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Mathematical Methodology in the Seismic Resilience Evaluation of the Water Supply System

Mathematical Methodology in the Seismic Resilience Evaluation of the Water Supply System This paper has collected more than one hundred papers about resilience at home and abroad, summarised and analysed the meaning and evaluation methods of resilience and then summarised the key issues. Through research on these papers, we found that the current evaluation method for the water supply system seismic resilience is divided into two aspects: on the one hand, it’s based on the seismic performance of water supply system, where the system itself evaluates the resilience absorbs the earthquake resistance level. On the other hand, the assessment is based on the post-earthquake recovery ability of the water supply system, namely the recovery time, recovery degree, recovery path and the resilience level of resources input. Based on past seismic safety of the water supply system and research on post-earthquake recovery ability, the meaning of water supply system in seismic resilience should include two aspects of seismic safety and post-earthquake recovery ability. From using the seismic safety evaluation method also and post-earthquake recovery ability in two angles, this paper gives a comprehensive multiple evaluation index framework of the evaluation model and points out that water system earthquake state resilience evaluative criteria and the requirements of the current seismic code should match. Keywords: water supply system, seismic resilience, seismic safety, post-earthquake recovery ability, mathematical methodology Resilience is a physical concept, defined as the ratio of the energy absorbed by material to the volume before its rupture [1]. It indicates the capacity of material for absorbing energy in the process of plastic deformation and fracture [2]. Seismic resilience reflects the property that a material can absorb a large amount of energy to produce deformation without suffering damage under the action of vibration load [3]. The water supply system is the most important infrastructure in cities, and thus, its resilience evaluation has attracted extensive attention from scholars and engineers. However, as the seismic resilience is a newly emerging research topic derived Corresponding author. Email address: 15546496559@163.com ISSN 2444-8656 doi:10.2478/amns.2021.2.00212 Open Access. © 2022 Qian et al., published by Sciendo. This work is licensed under the Creative Commons Attribution alone 4.0 License. 46 Qian et al. Applied Mathematics and Nonlinear Sciences 8(2023) 45–54 from the theory of resilient cities, its meaning, research content, research methods and techniques have not been clearly defined. To fill this gap, this paper summarises and analyses the meaning and evaluation methods of resilience using more than 100 references on resilience collected from China and other countries. The aim is to clarify the scope and feasible schemes for research on seismic resilience of the water supply system. 1 The meaning of seismic resilience of the water supply system 1.1 Research status of the concept of resilience Timmerma and Primm were the first to associate natural disasters with the concept of resilience. For the design of circulating water supply systems, Todini [4] proposed that the resilience of such water supply systems reflects their ability to resist excessive pressure and failure. By comparing simulated data with real-life data, Rose et al. [5] calculated the direct economic loss, indirect economic loss and the capital generated during the recovery period of lifeline engineering following natural disasters. Moreover, Rose developed the concept of disaster resilience of both the water supply system and the power supply system from the perspective of eco- nomics. Bruneau and Reinhorn [6] first proposed that resilience involves four attributes and four dimensions. These four attributes are robustness, rapidity, resourcefulness and redundancy; the four dimensions are physi- cal dimension, service dimension, social dimension and economic dimension. These attributes and dimensions can be applied to any engineering system. According to the United Nations International Disaster Reduction Strategy [7], resilience is an important attribute of engineering systems when facing the threat of natural disas- ters. Syed and Jayant [8] proposed that improving architectural standards and awareness of disaster prevention can improve the resilience of engineering systems. Crowley and Elliott [9] introduced the concept of seismic resilience and the major resilience affecting factors by analysing the losses to engineering systems caused by earthquakes. Wang [10] and Fujita [11] discussed robustness and resilience, identifying robustness as the struc- tural performance of a system which should meet norms and standards; they showed that resilience has many meanings, which mainly involve reducing the probability of accidents and improving resilience. Penny et al. [12] studied the seismic resilience of a city in Chile, and the results indicated that this design concept should be in- corporated into urban planning and construction. Penny also pointed out that effective urban planning, including a reasonable layout of water supply and power supply systems, can improve seismic resilience. 1.2 Analysis of key issues in the meaning of resilience After nearly 20 years of research on the concept of seismic resilience of water supply systems, the academic community holds that the resilience of water supply systems during earthquakes is reflected in three stages — resistance stage, absorption stage and recovery stage. The concept involves two aspects: (1) the resilience level of the seismic performance of the water supply system. In this aspect the resilience level is reflected in the resistance stage and the absorption stage. The resilience level at the resistance stage manifests as the probability of system failure (e.g., the vulnerability and earthquake damage rate of the water supply system). The resilience level at the absorption stage manifests as the degree of loss (e.g., function loss and economic loss) to the system caused by an earthquake. (2) The resilience level of the recovery capacity of the water supply system after an earthquake. The resilience level in this aspect is reflected at the recovery stage and manifests as the recovery time, recovery degree and recovery path after earthquakes. Based on these aspects, this paper uses the following definition of the safety of the water supply system [13]: Under the action of an earthquake with the relevant fortification intensity level for the area, the earthquake damage level of the water supply system should not exceed the moderate damage level, and a 70% capacity of the water supply service function should be ensured after an earthquake. The recovery capacity of the water supply system after an earthquake is defined of the water supply system to restore the water service function to 70% or above of the normal level within 2-3 days using local resources and manpower. Although these two definitions cannot be directly applied to study the seismic resilience of water supply systems, the underlying Mathematical methodology in seismic resilience evaluation 47 concepts can be used to define the meaning of resilience. Comprehensive considering the two aspects of the concept of resilience shows that the seismic resilience includes seismic safety and post-earthquake recovery capacity. In other words, the seismic resilience of the water supply system of a city manifests not only in the ability of the system to resist and absorb earthquake disasters but also in the ability to quickly recover from earthquakes by only using limited resources. Resilience is a combination of these two abilities. 