^{*}Correspondence to: Prof. Min Ouyang, School of Artificial Intelligence and Automation, Huazhong University of Science and Technology, 1037 Luoyu Road, Wuhan 430074, Hubei, China. E-mail:

Earthquakes are among the most devastating natural disasters, posing a significant threat to human life and property. With the rapid pace of urbanization, urban risk against earthquakes has increased, making them an increasingly pressing concern for human society. Urban infrastructure systems (UISs), such as electric power, water supply, and gas systems, are essential to the smooth functioning of modern society but are highly vulnerable to ground shaking, resulting in service interruptions to customers and triggering negative impacts on society. This article focuses on the seismic retrofit problem, which intends to enhance the resilience of UISs against seismic hazards. First, a two-stage stochastic programming model is developed for the seismic retrofit problem, where the first stage seeks an optimal seismic retrofit strategy under a limited budget, and the second stage attempts to identify a repair sequence to maximize the system resilience under the given retrofit strategy. Then, this article introduces a heuristic algorithm based on the scenario reduction method and integer L-shaped method to solve the formulated model. Finally, numerical experiments on the Qujing power transmission system are conducted to validate the proposed algorithm. Results show that they can be applied to the resilience-based seismic retrofit problem of large-scale UISs.

The battle against natural hazards and disasters has been a persistent theme throughout human history. Over time, mankind has made remarkable progress in eliminating or alleviating the threat posed by natural hazards and their resulting disasters. Earthquakes, as one of the common natural disasters, present significant risks to urban infrastructure, human life, and property. For instance, the 2008 Wenchuan earthquake collapsed 20 power plants and 170 substations, destroyed 80% of the buildings in Beichuan County, and caused approximately 150 billion dollars in economic losses and 69, 180 known deaths^{[1]}. Urban infrastructure systems (UISs), including electric power, water supply, and gas systems, are quite vulnerable to earthquakes as their physical facilities, such as substations, pipes, and roads, are sensitive to ground shaking^{[2]}. The collapse of these physical facilities not only threatens the lives of people but also affects post-disaster humanitarian relief due to the essential roles of UISs in the functioning of modern society. A successful solution to mitigate the impact of disaster events is to build more resilient UISs^{[3,4]}. Compared to other concepts such as reliability, risk, and safety, resilience emphasizes the comprehensive ability of UISs to resist and absorb negative impacts, recover rapidly from disasters, and adapt to better cope with future events^{[5,6]}.

Numerous researchers committed to the resilience enhancement problem of UISs against earthquakes and put forward a variety of constructive strategies across the four phases of disaster relief, including mitigation, preparedness, response, and recovery phases. The mitigation phase entails identifying risks and hazards to either substantially reduce or eliminate the impact of an incident usually through structural measures. The preparedness phase intends to reduce the system failure probability, and the relevant actions include deploying backup systems^{[7]}, extending system topology^{[8]}, and retrofitting components^{[9-11]}. The response phase attempts to resist the diffusion of failures and takes emergency actions to ensure the normal functioning of critical facilities. For example, operators can adjust the topology of electric power networks by the pre-installed switches on transmission lines to isolate the faulty section^{[12]}. The recovery phase aims to make an effective plan to repair damaged facilities and restore the services of UISs^{[13,14]}. Researchers established many mathematical models to describe the recovery process of UISs, in which various factors are taken into account, such as available resources, routes of repair crews, preferences of stakeholders, decision environments, and interdependencies across UISs^{[15-17]}. This article mainly focuses on the strategy of component retrofitting, which can decrease the failure probabilities of components facing disturbances and has been widely adopted in the literature and practice.

Generally, a UIS consists of thousands of components with different types, indicating that retrofitting each component is impracticable and costly. Hence, only partial components that are essential to system resilience are selected to be retrofitted. Researchers in the literature have proposed numerous methods to explore the critical components of UISs, and a common approach is based on the component importance index, which describes the importance of a component to the whole system^{[18-20]}. The component importance index could be measured in accordance with component types, topological characteristics (e.g., degree and betweenness), and geographical location^{[21-23]}. In the context of electric power systems, plants are the most critical facilities, followed by transmission substations and lines and distribution substations and lines. Also, a component with a large degree value has a high priority to be retrofitted as it connects many components in the system, and its failure might cause a large impact. Moreover, some studies measure the component importance index from the perspective of reliability and vulnerability. Espiritu ^{[24]}. Salman ^{[25]}. Here, risk achievement worth describes the "worth" of a component in achieving system reliability. Li ^{[26]}. Rocco ^{[27]}.

