*2.4. Negative Pressure Treatment*

Hydraulic analysis approaches for water distribution network can be classified as a DDA and a pressure-driven analysis (PDA). DDA assumes that the demand at individual node is satisfied regardless of the associated pressure, while PDA assumes that nodal discharge is dependent on the pressure head expressed by the head-outflow relationship (HOR). DDA often generates negative pressure when simulating abnormal conditions, such as multiple pipe breaks. PDA, on the other hand, provides more realistic hydraulics avoiding negative pressures. WaterGEMS [47], WDNetXL [48], and WaterNetGen [49] are well-known programs equipped with PDA option based on HOR. However, the system-specific HOR should be provided, which is spatially and temporally inconsistent and also operational dependent. Therefore, for PDA techniques used in pipe network analyses, the analyzer should assume a HOR for the whole network. Given that hydraulic states of pipe networks can vary greatly with changes in HORs, analysis results cannot guarantee high accuracy.

Quasi-PDA methods suppress the occurrence of negative pressure through repetitive DDA analyses. In general, if negative pressure occurs from the first DDA run, better hydraulic calculation results can be derived through quasi-PDA methods by resetting the nodal demands and the components' status.

Ballantyne *et al.* [21] used the KYPIPE model [50] for hydraulic analyses. KYPIPE, a DDA program similar to EPANET, can also produce negative pressures during simulation of pipeline destruction. Ballantyne *et al.* calculated system reliability by assuming that water could not be supplied to points with negative pressures, but the hydraulic results around the negative-pressure node still did not have realistic values. The method of Shinozuka *et al.* [22] and Shi [27] completely removed negative-pressure nodes and the connected pipelines from the original network, but the process required time-consuming repetitive hydraulic analyses and regeneration of input files.

Here, a quasi-PDA approach is proposed for realistic hydraulic simulation under multiple failure conditions to avoid the occurrence of negative pressure. That is, if negative pressure occurred after DDA simulation using EPANET, the updated base demand (after modification described in Chapter 2.3) of the negative-pressure node is set to zero and DDA is repeated. If negative pressure reappears, the pressure of the relevant node is assumed to zero. Compared to Shinozuka *et al.* [22] and Shi [27] approach, the proposed approach saves overhead processing time by avoiding regeneration of the EPANET input file for the new configuration. In addition to the negative pressure treatment, pressure-dependent water supply is considered in calculating the quantity of available water at each node. The available nodal demand at node j (Qavl,j) is estimated as

$$\mathbf{Q\_{nvl,j}} = \begin{cases} 0 & \text{if } \mathbf{P\_{j}} < 0 \\ \mathbf{Q\_{nev,j}} \times \sqrt{\frac{\mathbf{P\_{j}}}{\mathbf{P\_{min}}}} & \text{if } 0 < \mathbf{P\_{j}} < \mathbf{P\_{min}} \\ & \mathbf{Q\_{nev,j}} & \text{if } \mathbf{P\_{j}} > \mathbf{P\_{min}} \end{cases} \tag{7}$$

where Qnew,j = the updated base demand with pipe breakage consideration and negative pressure treatment at node j; Pj = the pressure head at node j; and Pmin " the minimum pressure requirement.

### *2.5. Seismic Reliability Indicator*

Various surrogate measures of WDS reliability have been proposed including capacity reliability [6,7], resilience [40], robustness [10,11], availability [51]. Bao and Mays [52] proposed three formulations of WDS system reliability which can be calculated from nodal reliability values: minimum nodal reliability, arithmetic mean reliability, and flow-weighted mean reliability. The first one concerns on the worst nodal reliability while the latter two values indicate system-wide reliability. In order to represent post-earthquake system performance in the water supply, a seismic reliability measure should be able to reflect system-wide water availability rather than local level of system performance.

To quantify system-wide seismic reliability, a new reliability indicator is proposed herein. The system seismic reliability (SS) is defined as the ratio of the total available system demand to the total required system demand:

$$\mathbf{S}\_{\mathbf{S}} = \frac{\sum\_{\mathbf{j=1}}^{\text{m}} \mathbf{Q}\_{\text{avl},\mathbf{j}}}{\sum\_{\mathbf{j=1}}^{\text{m}} \mathbf{Q}\_{\text{req},\mathbf{j}}} \tag{8}$$

where m = total number of demand nodes; and Qreq,j = the required demand at node j.

A water system planner would intend to design the most reliable network for earthquakes under a given budget condition. To that end, the proposed model here is a single-objective optimal design model that maximizes the system's seismic reliability with a constraint on economic cost. The optimization model is formulated as follows:

$$\begin{array}{l}\text{Maximize F} = \text{S}\_{\text{s}}\\\text{s.t.}\\\text{CC} \leqslant \text{CC}\_{\text{given}}\end{array}\tag{9}$$

where CC = the pipe construction cost; CCgiven = the available budget for pipe system.

Equation (9) is valid for each set of available commercial pipe diameters and in conditions where the minimum pressure head is guaranteed. The pipe construction cost accounting for the pipe material and installation is expressed as:

$$\text{CC} = \sum\_{\mathbf{i}=1}^{n} \left( \text{uc} \left( \mathbf{D}\_{\mathbf{i}} \right) \times \mathbf{L}\_{\mathbf{i}} \right) \tag{10}$$

where uc pDiq = the unit cost of the pipe with diameter Di determined for the ith pipe (USD/m); and Li = the length of the ith pipe.