2 Evaluation criteria for seismic resilience of the water supply system 2.1 Input criteria Albert et al. [14], Callaway et al. [15], Cohen et al. [16] and Paul et al. [17] abstracted the infrastructure network as a simplified model based on the graph theory. Moreover, they defined the key nodes in this network and established a resilience model by applying the water supply pressure on key nodes as criteria for measuring network resilience. Strogatz [18], Newman [19] and Boccaletti et al. [20] applied complex network theory to optimise the network topology of the water supply system. Jeon and O’Rourke [21], Wang et al. [22], Shi [23] and Tabucchi et al. [24] analysed the impact of the Northridge earthquake (with a magnitude of 6.6) on the water supply system of Los Angeles, USA, adopting the node negative pressure treatment method. They used GIRAFFE software to establish the corresponding seismic model for the water supply system and calculated the seismic reliability of important water consumption nodes in the water supply system. Based on the research results obtained by Toprak [25], Adachi [26], Mahmood [27] and Guo [28], Halfaya [29] evaluated the water supply capacity of the water supply system after an earthquake by analysing the size, characteristics and joint form of the pipeline. The research results showed that pipelines should be arranged using the frequency and intensity of earthquakes occurring in the region as criteria. Regarding input criteria, scholars mostly used one or more indexes to evaluate the seismic resilience of the water supply system in a region. However, this cannot be applied to all water supply systems, which prevents comparisons. The seismic resilience evaluation is divided into the two aspects of seismic safety evaluation and post-earthquake recovery ability evaluation. Scholars tried to combine these two research directions to comprehensively evaluate the seismic resilience level of water supply systems. However, without proposing corresponding evaluation criteria, the resilience level of water supply systems cannot be compared (neither hori- zontally nor vertically). For a specific water supply system, how large seismic action is used as a reference input is key for evaluating and classifying the seismic resilience of a water supply system. Existing studies indicated that the corresponding pipelines should be equipped according to the frequency and intensity of earthquakes in the focal area. According to the GB18306-2015 China Earthquake Parameter Zoning Map [30], all areas of China have their own inbuilt seismic fortification intensity. Based on this inherent intensity level, it is assumed that the resilience level of a specific region is realistic and comparable. In this paper, seismic intensity is taken as required input for seismic action, as it comprehensively considers various macroscopic effects of earthquakes and enables qualitative classification. The GB50032-2003 Code for Seismic Design of Outdoor Water Supply and Drainage and Gas Heating Engineering [31] stipulates that outdoor water supply engineering facilities in areas with seismic fortification intensity degree VI or above must be designed for seismic resistance. The seismic fortification intensity is calculated according to this criterion in an area, and the goal of implemented anti-seismic measures is to increase the seismic fortification intensity by one level. Therefore, by taking the seismic effect with the seismic fortification intensity level that should be adopted in an area as reference input, for the seismic resilience evaluation of a certain water supply system, the evaluation results are more reasonable and can be compared both horizontally and vertically. 2.2 Output criteria Wu et al. [32] and Mitrani et al. [33] developed evaluation software for evaluating the recovery ability of water supply systems. Davis [34] conducted extensive research on the seismic effects of the water supply 48 Qian et al. Applied Mathematics and Nonlinear Sciences 8(2023) 45–54 system and systematically analysed the research findings on reliability and applicability of the water supply system obtained by Ballantyne et al. [35], Taylor [36], Shinozuka et al. [37], Markov et al. [38] and Hwang et al. [39]. Davis divided the service functions of the water supply system into v fi e categories for comparing the changes of service functions before and after earthquakes. Li and Shen [40], Zhuang and Gao [41] and Zhou et al. [42] established a prediction model for the probability of earthquake damage to water supply systems based on Monte Carlo simulation and applied it to analyse the overall seismic performances of water supply systems. Liu et al. [43] applied modern combinatorial optimisation algorithms for optimising the seismic topology of water supply systems. Moreover, Liu compared and analysed the advantages and disadvantages of various algorithms through numerical examples, identifying the optimal algorithm a as criterion for evaluating the resilience level of water supply systems. Corresponding evaluation criteria should also be established for each output item of the seismic resilience evaluation of water supply systems. At present, regarding relevant criteria, most studies conducted by scholars outside of China are based on comparisons before and after earthquakes in a specific same area or use water supply pressure as criterion to discuss how the resilience of water supply systems can be improved. However, in China, most studies follow the concept of anti-seismic design of buildings and evaluate the resilience level of their water supply system according to the criterion of “suffering no damage during small earthquakes, being repairable during moderate earthquakes and not collapsing during strong earthquakes”. This criterion can be assumed as a prerequisite or framework for evaluating resilience. However, the boundary among “suffering no damage”, “being repairable” and “not collapsing” should be clearly defined. Only then can classifications be carried out and different regions be compared. Currently, applied standard specifications limit the performance level of different engineering structures in service, which should closely match the performance level required by seismic resilience. Otherwise, an imbalance of meeting the resilience requirements without meeting the re- quirements in the current specifications will emerge; alternatively, the requirements in the current specifications may be met, but the resilience requirements may not be met. For example, in GB/T24336-2009 Seismic Dam- age Level Classification of Lifeline Engineering [44], when the seismic damage level of the water supply pipe network reaches the level of medium damage, the average number of leakage points per 10 km is 2-5, and most of the pipe network functions are maintained. Moderate earthquake implies seismic action at the fortification intensity level; thus, the behaviour criterion for being repairable during moderate earthquakes is that the damage level is lower than medium. 