The aforementioned component importance-based methods are straightforward and easily implemented, but they ignore the synergistic effect between components, resulting in a low improvement performance to system resilience. Selecting a limited set of components for reinforcement is a typical combinatorial optimization problem, and researchers have established corresponding optimization models to seek the set of components that can bring the largest benefit to system resilience. Yuan ^{[28]}. Two resilience enhancement strategies, retrofitting components and deploying distributed generation resources, are incorporated into this model. Yan ^{[29]}. Ma ^{[30]}. Lu ^{[31]}. The first stage intends to seek the retrofitting strategy of highway bridges with the minimum retrofitting cost, whereas the second stage minimizes the travel cost given retrofitting decisions and hazard scenarios. Liu ^{[32]}. Numerical experiments were implemented on three electric power systems under seismic scenarios to illustrate the validity and superiority of this method. Several studies integrated the post-disaster restoration decision problem into the pre-disaster retrofit problem. Miller-Hooks ^{[33]}. Gomez and Baker also developed a two-stage stochastic model to address the coupled pre-disaster and post-disaster decision problem in a transportation network under seismic hazards^{[34]}.

The retrofit problem of UISs includes massive uncertainties, such as seismic hazard occurrence probabilities, components failure probabilities, and restoration time, which make those developed optimization models difficult to be exactly solved.Researchers utilized various methods to reduce the computation complexity, including robust programming, Monte Carlo simulation, and importance sampling. Miller-Hooks ^{[33]}. Romero ^{[35]}. Also, several studies put forward methods to generate damage scenarios. Adachi and Ellingwood used seismic hazard maps to determine component failure probabilities and generate component damage scenarios^{[7]}. Gomez and Baker utilized a probabilistic risk assessment of transportation networks to generate hazard-consistent scenarios^{[34]}.

This article studies the seismic retrofit problem of UISs and formulates a two-stage stochastic model. The first stage attempts to seek an optimum seismic retrofit strategy under a limited budget that takes its future benefit into account, whereas the future benefit is quantified by the expected system resilience to all possible seismic scenarios in the second stage. The restoration decision model for each seismic scenario is established in the second stage, and the system resilience describes the cumulative system functionality during the whole restoration process (i.e., from the initial time of a disaster event to the completion time of all repair actions). Then, a resilience-based heuristic method is introduced to solve this mathematical model. This heuristic method first generates a limited set of component damage scenarios, and then the original stochastic model is reformulated into an approximated model, which is solved by the integer L-shaped method. Also, the sample average approximation method is adopted to enhance the solution accuracy. Finally, this heuristic method is applied to the resilience-based seismic retrofit problem of the Qujing power transmission system to demonstrate its validity.

The main contributions of this article include (1) establishing a resilience-based seismic retrofit optimization model for UISs with the incorporation of post-disaster repair actions; (2) proposing an efficient heuristic method to solve the stochastic model; and (3) validating the heuristic method on an electric power transmission system. The proposed retrofit optimization model and efficient heuristic algorithm can be integrated into a decision support system and help the government officials with the seismic investment decision-making. The remainder of this article is organized as follows. Section 2 presents the resilience metric and models the seismic hazard scenarios. Section 3 formulates the resilience-based seismic retrofit problem. Section 4 introduces a heuristic solution method. Taking the Qujng power transmission system as an example, Section 5 presents the numerical results. Section 6 concludes and discusses the findings and provides directions for future research.

This article studies the seismic retrofit problem of UISs to maximize system resilience under a limited budget. A general UIS can be modeled as an undirected network

As the investment budget is limited, only some components can be retrofitted. Compared with un-retrofitted components, retrofitted components have lower failure probabilities of being damaged when facing a seismic event. For simplicity, this article only considers node retrofit, and the candidate of nodes to be retrofitted is denoted by

Note that UISs provide essential services (i.e., electricity, water, and transportation) to customers (i.e., residents, factories, and other UISs); the evaluation of UIS resilience needs to consider the expectations of customers, i.e., whether system service can be restored within expected critical times after a disruptive event. So, it is more reasonable and practical to quantify UIS resilience based on those time points that are critical to customers. As shown in ^{[36]}:

A typical restoration curve and four critical time points of concern.

where

To calculate the system functionality level at each time point, two factors should be known: the state of each component and the operation mechanisms of the UIS of concern. The state of each component depends on the initial component damage scenario ^{[37]}. This article assumes that a component will lose its functionality if it falls into the damage limit state exceeding "extensive" (i.e., extensive and complete). Let binary variables

Each type of UISs, transporting commodities (i.e., electricity, water, and gas) from the supply side (i.e., power and water plants) to the demand side (i.e., factories and residential districts) through lines (i.e., transmission lines and pipes), has a particular operating mechanism. The network flow model has been frequently used to simulate the operating mechanisms of UISs. However, for the electric power transmission system to be studied in the case study, the direct current power flow (DCPF) model is a better alternative and has been frequently used in the field of electrical engineering^{[38]}.

