*5.3. Results and Error Analysis*

The steady-state hydraulics of pipe network B were predicted by PNSOS in 309 s, and the transient event was then generated by closing the valve at N17. The transient head distributions for cases 1–3 were therefore measured at N17. Table 8 shows the results of fault detection for cases 1–3. In case 1, the information about deterioration reaches D1 and D2 was accurately determined with its corresponding parameters. The locations of three leaks and two blockages were also precisely detected by PEOS. It is noteworthy that leak L2 at node N22 was isolated by the proposed approach, indicating that PEOS was capable of handling the case of pipe junction leakage. In case 1, the *E* between the actual *CdLALs*/*CdBAB* values and the predicted ones was insignificant. Table 9 shows the values of the ME and SEE, which for case 1 were 3.41 <sup>×</sup> <sup>10</sup>−<sup>6</sup> m and 1.27 <sup>×</sup> <sup>10</sup>−<sup>4</sup> m, respectively. The results denote that the predicted heads were not affected by the use of limited observations. The results for case 1 and the small ME and SEE values indicate that PEOS had the potential to deliver moderately good results in a field survey even when only a few observations were available. The success of using fewer measurements indicates that PEOS may not be restricted by instrument limitations. In addition, the data measurement period can therefore be reduced, and the system impact due to a transient event may be slight while using PEOS.

Table 8 shows that PEOS provided relatively good results for deterioration detection in case 2. The locations of the deterioration segments, determined at 390 m for P62 and 610 m for P67, deviated slightly from the actual ones, which were instead located at 400 m for P62 and 600 m for P67. The lengths of D1 and D2 were accurately determined. The impedances for D1 and D2 were respectively estimated as 162.0 and 121.5 s/m2, with corresponding wave speeds of 794.3 and 595.8 m/s. For leak and blockage detection in case 2, the predicted locations of three leaks and two blockages were close to the real locations, implying that the measurement errors may not have affected location detection. There were errors in the predictions of *CdLAL* and *CdBAB* in case 2. The relative differences between the predicted *CdLAL* values and the actual ones were about 6%, 2%, and 5.83% for L1, L2, and L3, respectively. The relative differences between the determined *CdBAB* values and the real ones were about 5.25% for B1 and 4.17% for B2. The results showed that the predicted *CdLAL* values and *CdBAB* may have been more sensitive than location to measurement errors. This was due to the fact that the OFVs used in PEOS for fault detection were directly related to the head difference (i.e., Equation (14)), which may have been directly influenced by the change in leak area and blockage area. The MEs and SEEs for case 2 are listed in Table <sup>9</sup> and were respectively 1.73 <sup>×</sup> <sup>10</sup>−<sup>4</sup> m and 6.35 <sup>×</sup> <sup>10</sup>−<sup>2</sup> m, which were both two orders larger than those of case 1. Such a result indicates that measurement errors may have affected accuracy in determining the leak area and blockage area. Thus, data uncertainty should be of concern as an important issue in fault detection in a large-scale pipe network or in future field applications.

In case 3, leaks, blockages, and deterioration segments were also accurately determined by PEOS, with its associated parameters listed in Table 8. The locations of various faults were precisely detected by PEOS. The sizes of leaks and blockages were slightly overestimated compared to case 1, with the largest relative difference, 2.5%, for L3. The values for the ME and SEE for case 3 were respectively 3.29 <sup>×</sup> <sup>10</sup>−<sup>6</sup> <sup>m</sup> and 1.12 <sup>×</sup> <sup>10</sup>−<sup>4</sup> m, as shown in Table 9. The results indicate that the predicted heads were not affected, while the transient operation was inadequate. Note that the concept of ITA is to minimize errors between the measured and calculated system state variables. Measurements with an unsuitable transient operation still work well based on the objective function of ITA. The results of case 3 reveal that PEOS can provide good predictions when using different transient operation durations. However, a rapid transient operation is recommended, because it produces large system response data, thus improving the performance of the ITA [31].



**Table** 

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**Table 9.** The prediction errors for three cases. ME: mean error; SEE: standard error of the estimate.


#### *Water* **2019** , *11*, 1154