3 Evaluation indexes for seismic resilience of the water supply system 3.1 Current status of research on evaluation indexes As indicated by Todini, the most effective way for designing the pressure network of the water supply system is to apply loop topology. This can ensure sufficient water supply pressure in the system to overcome the impact of leakage points and effectively supply water for users. In contrast, in a tree-like distributed network, failure of the water supply system may exert a great impact on reliability because certain nodes will be out of service or only achieve poor service levels for a certain time. Therefore, adjusting the loop topology by adding pipes and closing valves is an effective way to improve the service level of the water supply system. Todini also proposed an index for measuring the ability of water supply systems to cope with failures. Prasad and Park [45] and Jayaram and Srinivasan [46] proposed improvements for specific shortcomings of the above mentioned index. Tierney and Trainor [47], Cagnan and Davidson [48], Scawthorn et al. [49], O’Rourke [50] and Cimellaro and Reinhorn [51] studied the interruption time, the restoration progress and how the interruption time of the water supply and power supply systems can be shortened during earthquakes. Based on the four attributes of resilience, Cutter et al. [52] established a set of evaluation indexes for water supply and power supply systems in communities and compared the resilience levels between different communities. Susan et al. [53], Jonas et al. [54] and Guo [55] analysed how the post-earthquake recovery time of the water supply system can be Mathematical methodology in seismic resilience evaluation 49 minimised from the three dimensions of maximising the use of groundwater sources, connecting to external reservoirs and rationing water supply. Based on the four dimensions of resilience, Wu et al. [56] proposed an evaluation index system for urban resilience. Pedcris and O’Mashahiko [57] proposed an evaluation index for resilience of urban communities based on the analytic hierarchy process. 3.2 Analysis of key issues with evaluation indexes At the start of the research on the resilience evaluation method of water supply systems, several scholars proposed to measure the resilience level of the water supply system by using the resistance of the water supply system to disturbance and the speed of system recovery to a balanced state as evaluation indexes. However, over the following 20 years, the research focus was separated into two aspects. On the one hand, the resilience level of the seismic safety of water supply system was evaluated based on the seismic performance of a water supply system. For this, vulnerability, earthquake damage rate, functional loss and economic loss were used as evaluation indicators; alternatively, the water supply network was used as research object, using the water pressure and reliability of key water nodes as evaluation indexes. On the other hand, the resilience level of the post-earthquake recovery ability of a water supply system was evaluated based on its recovery ability, using the recovery time, recovery speed, recovery path, recovery degree and input resources as evaluation indexes. It has been pointed out in the previous section that resilience evaluation indicates a comprehensive evaluation of the seismic safety and post-earthquake recovery ability of a water supply system. The next step is to determine the evaluation indexes for the seismic safety index and the post-earthquake recovery ability index. Evaluation indexes used to evaluate seismic safety include earthquake damage rate (R ), vulnerability (R ), functional loss 1 2 (R ), economic loss (R ), water reserve of the reservoir (R ), water quality parameters (R ), reliability of key 3 4 5 6 nodes (R ) and water pressure on key nodes (R ). Evaluation indexes for evaluating post-earthquake recovery 7 8 ability include post-earthquake water supply capacity (R ), water supply interruption time (R ), recovery time 9 10 (R ), recovery speed (R ), recovery path (R ), recovery degree (R ) and input resources (R ). These indexes 11 12 13 14 15 involve two key issues. First, the above mentioned evaluation indexes are not independent of each other. For example, earthquake damage rate (R ) is related to functional loss (R ) and economic loss (R ), and recovery 1 3 4 time (R ) is related to recovery speed (R ). How reasonable and operable evaluation indexes are selected, 11 12 how each evaluation index is introduced into the evaluation model and how the influence weight of each index is considered are key issues to be studied in the future. Second, the introduction of key evaluation indexes is inadequate. For example, input resources (R ) (i.e., post-earthquake recovery ability) are closely related to the input of human and material resources, materials, equipment, organisational management and other factors. However, few studies currently focus on this index. Related evaluation models mostly define this index either by estimation or hypothesis. Regarding research on input resources (R ), the author once proposed a formula for the relationship between the total length of the water supply network and the number of workers on duty, as well as a formula for calculating the recovery time based on the post-earthquake recovery efficiency. How previous studies are applied on the resilience evaluation is also a key issue that needs to be addressed in the future. 