**4. Application Results**

First, we applied two different commercial pipe sets for the pipe sizing of the study network. This analysis investigates the WDS's layout changes with respect to the seismic reliability increase. The same total cost constraint was applied for the two designs and the resultant pipe layouts were compared. Then, seven designs with a fixed layout and different redundancy levels were compared for the systems' seismic reliability. Finally, discussions and suggestions on improving WDS seismic reliability were provided at the end of this section.

### *4.1. Different Available Pipe Sizing Options*

The first application is intended to investigate the impact of seismic reliability on the network's layout. Two sets of commercial pipe sizes are assumed to be available for two design cases. In Case 1, all commercial pipes (152, 203, 254, 305, 356, 406, 457, 508, 610, and 762 mm) except the zero size option are available. In Case 2, zero diameter is available in addition to the commercial sizes in Case 1. Considering the same cost constraint (CCgiven = 18 million (M) USD) in Equation (9) provided a platform for consistent comparison of the resulting designs.

Figure 4 shows the optimal seismic reliability values of the two case designs. The corresponding optimal pipe layouts are presented in Figure 5. Contrary to expectations, the seismic reliability decreased with the availability of more pipe sizes. In Case 1, at least a 152 mm pipe should be installed because no pipe option is not available. By comparing Figures 5a,b and 6, we can observe that 152 mm pipes were installed in Case 1 for the link at which no pipe was constructed in Case 2. The 152 mm pipe, which is the most vulnerable pipe to earthquakes, almost always causes failure. Being able to have no pipe instead of a 152 mm pipe increased the system's seismic reliability by 0.2 (a 20% increase

in the amount of available water during an earthquake). The network layout becomes smaller from Case 2 to Case 1 (Figure 5a,b), while the overall pipe diameters also increase (Figure 6).

**Figure 4.** Maximum seismic reliability values of two case designs with different pipe sizing options.

Note that this is very different from what we observed in a traditional capacity reliability-based design. In the context of the capacity reliability field, it was believed that having additional paths and more loops would result in the increase of system reliability and redundancy [11,60]. However, the result of this study indicates that a different strategy should be available under the conditions where WDS component failures are affected by strong external forces (*i.e.*, earthquakes) and the components' physical characteristics (pipe's probability of failure as a function of RR).

**Figure 5.** Pipe layout comparison for the solutions obtained from Cases 1 and 2 (Figure 4); pipe diameters are in mm; the thicker and darker pipe is larger. (**a**) Case 1; (**b**) Case 2.

**Figure 6.** Histogram of pipes by the pipe diameter.

This plateau in seismic reliability was also observed in the optimal pipe designs of the Anytown network by applying different total cost constraints and using the available pipe sizes in Case 1. Figure 7 shows the Pareto optimal solutions' total cost and seismic reliability. The marginal cost becomes infinite for the designs whose cost is greater than 16.5 M USD. A reliability increase can no longer be achieved once a sufficient investment is made. Although the designs greater than 16.5 M USD have more large pipes compared to the solutions less than 16.5 M USD (the number of pipes equal to or larger than 508 mm is between 7 and 10, while the designs less than 16.5 M USD have three to five pipes), no benefit of having larger pipes was obtained with respect to seismic reliability. For effective and economical improvement of WDS seismic reliability, the threshold investment for a network should first be identified.

**Figure 7.** Trade-off relationship between total cost and seismic reliability in Case 1 where all pipes (152, 203, 254, 305, 356, 406, 457, 508, 610, and 762 mm) are available and without zero pipe option.

### *4.2. Constant Layout with a Single Pipe Sizing Option*

The impacts of having large pipes are also investigated through the seismic reliability evaluation of seven uniform designs. The Design 1 has 305 mm for all pipes in the study network. Design 2, 3, 4, 5, 6, and 7 have 356, 406, 457, 508, 610, and 762 mm, respectively, for all pipes. The seismic reliabilities of the seven designs are shown in Figure 8. For comparison, Todini's resilience is also calculated from the seven designs and plotted in Figure 8. While there is a large increase in seismic reliability from the 457 mm design (Design 4) to the 508 mm design (Design 5), reliability decreases from the 508 mm design to the 762 mm design (Design 7). This explains why we observed a plateau in seismic reliability in Figure 7. On the other hand, resilience (a traditional reliability measure) consistently increases with increasing pipe sizes. The marginal cost of improving resilience increases substantially for a resilience value of 0.3–0.8 and stabilized for a value higher than 0.8.

**Figure 8.** Seismic reliability of the seven uniform designs; all pipes are 508 mm in Case 5.

As the pipe sizes decrease, the correction factor for RR (Equation (4)) increases, finally resulting in high pipe breakage and leakage probability. However, although the pipes failed, the resulting impact is not significant to the system's seismic reliability because the calculated discharge coefficient, which is a function of the pipe's cross-sectional area (Table 2), is not large. However, as seen in Table 2, large pipes such as 508, 610, and 762 mm have a smaller failure probability compared to small pipes, but the resulting discharge coefficient is more than 10 times as big as that of small pipes. The failure effects are more significant for system seismic reliability compared to small pipes.


**Table 2.** Correction factors and discharge coefficients (Equation (6)) for the pipe sizes considered.
