**1. Introduction**

The expected performance of an infrastructure system can be interpreted through the concept of "system reliability", which quantifies marginal capacity to fulfil the users' requirements. In a water distribution network (WDN), the system reliability indicates the stable performance of supplying required water with adequate service pressure. Here, the specific performance of WDN could be assessed by representative hydraulic measures.

Wildavsky [1] defined "resilience", one of the most important performance parameters of WDNs, as the capacity to cope with unanticipated dangers after they have become manifest and learning to bounce back. Subsequently, Comfort [2] defined resilience as the capacity to adapt existing resources and skills to new situations and operating conditions. For theoretical concepts of reliability, Maier et al. [3] suggested first-order estimators such as reliability, vulnerability, and resilience of water quality service in rivers, and Bruneau et al. [4] summarized the seismic resilience of an infrastructure system into 4 R's: robustness, redundancy, resourcefulness, and rapidity. For reliability assessment of WDNs, several studies [5–7] compared the performance of different WDNs using simple types of representative hydraulic measure such as average surplus head, minimum surplus head, and supplied demand. Moreover, Marlim et al. [8] divided and formulated the reliability objectives of a WDN's user service into social, economic, hydraulic, and water quality, and Markov et al. [9] also found that

the performance of a WDN could be measured via users' satisfaction and proposed a serviceability indicator to quantify this.

However, any individual hydraulic measurement is too fragmentary to be applied as the objective for WDN design and operation; hence, a large number of studies have been attempting to formulate a single "synthetic" index using various theoretical approaches. Wagner et al. [10] were the first to introduce and apply the concepts of mechanical and hydraulic reliability approaches to WDNs. Mays [11] also defined mechanical reliability as network topology evaluating system connectivity, given failure conditions, and hydraulic reliability as the ability of a system to meet the required water demand and pressure under normal and abnormal conditions. Later, Ostfeld [12] categorized WDN reliability evaluations into topological, hydraulic, and entropic backgrounds.

With regard to the hydraulic approach, Todini [13] developed the resilience index (RI), which represents the surplus and required energy in a WDN, whereas Jayaram and Srinivasan [14] developed a modified resilience index (MRI) with a different energy composition. Later, Liu et al. [15] and Jeong et al. [16] identified that a topographical relationship alters the reliability of the network performance, and proposed mixed reliability indices, a pipe hydraulic resilience index (PHRI) and a revised resilience index (RRI) by incorporating hydraulic and topographical approaches.

Within topological methods, research using a geometric approach [17–19] was performed, leading to different measures for estimating network reliability such as network efficiency (NE), average degree (AD), and link density. Creaco et al. [7] found that network performance is represented by the uniformity of pipe diameters in loop structures and developed a uniformity coefficient as the topological index. Moreover, Prasad and Park [6] also proposed a mixed reliability index, namely network resilience index (NRI), considering diameter uniformity along with the existing resilience index.

Regarding entropic reliability approaches, Awumah et al. [20] proposed an entropy reliability index by formulating water supply diversity in a WDN, and Tanyimboh and Templeman [21] developed and applied flow entropy (FE) into a WDN study based on the entropy concept of Shannon [22]. Raad et al. [23] suggested another mixed reliability index incorporating hydraulic and entropic approaches and compared four different reliability indices using performance measures in a benchmark network. Moreover, Jeong and Kang [24] suggested a hydraulic uniformity index (HUI), which is a mixed reliability index considering uniformity of the hydraulic gradients of pipes within a WDN.

However, the previously mentioned reliability indices have a bias towards certain system performances as influenced by their theoretical background. For example, in a recent study by Paez et al. [25], the correlation between different indices was analyzed through five arbitrary network designs. In addition, Tanyimboh et al. [26] investigated the correlations between surrogate reliability/redundancy measures (e.g., FE, RI, NRI) and surplus power factor with hydraulic reliability in hypothetical WDNs. In the most recent study of Sirsant and Reddy [27], the correlation between a reliability index and hydraulic and mechanical performance was also analyzed based on an optimally designed network and multipurpose functions of design cost, entropy, resiliency, and combined indices.

Eventually, it is necessary to appropriately examine which reliability index best reflects each type of system performance according to various situations and purposes required in the design and operation of WDNs. To that end, in this study, the correlations between representative reliability indices and hydraulic measures under three abnormal conditions (pipe failure; demand increase; fire flow) are analyzed for various types of application networks. Through these simulations, it was intended to determine the most adequate reliability index for evaluation of WDN performance in various abnormal water supply conditions.

The rest of this paper is organized as follows. The following section provides an overview of the proposed correlation analysis, and details of the hydraulic measures and various reliability indices are also described. Section 3 explains the design process of application networks and three application scenarios, and the application results and analyses are provided in Section 4. Finally, the conclusions of the study are summarized in Section 5.