4 Evaluation model for seismic resilience of the water supply system 4.1 Current status of research on evaluation model Rossman [58] proposes a resilience index for the water supply system (R), which is defined as the product of three indexes. R = R R R (1) α β γ In Rossman’s formula, R describes the demand for water supply service, which is determined based on the number of users temporarily without water supply; R denotes the water reserve, which is determined according to the depth of the water in the reservoir; R is determined based on water quality parameters. γ 50 Qian et al. Applied Mathematics and Nonlinear Sciences 8(2023) 45–54 These indexes can be used to evaluate the service level of a water supply system. Rossman conducted a case study on a small town in an earthquake-prone zone of Italy, using software EPANET to analyse its water supply system. The analysis involved the relationship between water supply demand and resilience index under the premise of a certain degree of water supply pressure and water quality requirements. Miles and Chang [59] and Chang et al. [60] comprehensively addressed the four attributes of resilience, constructed an evaluation model for disaster losses and quantitatively evaluated the seismic resilience of the water supply system. They suggested that the relevance between lifeline projects is of great significance for resilience evaluation. Therefore, for improving the seismic resilience of lifeline projects, it is necessary to strengthen inter-departmental cooperation and information sharing. Resilience = Robustness + Rapidity + Resourcefulness + Redundancy (2) Wang, Shi, Dominic and Paolo held that resilience is not a system attribute but an emergency attribute. There- fore, it is difficult to make horizontal or vertical comparisons of resilience. They analysed the post-earthquake recovery ability of a water supply system from the perspectives of system reliability and post-earthquake water supply rate and propose an evaluation framework for resilience based on the recovery level and recovery path. Using spatial network topology, Barth quantitatively analysed infrastructure, e.g., roads, power transmission lines and communications facilities and presented the following model: Seismic resilience = recovery ability/(risk factor× vulnerability factor× exposure factor) (3) = resilience/seismic vulnerability Lhomme and Denis divided the resilience of the water supply system into resistance, absorbability and resilience and used the “redundancy rate” of the system network as resilience index for measuring resilience. Li Tongyue established an evaluation model for seismic resilience of the water supply system based on the network topology theory. 4.2 Analysis of key issues in the evaluation model Regarding the seismic resilience evaluation of water supply systems, it is most important to select appropriate evaluation indexes (R , R , R ,... , R ) to define the resilience index ( R) and measure its performance. Among the 1 2 3 n three evaluation models currently proposed, model (1) evaluates the resilience level of seismic safety based on the seismic performance of the water supply system but without an evaluation index that reflects recovery ability; models (2) and (3) are theoretical frameworks for calculating the seismic resilience of water supply systems but without practical operability. In 2013, the resilience of the water supply system was divided into resistance, absorbability and resilience, but its evaluation index is only the “redundancy rate” of the water supply network, i.e., redundant pipes laid out in the water supply network with safety are considered. This type of pipeline ensures that the water supply system can still operate normally when the main pipeline suffers failure. In this paper, it is held that the redundancy rate does not reflect the post-earthquake recovery ability of the water supply system nor that it can fully reflect its resistance and absorbability. This index can only reflect one of the four attributes of the water supply system. Therefore, the key issue with the seismic resilience evaluation method of the water supply system is that there is no evaluation method or model for comprehensively evaluating seismic safety and post-earthquake resilience of the water supply system. Based on previous experience and theoretical orientation, in this paper, resilience indexes are divided into seismic safety index and post-earthquake recovery ability index. The assumption is that resilience is the superposition of the behavioural capabilities of seismic safety and post-earthquake recovery ability. Taking the limit state as example, if the water supply system in a certain area can still operate normally during strong earthquakes, its resilience level is acceptable even if it has poor recovery ability. In contrast, if the water supply system in a certain area has poor seismic safety but can recover rapidly, its resilience level is acceptable. Therefore, the seismic resilience index of the water supply system can be expressed as follows: Resilience = Safety + Recovery (4) Mathematical methodology in seismic resilience evaluation 51 5 Conclusion Through a review of the literature, the meaning and evaluation methods of the seismic resilience of water supply systems have been summarised and analysed. The following four key issues have been identified: 1. The seismic resilience of the water supply system manifests not only as the ability of the system itself to resist and absorb earthquake disasters but also as the ability of the system to quickly recover after earthquakes using limited resources. Therefore, the seismic resilience of the water supply system involves the two aspects of seismic safety and post-earthquake resilience, and therefore, resilience should be a combination of these two abilities. 2. To arrive at more reasonable evaluation results of the seismic resilience of the water supply system and enable both horizontal and vertical comparisons, for the first time, it is proposed that seismic action with seismic fortification intensity level should be adopted as reference input. Based on this, a seismic resilience evaluation model of the water supply system is established, and grading classification is car- ried out. Corresponding evaluation criteria should be established for various indexes regarding seismic resilience evaluation of water supply systems. The existing standards and specifications limit the per- formance levels of different engineering structures currently in service, which should match the required behaviour levels of seismic resilience. 3. The evaluation indexes that can be used to define the seismic safety index and the post-earthquake re- silience index are summarised. Inductive analyses of the evaluation indexes for seismic resilience of the water supply system showed that the evaluation indexes are mutually correlated to a certain extent. In ad- dition, studies on important evaluation indexes are rare. How to select reasonable and operable evaluation indexes, introduce various evaluation indexes to the evaluation model and conduct in-depth research on important evaluation indexes are key topics that need to be explored in the future. 4. At present, the seismic resilience evaluation methods and models for water supply systems are limited by many problems. There is currently no method for comprehensively evaluating seismic safety and post- earthquake resilience. In this paper, resilience indexes comprise seismic safety index and post-earthquake resilience index. References [1] Huang Lang, Wu Chao, Yang Mian, Wang Bing. Application of Resilience Theory in the Field of Safety Science[J]. China Safety Science, 2017, 27(3): 1-6. [2] Zheng Yuxuan. 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International Journal of Disaster Risk Reduction, 2013, 3:62-75. [58] Rossman LA. Epanet users manual[C]//Report EPA/600/R-00/057, National Risk Management Research Laboratory, [59] Miles S B, Chang S E. Modeling community recovery from earthquake[J]. Earthquake Spectra, 2006, 22(2):439-458. [60] Chang S E, McDaniels T, Longstaff H. Fostering disaster resilience through addressing infrastructure inter dependen- cies[J]. Plan Canada, 2006, 46(4):33-36. 54 Qian et al. Applied Mathematics and Nonlinear Sciences 8(2023) 45–54 This page is intentionally left blank http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Nonlinear Sciences de Gruyter