This section proposes a two-stage stochastic optimization model for the resilience-based seismic retrofit to maximize the seismic resilience of UISs under a limited retrofit budget. A graphical representation of the optimization model is shown in

A graphical representation of the two-stage stochastic optimization model.

Denote a binary variable

The objective function (2) is to maximize the expected resilience under all generated component damage scenarios, where

where

Constraint (3) limits the retrofit budget, and Constraint (4) enforces binary retrofit decision variables. Constraints (5)-(10) describe the DCPF model. Constraint (5) sets the phase angle of the reference node as zero. Constraint (6) states that the flow of each edge is determined by its susceptance and the phase angles and operation states of its origin and destination nodes. Constraint (7) ensures flow conservation, and Constraint (8) states the flow capacity of each edge. Constraints (9)-(10) state the maximum output of each source node and the target demand of each demand node at different critical time points. The constraints for the recovery decision variables are described by Constraints (11)-(16). Constraint (11) ensures that if node

This section introduces an efficient heuristic method that takes advantage of several existing methods. The proposed method mainly includes the following three steps: (1) generates limited component damage scenarios to reformulate the original problem as an approximated model; (2) adopts a retrofit efficacy-based method to reduce the solution space and applies the integer L-shaped method to solve the approximated model; (3) employs the sample average approximation method to enhance the solution quality.

For the first step, this article adopts the following procedures to generate the limited component damage scenarios: (1) randomly generates a large number

Define a binary decision variable

The objective function (18) minimizes the sum of gap errors for component failure probability and system resilience. Constraints (19)-(20) define the gap errors with respect to pre-retrofit and post-retrofit failure probabilities of components. Constraints (21)-(22) define the gap errors with respect to pre-retrofit and post-retrofit system resilience. Constraint (23) ensures the number of selected scenarios under seismic scenario

Based on limited component damage scenarios, the original seismic retrofit optimization problem can be reformulated as an approximated model, which is a standard integer program with linear constraints, with the objective function (2) updated as follows:

For the second step, this article adopts a retrofit efficacy-based method to reduce the solution space of the seismic retrofit optimization problem and then solves the updated problem using the integer L-shaped method. In the retrofit efficacy-based method, the original objective function is replaced by some leading terms of its Taylor series expansion. Peeta ^{[9]} and then reformulated the problem as a knapsack problem, with its solution being a local optimum of the original problem. Denote the objective function in Equation (29) by

where

Where ^{[39]} or by the branch-and-bound algorithm.

The solution space reduction is realized through the following procedures: (1) enlarges the retrofit budget to be

Constraint (32) ensures the solution space is limited to the space solved by the retrofit efficacy-based method with

The third step applies the sample average approximation method to enhance the solution quality. Kleywegt ^{[40]}. Hence, the proposed method replicates generating and solving several sample average approximation problems with middle sample sizes (a small number of component damage scenarios) through the above two steps to return several candidate retrofit strategies. Among those strategies, the optimum retrofit strategy is determined by comparing their performance gain using large-scale Monte Carlo simulations.

This article adopts the electric power transmission system in the central area of Qujing, Yunnan province, China, for a case study, which contains 8 gate stations, 35 substations (twenty-six 110 kV substations and nine 35 kV substations), and 56 transmission lines (forty-five 110 kV transmission lines and eleven 35 kV transmission lines), as shown in ^{[41]}.

The electric power transmission system in the central area of Qujing, Yunnan province, China.

Yunnan Province (21-29

Peak ground acceleration (PGA) distribution for rare ground motion in the central area of Qujing.

The comparison of the "true" fragilities and estimated fragilities of components in the electric power transmission system located in the central area of Qujing under the rarely occurred earthquake is shown in

Comparison of the "true" (circle) and the estimated (dot) fragilities of components in the electric power transmission system located in the central area of Qujing under the rarely occurred earthquake.

