2.3.4. Mixed Reliability Index

The pipe hydraulic resilience index (PHRI) focuses on nodal water head and simultaneously considers the hydraulic gradient between upstream and downstream nodes; hence, it can be categorized as a mixed reliability index based on hydraulic and topographical aspects. The detailed calculation method for PHRI can be presented as Equations (16)–(19).

$$\text{PHRI} = \frac{\sum\_{i=1}^{npip\epsilon} (S\_i)}{\sum\_{i=1}^{npip\epsilon} (A\_i + S\_i)} \tag{16}$$

$$S\_i = \frac{1}{2} (H\_{ds,i} - H\_{ds,req,i}) L\_{pro,i} \tag{17}$$

$$S\_i + A\_i = \frac{1}{2} (H\_{\text{ns},i} - H\_{\text{ds},\text{raq},i}) L\_{\text{pro},i} \tag{18}$$

$$L\_{\rm pro,i} = \sqrt{L\_i^2 - \left(Z\_{\rm us,i} - Z\_{\rm ds,i}\right)^2} \tag{19}$$

where *Hds,i* is the total head at the downstream node of pipe *i*; *Hus,i* is the total head at the upstream node of pipe *i*; *Hds,req,i* is the minimum required head at the downstream node of pipe *i*; *Li* is the length of pipe *i*; *Zds,i* is the elevation at the downstream node of pipe *i*; and *Zus,i* is the elevation at the upstream node of pipe *i*.

The revised resilience index (RRI) is the mixed reliability index based on hydraulic and topographical approaches. Although the calculation method for RRI is identical with MRI, RRI applies the hydraulic gradient representing network topography when calculating the minimum required head at downstream nodes [16]. The calculation method for RRI is as shown in Equation (20).

$$\text{RRI} = \frac{\text{\textdegree } \sum\_{j=1}^{\text{numdc}} Q\_j \text{\textdegree} \left( H\_j - H\_{\text{req},j}^\* \right)}{\text{\textdegree } \sum\_{j=1}^{\text{numdc}} Q\_j H\_{\text{req},j}^\*} \tag{20}$$

where *H*\* *req,j* denotes the actual minimum required head at node *j*.

The network resilience index (NRI) is another mixed reliability index incorporating network topology into the formulation of RI. In WDNs, reliable loops can be ensured, if the pipes connected to a node are not widely varying in diameter. NRI incorporates the effects of both surplus power and reliable loops [6]. The detailed calculation method of NRI is as shown in Equations (21) and (22):

$$\text{NRI} = \frac{\text{\textdegree NIR} - \text{\textdegree C}\_{j=1} \text{\textdegree C}\_{j} \text{Q}\_{j} \text{\textdegree \text{\textdegree C}\_{j} \text{\textdegree C}\_{j} \text{\textdegree C}\_{j} \text{\textdegree C}\_{j} \text{\textdegree C}\_{j}}{\text{\textdegree \text{\textdegree C}\_{s=1}^{\text{\textdegree NIR}} \text{\textdegree Q}\_{s} H\_{s} + \text{\textdegree C}\_{p=1} \text{\textdegree Q}\_{p} H\_{p} - \text{\textdegree D}\_{j=1}^{\text{\textdegree}D} \text{Q}\_{j} H\_{\text{req},j}} \tag{21}$$

$$C\_j = \frac{\sum\_{i=1}^{nippic\_j} D\_i}{nippic\_j \times \max\{D\_i\}}\tag{22}$$

where *npipej* is the number of pipes connected with node *j*; and *Di* denotes the diameter of pipe *i*.