Mathematical Methodology in the Seismic Resilience Evaluation of the Water Supply System

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Abstract

This paper has collected more than one hundred papers about resilience at home and abroad, summarised and analysed the meaning and evaluation methods of resilience and then summarised the key issues. Through research on these papers, we found that the current evaluation method for the water supply system seismic resilience is divided into two aspects: on the one hand, it’s based on the seismic performance of water supply system, where the system itself evaluates the resilience absorbs the earthquake resistance level. On the other hand, the assessment is based on the post-earthquake recovery ability of the water supply system, namely the recovery time, recovery degree, recovery path and the resilience level of resources input. Based on past seismic safety of the water supply system and research on post-earthquake recovery ability, the meaning of water supply system in seismic resilience should include two aspects of seismic safety and post-earthquake recovery ability. From using the seismic safety evaluation method also and post-earthquake recovery ability in two angles, this paper gives a comprehensive multiple evaluation index framework of the evaluation model and points out that water system earthquake state resilience evaluative criteria and the requirements of the current seismic code should match. Keywords: water supply system, seismic resilience, seismic safety, post-earthquake recovery ability, mathematical methodology Resilience is a physical concept, defined as the ratio of the energy absorbed by material to the volume before its rupture [1]. It indicates the capacity of material for absorbing energy in the process of plastic deformation and fracture [2]. Seismic resilience reflects the property that a material can absorb a large amount of energy to produce deformation without suffering damage under the action of vibration load [3]. The water supply system is the most important infrastructure in cities, and thus, its resilience evaluation has attracted extensive attention from scholars and engineers. However, as the seismic resilience is a newly emerging research topic derived Corresponding author. Email address: 15546496559@163.com ISSN 2444-8656 doi:10.2478/amns.2021.2.00212 Open Access. © 2022 Qian et al., published by Sciendo. This work is licensed under the Creative Commons Attribution alone 4.0 License. 46 Qian et al. Applied Mathematics and Nonlinear Sciences 8(2023) 45–54 from the theory of resilient cities, its meaning, research content, research methods and techniques have not been clearly defined. To fill this gap, this paper summarises and analyses the meaning and evaluation methods of resilience using more than 100 references on resilience collected from China and other countries. The aim is to clarify the scope and feasible schemes for research on seismic resilience of the water supply system. 1 The meaning of seismic resilience of the water supply system 1.1 Research status of the concept of resilience Timmerma and Primm were the first to associate natural disasters with the concept of resilience. For the design of circulating water supply systems, Todini [4] proposed that the resilience of such water supply systems reflects their ability to resist excessive pressure and failure. By comparing simulated data with real-life data, Rose et al. [5] calculated the direct economic loss, indirect economic loss and the capital generated during the recovery period of lifeline engineering following natural disasters. Moreover, Rose developed the concept of disaster resilience of both the water supply system and the power supply system from the perspective of eco- nomics. Bruneau and Reinhorn [6] first proposed that resilience involves four attributes and four dimensions. These four attributes are robustness, rapidity, resourcefulness and redundancy; the four dimensions are physi- cal dimension, service dimension, social dimension and economic dimension. These attributes and dimensions can be applied to any engineering system. According to the United Nations International Disaster Reduction Strategy [7], resilience is an important attribute of engineering systems when facing the threat of natural disas- ters. Syed and Jayant [8] proposed that improving architectural standards and awareness of disaster prevention can improve the resilience of engineering systems. Crowley and Elliott [9] introduced the concept of seismic resilience and the major resilience affecting factors by analysing the losses to engineering systems caused by earthquakes. Wang [10] and Fujita [11] discussed robustness and resilience, identifying robustness as the struc- tural performance of a system which should meet norms and standards; they showed that resilience has many meanings, which mainly involve reducing the probability of accidents and improving resilience. Penny et al. [12] studied the seismic resilience of a city in Chile, and the results indicated that this design concept should be in- corporated into urban planning and construction. Penny also pointed out that effective urban planning, including a reasonable layout of water supply and power supply systems, can improve seismic resilience. 1.2 Analysis of key issues in the meaning of resilience After nearly 20 years of research on the concept of seismic resilience of water supply systems, the academic community holds that the resilience of water supply systems during earthquakes is reflected in three stages — resistance stage, absorption stage and recovery stage. The concept involves two aspects: (1) the resilience level of the seismic performance of the water supply system. In this aspect the resilience level is reflected in the resistance stage and the absorption stage. The resilience level at the resistance stage manifests as the probability of system failure (e.g., the vulnerability and earthquake damage rate of the water supply system). The resilience level at the absorption stage manifests as the degree of loss (e.g., function loss and economic loss) to the system caused by an earthquake. (2) The resilience level of the recovery capacity of the water supply system after an earthquake. The resilience level in this aspect is reflected at the recovery stage and manifests as the recovery time, recovery degree and recovery path after earthquakes. Based on these aspects, this paper uses the following definition of the safety of the water supply system [13]: Under the action of an earthquake with the relevant fortification intensity level for the area, the earthquake damage level of the water supply system should not exceed the moderate damage level, and a 70% capacity of the water supply service function should be ensured after an earthquake. The recovery capacity of the water supply system after an earthquake is defined of the water supply system to restore the water service function to 70% or above of the normal level within 2-3 days using local resources and manpower. Although these two definitions cannot be directly applied to study the seismic resilience of water supply systems, the underlying Mathematical methodology in seismic resilience evaluation 47 concepts can be used to define the meaning of resilience. Comprehensive considering the two aspects of the concept of resilience shows that the seismic resilience includes seismic safety and post-earthquake recovery capacity. In other words, the seismic resilience of the water supply system of a city manifests not only in the ability of the system to resist and absorb earthquake disasters but also in the ability to quickly recover from earthquakes by only using limited resources. Resilience is a combination of these two abilities. 