to be repaired randomly. The process would be iterated until all the isolated nodes were connected to the functional network, as shown in Figure 2(C3). At last, the yellow edges are repaired randomly one by one until all are repaired.
- Preferential recovery based on nodal weight (PRNW). In this method, the repair preference is given to the links that could connect the most populated isolated nodes to the functional component of the network [4]. In Figure 2(D1), the red node n3 has the largest population among all the isolated nodes, and edge m5 connects edge n3 to the network. According to the PRNW algorithm, edge m5 is repaired first and colored green. Following the same procedure, the most populated node—n5—is connected to the functional network through the edges m6 and m7 in Figure 2(D2) and m8 in Figure 2(D3). The steps are iterated until all the isolated nodes are connected to the network, as shown in Figure 2(D4). At last, the yellow edges are repaired randomly one by one until all edges are repaired. PRNW shows high efficiency in connecting the most populated area reducing the recovery time. It can also provide a rational solution while limited resources are available.
- Localized recovery (LR). A localized recovery is where the priority of being recovered is given to the edges of a root node as well as its neighboring nodes, respectively [35]. This recovery process begins with the selection of root nodes. The rest of the nodes are listed in order of their distance from the root node as shown in Figure 2(E1). Nodes being in the same distance from the root node are placed in the same shell. The edges of the root node are recovered first with the edges connected to it. Then the nodes in the same shell h are randomly selected and their edges are further recovered. After all the nodes in the first shell h = 1 are recovered, recovery in the next shell h + 1 starts. The recovery process stops when all the edges are recovered, as shown in Figure 2(E2).

#### 2.3. Resilience-Based Recovery Assessments Framework

## 3. Infrastructure Resilience

#### 3.1. Resilience Metric

_{0}≤ t ≤ t

_{i}), (2) a damage propagation stage (t

_{i}≤ t ≤ t

_{i}), (3) an assessment and recovery stage (t

_{d}≤ t ≤ t

_{r}), and (4) a stable state after the recovery process is fully completed (t

_{r}≤ t ≤ T), as shown in Figure 5 [9]. From this framework, resilience value can be quantified according to the targeted performance curve P

_{T}(t) and the real performance curve P

_{R}(t) as

#### 3.2. Resilience Optimization

_{R}) and targeted performance curve (A

_{T}), respectively. These two terms are used in quantifying the resilience metric in the objective function Equation (2). Equation (7) defines the total recovery time required when applying recovery strategy r. Equation (8) indicates that the total time cannot exceed the maximum allowable given time, T. Equation (9) is the constraint that measures the cost of recovery at time step t while implementing strategy r. The total cost of recovery includes a time-dependent fixed cost, C

_{f}(for example, labor cost or instrumental cost), and the cost of repairing edges, C

_{e}, which depends on the edge weight (for example, cost to repair each unit of pipe). Equation (10) is the total cost of recovery and Equation (11) is the budget constraint. Finally, Equation (12) is the binary decision variable constraint.

## 4. Water Supply Network Case Study

#### 4.1. Case Study Description

#### 4.2. Multiobjective Optimization

- Objective Function 1: Maximize resilience. The aim of the first objective function is to maximize the overall system resilience by applying the recovery strategy. Here, R was used as the resilience metric. Both maximum flow and the shortest path were considered as the performance measure. For all three recovery strategies, system resilience was quantified, and a recovery strategy was selected based on the highest resilience level.
- Objective Function 2: Minimize cost. The overall recovery cost is minimized by this objective function. For this purpose, the recovery cost at each time step was calculated based on the weight of the edges that need to be recovered. An amount of $200 was assumed to be the fixed cost for each time step of the recovery process. Additionally, the cost of $100 for repairing each unit of edges was added to find the total cost. The strategy with the lowest cost was selected.
- Objective Function 3: Minimize time or number of iterations. Through the third objective function the fastest recovery process was selected (the smallest number of iterations to full recovery).
- Integrated objective: To solve the multiobjective formulation, an integrated objective function, combining Objectives 1–3, is shown in Equation (13).$$Minimize-R+C+{t}_{t}$$

#### 4.3. Results and Discussion

_{R}and A

_{T}refer to the area under the real performance curve and the area under the targeted performance curve respectively. With maximum flow, the A

_{R}values are 1413, 1392, and 1473 for PRNW, PR, and LR, respectively, and the A

_{T}value is 1500. It is shown that A

_{R}is always lesser than A

_{T}resulting in resilience values of 0.94, 0.93, and 0.98 for PRNW, PR, and LR, respectively. This indicates that LR shows the highest resilience as the maximum flow follows ‘the larger the better’ concept. On the other hand, with shortest path A

_{R}values are 4312, 4238, and 4104 for PRNW, PR, and LR, respectively, and the A

_{T}value is 3840. A

_{R}is always greater than A

_{T}, resulting in resilience values of 1.12, 1.1, and 1.07. As the shortest path follows “the smaller, the better” concept, LR is the most resilient strategy in this case as well.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**A typical aftermath of localized attacks [3].

