2.3.1. Hydraulic Reliability Index

The resilience index (RI) quantifies WDN reliability based on system power such as input, dissipated, and surplus power. In a looped network, the goal is to provide more power (energy per unit time) at each node than is required, in order to have a sufficient surplus to be dissipated internally in case of failures. This surplus can be used to characterize the resilience of the looped network, i.e., its intrinsic capability for overcoming sudden failures [13]. In other words, RI is the ratio of surplus power excluding the minimum required power at the node with the total system power supplied; the detailed calculation method can be expressed as

$$\text{RI} = \frac{\text{\textdegree\textdegree\textdegree\textdegree\textdegree\textdegree\textdegree\textdegree\textdegree\textdegree\textdegree\textdegree\textdegree\textdegree\textdegree\textdegree\textbot\textyellow\textdegree\textdegree\textdegree\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\textbot\text1$$

where γ is the specific weight of water; *Qs* is the inflow at source *s*; *Qp* is the pumping flow at pump *p*; *Hs* is the total head at source *s*; *Hp* is the pumping head at pump *p*; *nsource* is the number of sources; and *npump* is the number of pumps.

The modified resilience index (MRI) quantifies the reliability of a WDN based on system power, similarly to RI. In particular, MRI uses only required and surplus power to calculate the index and could be used to compare the uncertainty handling of one network relative to another, which is essential in design and rehabilitation problems [14]. A detailed calculation method for MRI is given as follows:

$$\text{MRI} = \frac{\text{\textdegree } \sum\_{j=1}^{\text{m node}} \text{Q}\_{j} (H\_{j} - H\_{\text{raq},j})}{\text{\textdegree } \sum\_{j=1}^{\text{m node}} \text{Q}\_{j} H\_{\text{raq},j}} \tag{8}$$

The available power index (API) quantifies the WDN reliability using total available power and input power. Here, the available power represents the output power at demand nodes; while the unavailable power includes the power dissipated due to pipe friction losses and various minor losses in the network [15]. The detailed calculation method of API is given as

$$\text{API} = \frac{\text{\ $ \cdot \sum\_{j=1}^{\text{mondet}} Q\_j H\_j}{\text{\$  \cdot \sum\_{s=1}^{\text{nsour}} Q\_s H\_s + \text{\ $ \cdot \sum\_{p=1}^{\text{npump}} Q\_p H\_p + \text{\$  \cdot \sum\_{t=1}^{\text{ntank}} Q\_t H\_t}}} \tag{9}$$

where *Qt* denotes the inflow at tank *t*; *Ht* is the total head at tank *t*; and *ntank* is the number of tanks.

### 2.3.2. Topological Reliability Index

The average degree (AD) is a convenient geometric index for quantifying system reliability based on the number of node and pipe elements. It implies that network reliability is proportional to the diversity of link (i.e., pipe) elements and paths. If a network has too few pipes, there will be many isolated nodes and clusters with a small number of nodes. As more pipes are added to the network, the small clusters are connected to larger clusters [18]. The calculation method of AD can be presented as follows:

$$\text{AD} = \frac{2 \times npipe}{mode} \tag{10}$$

While many topological indices, including AD, quantify connectivity of networks using only the number of nodes and pipes without distinction of their layouts, Latora and Marchiori [17] suggested an index, called network efficiency (NE), that quantifies the average efficiency of paths. The NE can be calculated as an average distance between two generic nodes and is expressed by Equation (11).

$$NE = \frac{1}{mnode(mode - 1)} \sum\_{j=1}^{mnode} \sum\_{\substack{j^\* = 1 \\ j^\* \neq \ j^\*}}^{mnode} \frac{1}{d\_{jj^\*}} \tag{11}$$

where *djj\** is the shortest path length from node *j* to node *j\**.

#### 2.3.3. Entropic Reliability Index

Shannon [22] derived the informational entropy function as a statistical measure of the amount of uncertainty that a probability distribution represents. The flow entropy (FE) proposed by Tanyimboh and Templeman [21] is another representative entropic reliability index available for WDNs. Prasad and Tanyimboh [30] suggested that this measure can better represent multi-source networks, and it was observed that, as the FE increases, the network becomes more reliable. The detailed calculation method of FE can be presented by Equations (12)–(15).

$$\text{FE} = Eo + \sum\_{j=1}^{mndc} P\_j E\_j \tag{12}$$

$$P\_j = \frac{T\_j}{T} \tag{13}$$

$$E\_0 = -\sum\_{s=1}^{\text{measure}} \frac{Q\_s}{T} \ln\left(\frac{Q\_s}{T}\right) \tag{14}$$

$$E\_j = -\frac{Q\_j}{T\_j} \ln \left( \frac{Q\_j}{T\_j} \right) - \sum\_{ji \in \text{ND}\_j} \frac{Q\_{ji}}{T\_j} \ln \left( \frac{Q\_{ji}}{T\_j} \right) \tag{15}$$

where *Tj* is the total flow reaching node *j*; *T* is the sum of the nodal demands; *NDj* denotes the set of all pipe flows emanating from node *j*; and *Qji* represents the flow rate at pipe *i* from node *j*.