2 Evaluation criteria for seismic resilience of the water supply system 2.1 Input criteria Albert et al. [14], Callaway et al. [15], Cohen et al. [16] and Paul et al. [17] abstracted the infrastructure network as a simplified model based on the graph theory. Moreover, they defined the key nodes in this network and established a resilience model by applying the water supply pressure on key nodes as criteria for measuring network resilience. Strogatz [18], Newman [19] and Boccaletti et al. [20] applied complex network theory to optimise the network topology of the water supply system. Jeon and O’Rourke [21], Wang et al. [22], Shi [23] and Tabucchi et al. [24] analysed the impact of the Northridge earthquake (with a magnitude of 6.6) on the water supply system of Los Angeles, USA, adopting the node negative pressure treatment method. They used GIRAFFE software to establish the corresponding seismic model for the water supply system and calculated the seismic reliability of important water consumption nodes in the water supply system. Based on the research results obtained by Toprak [25], Adachi [26], Mahmood [27] and Guo [28], Halfaya [29] evaluated the water supply capacity of the water supply system after an earthquake by analysing the size, characteristics and joint form of the pipeline. The research results showed that pipelines should be arranged using the frequency and intensity of earthquakes occurring in the region as criteria. Regarding input criteria, scholars mostly used one or more indexes to evaluate the seismic resilience of the water supply system in a region. However, this cannot be applied to all water supply systems, which prevents comparisons. The seismic resilience evaluation is divided into the two aspects of seismic safety evaluation and post-earthquake recovery ability evaluation. Scholars tried to combine these two research directions to comprehensively evaluate the seismic resilience level of water supply systems. However, without proposing corresponding evaluation criteria, the resilience level of water supply systems cannot be compared (neither hori- zontally nor vertically). For a specific water supply system, how large seismic action is used as a reference input is key for evaluating and classifying the seismic resilience of a water supply system. Existing studies indicated that the corresponding pipelines should be equipped according to the frequency and intensity of earthquakes in the focal area. According to the GB18306-2015 China Earthquake Parameter Zoning Map [30], all areas of China have their own inbuilt seismic fortification intensity. Based on this inherent intensity level, it is assumed that the resilience level of a specific region is realistic and comparable. In this paper, seismic intensity is taken as required input for seismic action, as it comprehensively considers various macroscopic effects of earthquakes and enables qualitative classification. The GB50032-2003 Code for Seismic Design of Outdoor Water Supply and Drainage and Gas Heating Engineering [31] stipulates that outdoor water supply engineering facilities in areas with seismic fortification intensity degree VI or above must be designed for seismic resistance. The seismic fortification intensity is calculated according to this criterion in an area, and the goal of implemented anti-seismic measures is to increase the seismic fortification intensity by one level. Therefore, by taking the seismic effect with the seismic fortification intensity level that should be adopted in an area as reference input, for the seismic resilience evaluation of a certain water supply system, the evaluation results are more reasonable and can be compared both horizontally and vertically. 2.2 Output criteria Wu et al. [32] and Mitrani et al. [33] developed evaluation software for evaluating the recovery ability of water supply systems. Davis [34] conducted extensive research on the seismic effects of the water supply 48 Qian et al. Applied Mathematics and Nonlinear Sciences 8(2023) 45–54 system and systematically analysed the research findings on reliability and applicability of the water supply system obtained by Ballantyne et al. [35], Taylor [36], Shinozuka et al. [37], Markov et al. [38] and Hwang et al. [39]. Davis divided the service functions of the water supply system into v fi e categories for comparing the changes of service functions before and after earthquakes. Li and Shen [40], Zhuang and Gao [41] and Zhou et al. [42] established a prediction model for the probability of earthquake damage to water supply systems based on Monte Carlo simulation and applied it to analyse the overall seismic performances of water supply systems. Liu et al. [43] applied modern combinatorial optimisation algorithms for optimising the seismic topology of water supply systems. Moreover, Liu compared and analysed the advantages and disadvantages of various algorithms through numerical examples, identifying the optimal algorithm a as criterion for evaluating the resilience level of water supply systems. Corresponding evaluation criteria should also be established for each output item of the seismic resilience evaluation of water supply systems. At present, regarding relevant criteria, most studies conducted by scholars outside of China are based on comparisons before and after earthquakes in a specific same area or use water supply pressure as criterion to discuss how the resilience of water supply systems can be improved. However, in China, most studies follow the concept of anti-seismic design of buildings and evaluate the resilience level of their water supply system according to the criterion of “suffering no damage during small earthquakes, being repairable during moderate earthquakes and not collapsing during strong earthquakes”. This criterion can be assumed as a prerequisite or framework for evaluating resilience. However, the boundary among “suffering no damage”, “being repairable” and “not collapsing” should be clearly defined. Only then can classifications be carried out and different regions be compared. Currently, applied standard specifications limit the performance level of different engineering structures in service, which should closely match the performance level required by seismic resilience. Otherwise, an imbalance of meeting the resilience requirements without meeting the re- quirements in the current specifications will emerge; alternatively, the requirements in the current specifications may be met, but the resilience requirements may not be met. For example, in GB/T24336-2009 Seismic Dam- age Level Classification of Lifeline Engineering [44], when the seismic damage level of the water supply pipe network reaches the level of medium damage, the average number of leakage points per 10 km is 2-5, and most of the pipe network functions are maintained. Moderate earthquake implies seismic action at the fortification intensity level; thus, the behaviour criterion for being repairable during moderate earthquakes is that the damage level is lower than medium. 