To demonstrate the solution quality of the PHM, this article adopts a component importance-based simple heuristic method (SHM) for comparison. The SHM identifies a set of critical components to be retrofitted in terms of the retrofit efficacy, which is the resilience difference between the two cases when the component is retrofitted and the component is not retrofitted. The component with larger retrofit efficacy is retrofitted in priority. ^{[42]}. For example, when the retrofit budget and the number of restoration resources

Estimated resilience under varied retrofit budgets and different amounts of restoration resources

SHM | 1 | 0.7123 | 0.7215 | 0.7305 | 0.7382 | 0.7431 |

PHM | 0.7123 | 0.7218 | 0.7308 | 0.7384 | 0.7448 | |

SHM | 2 | 0.7596 | 0.7669 | 0.7749 | 0.7810 | 0.7851 |

PHM | 0.7596 | 0.7676 | 0.7749 | 0.7812 | 0.7864 | |

SHM | 3 | 0.7808 | 0.7872 | 0.7944 | 0.7999 | 0.8036 |

PHM | 0.7808 | 0.7880 | 0.7944 | 0.8000 | 0.8047 |

Computational cost (s) under varied retrofit budgets and different amounts of restoration resources

SHM | 1 | 2, 161.42 | 2, 161.42 | 2, 161.42 | 2, 161.42 | 2, 161.42 |

PHM | 488.58 | 521.11 | 527.25 | 534.82 | 546.72 | |

SHM | 2 | 2, 743.95 | 2, 743.95 | 2, 743.95 | 2, 743.95 | 2, 743.95 |

PHM | 547.35 | 552.66 | 565.11 | 577.05 | 596.27 | |

SHM | 3 | 5, 074.83 | 5, 074.83 | 5, 074.83 | 5, 074.83 | 5, 074.83 |

PHM | 1, 014.95 | 1.015.97 | 1, 092.25 | 1, 122.53 | 1, 142.70 |

This article further analyzes the impact of restoration resources on retrofit strategy.

Retrofit strategies under varied retrofit budgets and different amounts of restoration resources

1, 800 | 3, 5, 6, 26, 32, 36 | 3, 5, 6, 26, 32, 36 | 3, 5, 6, 26, 32, 36 |

2, 400 | 3, 5, 6, 26, 28, 32, 34, 36 | 3, 5, 6, 26, 28, 32, 36, 38 | 3, 5, 6, 26, 28, 32, 34, 36 |

3, 000 | 3, 5, 6, 26, 28, 32, 33, 34, 36, 38 | 3, 5, 6, 7, 26, 28, 32, 34, 36, 38 | 3, 5, 6, 7, 26, 28, 32, 34, 36, 38 |

3, 600 | 3, 5, 6, 7, 26, 28, 30, 32, 33, 34, 36, 38 | 3, 5, 6, 7, 26, 28, 30, 32, 33, 34, 36, 38 | 3, 5, 6, 7, 26, 28, 30, 32, 33, 34, 36, 38 |

4, 200 | 3, 5, 6, 7, 19, 26, 28, 30, 32, 33, 34, 35, 36, 38 | 3, 5, 6, 7, 19, 26, 28, 30, 32, 33, 34, 35, 36, 38 | 3, 5, 6, 7, 19, 21, 26, 28, 30, 32, 33, 34, 36, 38 |

Resilience curve under varied retrofit budgets and different amounts of restoration resources.

In addition, to illustrate the regional differentiation of resilience,

Spatial distribution of resilience for each street district in the central area of Qujing under varied retrofit budgets and different amounts of restoration resources.

This article proposes a resilience-based seismic retrofit optimization model for UISs under a limited retrofit budget and an efficient heuristic algorithm for its solution and also analyzes the impact of post-earthquake restoration resources on pre-earthquake retrofit strategies. Results show that the PHM performs better than the existing SHM. In addition, the amount of post-earthquake restoration resources not only influences the calculated resilience level but also affects the pre-earthquake retrofit strategies when the retrofit budgets are identical. Also, the retrofit budget and the amount of restoration resources influence the spatial distribution of the resilience at the street district levels served by the UIS. The proposed model and the solution algorithm can be used by local and central government agencies to aid investment decisions to upgrade UISs for disaster response.

However, this study still has certain limitations, and there are several areas that can be explored in future research directions. First, apply the proposed method to the seismic retrofit optimization of interdependent infrastructure systems and collect hazard scenario data with a higher resolution. Second, formulate and solve the seismic retrofit optimization problem from a life-cycle perspective with the consideration of different types of hazards. Third, as the disruptions outside the system of concern significantly affect the system performance, the system boundary issue needs to be integrated into the problem formulation and taken into account for its solution.

Conceptualization, methodology, validation, formal analysis, writing - original draft writing - review & editing, resources: Liu C

Software, writing - review & editing: Xu M, Hu S

Writing - review & editing, project administration: Ouyang M

Not applicable.

This work was supported by the National Science Foundation of China (No. 71671074; No. 61572212; No. 51938004).

All authors declared that there are no conflicts of interest.

Not applicable.

Not applicable.

© The Author(s) 2023.

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