*2.2. Hydraulic Measures*

Hydraulic measures are an indication of the water supply's status in the system, which directly represents the performance of a WDN. In this study, the properties that can ensure the performance of a WDN were classified as redundancy, robustness, and serviceability, and change in performance was quantified based on two hydraulic measures of each property. Detailed descriptions of the concept of each performance property and the selected hydraulic measures are presented below.

## 2.2.1. Redundancy Measures

Redundancy used to be defined as substitutable capacity such as excessive backup [4] and the extent captured by the loops [28] in WDN design. In this study, network average surplus head (Red1) and minimum surplus head (Red2) under abnormal conditions were used as measures to indicate the redundancy of the system. Here, the surplus head refers to the head supplied in excess of the required head at each node (i.e., total head, subtracting elevation and minimum required pressure

head); it can be shown that the higher the values of Red1 and Red2, the higher the network redundancy. The detailed calculation of redundancy measures can be presented as follows:

$$\text{Red1} = \frac{\sum\_{j=1}^{\text{model}} \left( H\_j - H\_{\text{req},j} \right)}{\text{mode}} \tag{1}$$

$$\text{Red2} = \min \{ H\_j - H\_{\text{req},j} \}, j = 1, \ 2, \ \dots, \text{mode} \tag{2}$$

where *nnode* is the number of nodes; *Hj* is the total head at node *j*; and *Hreq,j* is the minimum required head at node *j*.

## 2.2.2. Robustness Measures

Robustness represents the capacity of a system to withstand a given level of stress or demand without suffering degradation or loss of function [4]. Hence, robustness performance keeps the variability of losses within a narrow band [29]. In this study, network average pressure maintenance (Rob1) and minimum pressure maintenance (Rob2) under abnormal conditions (as opposed to normal conditions) were used as measures to reflect the robustness of the system. Herein, pressure maintenance refers to the ratio between nodal pressure under abnormal and normal conditions (i.e., pressure under abnormal conditions divided by the pressure under normal conditions), and it can be understood that the higher the values of Rob1 and Rob2, the stronger the robustness of the network. The detailed calculation of robustness measures can be expressed as follows:

$$\text{Rob1} = \frac{1}{m \, \text{node}} \sum\_{j=1}^{m \, \text{node}} \frac{H\_{j, \, \text{bar}}}{H\_{j, \, \text{normal}}} \, \text{s} \tag{3}$$

$$\text{Rob2} = \min \left\{ \frac{H\_{\text{j, abnormal}}}{H\_{\text{j, normal}}} \right\}, j = 1, 2, \dots, \text{ mode} \tag{4}$$

where *Hj,normal* and *Hj,abnormal* are the total heads at node *j* in normal and abnormal water supply conditions, respectively.

## 2.2.3. Serviceability Measures

Serviceability represents users' satisfaction with a system's functionality, depending on several factors such as the vulnerability of system components, topology, and operation scenarios [9]. In this study, the available demand ratio (Ser1) and number of supplied nodes (Ser2) under abnormal conditions were used as measures to represent the serviceability of the system. Herein, the available demand ratio indicates the ratio between the nodal supplied demand under abnormal and normal conditions (i.e., supplied demand under abnormal conditions divided by the desired demand), and supplied node refers to a node in which all desired demands are satisfied. Therefore, it can be understood that the higher the values of Ser1 and Ser2, the better the serviceability of the network. The detailed calculation of serviceability measures can be presented as follows:

$$\text{Ser1} = \frac{\sum\_{j=1}^{m \text{node}} Q\_{j,\text{ avl}}}{\sum\_{j=1}^{m \text{node}} Q\_j} \tag{5}$$

$$\text{Ser2} = \sum\_{j=1}^{m adc} A\_j \text{ where } A\_j = \begin{cases} 1 & \text{if } Q\_{j, \, avl} = Q\_j \\ 0 & \text{otherwise} \end{cases} \tag{6}$$

where *Qj* is the water demand at node *j*; *Qj,avl* is the supplied water demand at node *j*; and *Aj* is the water availability indicator of node *j*.