3 Evaluation indexes for seismic resilience of the water supply system 3.1 Current status of research on evaluation indexes As indicated by Todini, the most effective way for designing the pressure network of the water supply system is to apply loop topology. This can ensure sufficient water supply pressure in the system to overcome the impact of leakage points and effectively supply water for users. In contrast, in a tree-like distributed network, failure of the water supply system may exert a great impact on reliability because certain nodes will be out of service or only achieve poor service levels for a certain time. Therefore, adjusting the loop topology by adding pipes and closing valves is an effective way to improve the service level of the water supply system. Todini also proposed an index for measuring the ability of water supply systems to cope with failures. Prasad and Park [45] and Jayaram and Srinivasan [46] proposed improvements for specific shortcomings of the above mentioned index. Tierney and Trainor [47], Cagnan and Davidson [48], Scawthorn et al. [49], O’Rourke [50] and Cimellaro and Reinhorn [51] studied the interruption time, the restoration progress and how the interruption time of the water supply and power supply systems can be shortened during earthquakes. Based on the four attributes of resilience, Cutter et al. [52] established a set of evaluation indexes for water supply and power supply systems in communities and compared the resilience levels between different communities. Susan et al. [53], Jonas et al. [54] and Guo [55] analysed how the post-earthquake recovery time of the water supply system can be Mathematical methodology in seismic resilience evaluation 49 minimised from the three dimensions of maximising the use of groundwater sources, connecting to external reservoirs and rationing water supply. Based on the four dimensions of resilience, Wu et al. [56] proposed an evaluation index system for urban resilience. Pedcris and O’Mashahiko [57] proposed an evaluation index for resilience of urban communities based on the analytic hierarchy process. 3.2 Analysis of key issues with evaluation indexes At the start of the research on the resilience evaluation method of water supply systems, several scholars proposed to measure the resilience level of the water supply system by using the resistance of the water supply system to disturbance and the speed of system recovery to a balanced state as evaluation indexes. However, over the following 20 years, the research focus was separated into two aspects. On the one hand, the resilience level of the seismic safety of water supply system was evaluated based on the seismic performance of a water supply system. For this, vulnerability, earthquake damage rate, functional loss and economic loss were used as evaluation indicators; alternatively, the water supply network was used as research object, using the water pressure and reliability of key water nodes as evaluation indexes. On the other hand, the resilience level of the post-earthquake recovery ability of a water supply system was evaluated based on its recovery ability, using the recovery time, recovery speed, recovery path, recovery degree and input resources as evaluation indexes. It has been pointed out in the previous section that resilience evaluation indicates a comprehensive evaluation of the seismic safety and post-earthquake recovery ability of a water supply system. The next step is to determine the evaluation indexes for the seismic safety index and the post-earthquake recovery ability index. Evaluation indexes used to evaluate seismic safety include earthquake damage rate (R ), vulnerability (R ), functional loss 1 2 (R ), economic loss (R ), water reserve of the reservoir (R ), water quality parameters (R ), reliability of key 3 4 5 6 nodes (R ) and water pressure on key nodes (R ). Evaluation indexes for evaluating post-earthquake recovery 7 8 ability include post-earthquake water supply capacity (R ), water supply interruption time (R ), recovery time 9 10 (R ), recovery speed (R ), recovery path (R ), recovery degree (R ) and input resources (R ). These indexes 11 12 13 14 15 involve two key issues. First, the above mentioned evaluation indexes are not independent of each other. For example, earthquake damage rate (R ) is related to functional loss (R ) and economic loss (R ), and recovery 1 3 4 time (R ) is related to recovery speed (R ). How reasonable and operable evaluation indexes are selected, 11 12 how each evaluation index is introduced into the evaluation model and how the influence weight of each index is considered are key issues to be studied in the future. Second, the introduction of key evaluation indexes is inadequate. For example, input resources (R ) (i.e., post-earthquake recovery ability) are closely related to the input of human and material resources, materials, equipment, organisational management and other factors. However, few studies currently focus on this index. Related evaluation models mostly define this index either by estimation or hypothesis. Regarding research on input resources (R ), the author once proposed a formula for the relationship between the total length of the water supply network and the number of workers on duty, as well as a formula for calculating the recovery time based on the post-earthquake recovery efficiency. How previous studies are applied on the resilience evaluation is also a key issue that needs to be addressed in the future. 