**Figure 5.**Performance process of an infrastructure system during disruptive events [15].

**Figure 8.**Objective trade-offs between (

**a**) total recovery cost and resilience, (

**b**) total recovery cost and recovery time steps, and (

**c**) resilience and recovery time steps.

Sets | Parameters | Decision Variables | |||
---|---|---|---|---|---|

r | Set of recovery strategies | R | Resilience | γ_{r} | Binary variable (1 if strategy r is selected, 0 otherwise) |

P_{T} | Targeted performance | ||||

P_{R} | Real performance | ||||

A_{T} | The area under the targeted performance curve | ||||

A_{R} | The area under the real performance curve | ||||

t | Set of time steps | t_{d} | Time after the disaster stop propagating or the start of the recovery strategy | ||

t_{r} | Time at which the recovery is completed | ||||

t_{t} | Total recovery time required | ||||

T | Maximum allowable time | ||||

C_{tr} | Cost of recovery at time t for strategy r | ||||

e | Set of edges to be repaired | C_{f} | Fixed cost to repair at each time step | ||

C_{e} | Cost for repairing edge e | ||||

W_{et} | Total edge weight recovered at time t | ||||

C | Total cost for strategy r | ||||

B | Budget |

Time | Critical Performance | Description | |
---|---|---|---|

Max flow | Shortest Path | ||

0 | 75 | 192 | Original state |

1 * | 75 | 206 | 1 node was isolated |

2 | 75 | 206 | 2 nodes were isolated |

3 | 75 | 206 | 3 nodes were isolated |

4 | 75 | 229 | 4 nodes were isolated |

5 | 75 | 229 | 5 nodes were isolated |

6 | 75 | 229 | 6 nodes were isolated |

7 | 75 | 229 | 7 nodes were isolated |

8 ** | 48 | 229 | 8 nodes were isolated |

**Table 3.**Changes of max flow and shortest path distance during recovery (shaded area indicates that the recovered state was reached).

Time | PRNW ** | PR ** | LR *** | |||
---|---|---|---|---|---|---|

Max Flow | Shortest Path | Max Flow | Shortest Path | Max Flow | Shortest Path | |

9 * | 48 | 229 | 48 | 229 | 75 | 229 |

10 | 48 | 229 | 48 | 229 | 75 | 192 |

11 | 73 | 229 | 48 | 229 | 75 | 192 |

12 | 73 | 229 | 75 | 229 | 75 | 192 |

13 | 73 | 229 | 75 | 215 | 75 | 192 |

14 | 75 | 229 | 75 | 192 | 75 | 192 |

15 | 75 | 215 | 75 | 192 | 75 | 192 |

16 | 75 | 192 | 75 | 192 | 75 | 192 |

17 | 75 | 192 | 75 | 192 | 75 | 192 |

18 | 75 | 192 | 75 | 192 | 75 | 192 |

19 | 75 | 192 | 75 | 192 | 75 | 192 |

20 | 75 | 192 | 75 | 192 | 75 | 192 |

PRNW | PR | LR | ||||
---|---|---|---|---|---|---|

Max Flow | Shortest Path | Max Flow | Shortest Path | Max Flow | Shortest Path | |

A_{R} | 1413 | 4312 | 1392 | 4238 | 1473 | 4104 |

A_{T} | 1500 | 3840 | 1500 | 3840 | 1500 | 3840 |

R | 0.94 | 1.12 | 0.93 | 1.1 | 0.98 | 1.07 |

Time Steps | Cost ($) | ||
---|---|---|---|

PRNW | PR | LR | |

1–8 | Damage Propagation Stage | ||

9 | 7300 | 7300 | 42,300 |

10 | 4700 | 7300 | 24,400 |

11 | 13,100 | 7700 | 3400 |

12 | 3300 | 8700 | 0 |

13 | 6300 | 9800 | 0 |

14 | 12,600 | 4700 | 0 |

15 | 15,100 | 10,000 | 0 |

16 | 8,700 | 15,600 | 0 |

Total | 71,100 | 71,100 | 70,100 |

Time Step | PRNW | PR | LR | |||
---|---|---|---|---|---|---|

Recovered Edges * | Sum of Weights | Recovered Edges * | Sum of Weights | Recovered Edges * | Sum of Weights | |

9 | 23, 15 | 71 | 23, 15 | 71 | 28, 36, 38, 39, 25, 27, 26, 30, 37, 47, 41, 49 | 421 |

10 | 26, 36 | 45 | 40, 41 | 71 | 14, 16, 15, 17, 19, 23, 29, 34, 40, 50 | 242 |

11 | 38, 40, 41 | 129 | 26, 34, 37 | 75 | 6 | 32 |

12 | 34, 37 | 31 | 47, 49, 50 | 85 | ||

13 | 25, 28 | 61 | 6, 14, 16 | 96 | ||

14 | 39, 47, 49, 50 | 124 | 19, 29, 30 | 45 | ||

15 | 6, 14, 16, 17 | 149 | 36, 38, 39 | 98 | ||

16 | 19, 27, 29, 30 | 85 | 17, 25, 27, 28 | 154 | ||

17–20 | Stable State |

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**MDPI and ACS Style**

Afrin, T.; Yodo, N. Resilience-Based Recovery Assessments of Networked Infrastructure Systems under Localized Attacks. *Infrastructures* **2019**, *4*, 11.
https://doi.org/10.3390/infrastructures4010011

**AMA Style**

Afrin T, Yodo N. Resilience-Based Recovery Assessments of Networked Infrastructure Systems under Localized Attacks. *Infrastructures*. 2019; 4(1):11.
https://doi.org/10.3390/infrastructures4010011

**Chicago/Turabian Style**

Afrin, Tanzina, and Nita Yodo. 2019. "Resilience-Based Recovery Assessments of Networked Infrastructure Systems under Localized Attacks" *Infrastructures* 4, no. 1: 11.
https://doi.org/10.3390/infrastructures4010011