**1. Introduction**

An earthquake occurs as a result of a sudden release of energy in the earth's crust. The released massive energy creates seismic waves, which cause deformation of the water distribution system (WDS) of which components are mainly connected beneath the ground. The damage of an earthquake is the multiple failure of system components. For example, for an earthquake occurred in Kobe, Japan in 1995, many pipes were ruptured, causing water to flow out of the system, while some of the pump stations stopped working completely due to power outage. However, the likelihood of concurrence of multiple failures within a system is low under normal conditions. Therefore, a different strategy should be adopted for WDS design of the regions with the risk of earthquake occurrence, such as Japan, Korean peninsula, and West Coast cities of the United States.

During the last two decades, many optimal design approaches have been proposed for WDSs. The early works have mainly taken into account the economic cost as a single objective [1–5]. Thereafter, some studies have included the system's performance index within the optimization framework. Lansey *et al.* [6] developed a chance-constrained least-cost model in which the capacity reliability is

defined as the probability that stochastic pressures are equal to or higher than pressure requirement and are constrained at a certain level while minimizing cost. The nodal demands, pressure requirements, and pipe roughness coefficients were assumed to be uncertain while optimizing the pipe size. Several later studies also used chance-constrained optimization models for WDS design [7,8].

Kapelan *et al.* [9] was one of the earliest studies that proposed the multi-objective optimal design of WDS. Since then, most design approaches have adopted two objective functions to minimize economic cost and to maximize system reliability [10,11]. The various system reliability indices were suggested to reflect the uncertainties of pipe roughness and system demands when sizing the system. Component failures have been rarely considered in this stream of optimal WDS design.

Su *et al.* [12] is one of the few studies that considered component failure in WDS design. In their study, reliability was defined as the probability of water demand provision under pipe break (failure) conditions and is used as a constraint for the least-cost design of a WDS. The minimum cut-set (*i.e.*, the most critical set of pipes) was identified by closing the pipe individually for calculating the system reliability. Note that the pipe failure conditions were not considered in hydraulic simulation within the optimization framework, since the computational intensity is overwhelming.

The seismic hazard assessment models have been developed recently. HAZUS [13] was the first model to assess the economic losses of an infrastructure by earthquake. HAZUS is mainly intended to assess the seismic damage but not detailed simulation of the system behavior. Later, the Mid-America Earthquake Center developed a seismic impact assessment model and investigated interdependencies between water and power systems [14].

Early earthquake studies in the water domain focused on investigating individual components' physical behavior under earthquakes rather than quantifying the system-wide performance by modeling the WDS and earthquakes [15–17]. In a recent study, Fragiadakis *et al.* [18] proposed a seismic reliability assessment model of a WDS using survival curves of pipes based on general seismic assessment standards and American Lifelines Alliance (ALA) [19] guidelines. However, detailed hydraulic simulations were not conducted in this paper. Later studies began proposing methodologies to evaluate seismic reliability with hydraulic simulations using well-known hydraulic solvers and seismic simulations [20–29].

The latest and most popular assessment model is the graphical iterative response analysis for flow following earthquakes (GIRAFFE [30]) developed by a research team at Cornell University. GIRAFFE can simulate various pipe leakage and breakage conditions by using EPANET [31]. The model's graphical user interface helps to visualize the model results, which is compatible with other geographic information system tools. After the initial development in 2008, the model has been improved and validated through many case studies [27,32–36]. However, GIRAFFE uses a controversial approach to treat negative pressure that removes the nodes' negative pressure and connected pipes. This process is repeated until the negative-pressure nodes are no longer produced. This approach can be time-consuming because the system file must be revised iteratively.

Yoo *et al.* [37] developed seismic reliability assessment model under stochastic earthquake events. The model quantifies the seismic reliability of a WDS through a series of procedures: stochastic earthquake generation, seismic intensity attenuation, determination of the pipe failure status (normal, leakage, and breakage), pipe failure modeling in hydraulic simulation, and negative pressure treatment.

To the authors' best knowledge, this study is the first attempt to develop an economic, cost-constrained optimal WDS design approach that takes into account seismic reliability based on detailed hydraulic simulations. The proposed model maximizes the system's seismic reliability while satisfying the constraints on economic cost and node pressure requirements. The physical impacts of a seismic wave to WDS components are simulated to determine the failure conditions. The seismic reliability is defined as the ratio of the supplied water to the required demand under stochastic earthquake events. A well-known benchmark network, the Anytown network, is used for the applications of the optimal design and layout maximizing the seismic reliability.