4 Evaluation model for seismic resilience of the water supply system 4.1 Current status of research on evaluation model Rossman [58] proposes a resilience index for the water supply system (R), which is defined as the product of three indexes. R = R R R (1) α β γ In Rossman’s formula, R describes the demand for water supply service, which is determined based on the number of users temporarily without water supply; R denotes the water reserve, which is determined according to the depth of the water in the reservoir; R is determined based on water quality parameters. γ 50 Qian et al. Applied Mathematics and Nonlinear Sciences 8(2023) 45–54 These indexes can be used to evaluate the service level of a water supply system. Rossman conducted a case study on a small town in an earthquake-prone zone of Italy, using software EPANET to analyse its water supply system. The analysis involved the relationship between water supply demand and resilience index under the premise of a certain degree of water supply pressure and water quality requirements. Miles and Chang [59] and Chang et al. [60] comprehensively addressed the four attributes of resilience, constructed an evaluation model for disaster losses and quantitatively evaluated the seismic resilience of the water supply system. They suggested that the relevance between lifeline projects is of great significance for resilience evaluation. Therefore, for improving the seismic resilience of lifeline projects, it is necessary to strengthen inter-departmental cooperation and information sharing. Resilience = Robustness + Rapidity + Resourcefulness + Redundancy (2) Wang, Shi, Dominic and Paolo held that resilience is not a system attribute but an emergency attribute. There- fore, it is difficult to make horizontal or vertical comparisons of resilience. They analysed the post-earthquake recovery ability of a water supply system from the perspectives of system reliability and post-earthquake water supply rate and propose an evaluation framework for resilience based on the recovery level and recovery path. Using spatial network topology, Barth quantitatively analysed infrastructure, e.g., roads, power transmission lines and communications facilities and presented the following model: Seismic resilience = recovery ability/(risk factor× vulnerability factor× exposure factor) (3) = resilience/seismic vulnerability Lhomme and Denis divided the resilience of the water supply system into resistance, absorbability and resilience and used the “redundancy rate” of the system network as resilience index for measuring resilience. Li Tongyue established an evaluation model for seismic resilience of the water supply system based on the network topology theory. 4.2 Analysis of key issues in the evaluation model Regarding the seismic resilience evaluation of water supply systems, it is most important to select appropriate evaluation indexes (R , R , R ,... , R ) to define the resilience index ( R) and measure its performance. Among the 1 2 3 n three evaluation models currently proposed, model (1) evaluates the resilience level of seismic safety based on the seismic performance of the water supply system but without an evaluation index that reflects recovery ability; models (2) and (3) are theoretical frameworks for calculating the seismic resilience of water supply systems but without practical operability. In 2013, the resilience of the water supply system was divided into resistance, absorbability and resilience, but its evaluation index is only the “redundancy rate” of the water supply network, i.e., redundant pipes laid out in the water supply network with safety are considered. This type of pipeline ensures that the water supply system can still operate normally when the main pipeline suffers failure. In this paper, it is held that the redundancy rate does not reflect the post-earthquake recovery ability of the water supply system nor that it can fully reflect its resistance and absorbability. This index can only reflect one of the four attributes of the water supply system. Therefore, the key issue with the seismic resilience evaluation method of the water supply system is that there is no evaluation method or model for comprehensively evaluating seismic safety and post-earthquake resilience of the water supply system. Based on previous experience and theoretical orientation, in this paper, resilience indexes are divided into seismic safety index and post-earthquake recovery ability index. The assumption is that resilience is the superposition of the behavioural capabilities of seismic safety and post-earthquake recovery ability. Taking the limit state as example, if the water supply system in a certain area can still operate normally during strong earthquakes, its resilience level is acceptable even if it has poor recovery ability. In contrast, if the water supply system in a certain area has poor seismic safety but can recover rapidly, its resilience level is acceptable. Therefore, the seismic resilience index of the water supply system can be expressed as follows: Resilience = Safety + Recovery (4) Mathematical methodology in seismic resilience evaluation 51 5 Conclusion Through a review of the literature, the meaning and evaluation methods of the seismic resilience of water supply systems have been summarised and analysed. The following four key issues have been identified: 1. The seismic resilience of the water supply system manifests not only as the ability of the system itself to resist and absorb earthquake disasters but also as the ability of the system to quickly recover after earthquakes using limited resources. Therefore, the seismic resilience of the water supply system involves the two aspects of seismic safety and post-earthquake resilience, and therefore, resilience should be a combination of these two abilities. 2. To arrive at more reasonable evaluation results of the seismic resilience of the water supply system and enable both horizontal and vertical comparisons, for the first time, it is proposed that seismic action with seismic fortification intensity level should be adopted as reference input. Based on this, a seismic resilience evaluation model of the water supply system is established, and grading classification is car- ried out. Corresponding evaluation criteria should be established for various indexes regarding seismic resilience evaluation of water supply systems. The existing standards and specifications limit the per- formance levels of different engineering structures currently in service, which should match the required behaviour levels of seismic resilience. 3. The evaluation indexes that can be used to define the seismic safety index and the post-earthquake re- silience index are summarised. Inductive analyses of the evaluation indexes for seismic resilience of the water supply system showed that the evaluation indexes are mutually correlated to a certain extent. In ad- dition, studies on important evaluation indexes are rare. How to select reasonable and operable evaluation indexes, introduce various evaluation indexes to the evaluation model and conduct in-depth research on important evaluation indexes are key topics that need to be explored in the future. 4. At present, the seismic resilience evaluation methods and models for water supply systems are limited by many problems. There is currently no method for comprehensively evaluating seismic safety and post- earthquake resilience. In this paper, resilience indexes comprise seismic safety index and post-earthquake resilience index. References [1] Huang Lang, Wu Chao, Yang Mian, Wang Bing. Application of Resilience Theory in the Field of Safety Science[J]. China Safety Science, 2017, 27(3): 1-6. [2] Zheng Yuxuan. 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Journal

Applied Mathematics and Nonlinear Sciences – de Gruyter

Published: Jan 1, 2023

Keywords: water supply system; seismic resilience; seismic safety; post-earthquake recovery ability; mathematical methodology

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