**6. Conclusions**

This paper demonstrates an inverse transient-based heuristic optimization approach called PEOS for pipe examination in a pipeline or pipe network system. The application of PEOS was verified by two experimental RPV systems in the literature, and PEOS was further applied to identify fault information in synthetic pipe networks. PEOS was used to detect faults in an experimental pipeline (carried out at Imperial College London) and in a pipeline at the Water Engineering Laboratory at the University of Perugia. The head distributions predicted by PEOS agreed well with those from the experiments reported in the literature. The leak and blockage information in both systems was accurately determined by the proposed approach. The results indicated that PEOS provided good predictions in fault detection in a real pipeline system.

The proposed approach was further compared to three evolutionary-based algorithms in fault detection in a synthetic benchmark pipe network. Temporal head distribution and fault information were accurately predicted by PEOS and agreed well with the actual ones, even when using only 10 initial input organisms. PEOS on average took about 50.6 min and 1382 iterations to obtain the optimal results, which is significantly faster than other algorithms. The results indicated that the OOA made the proposed approach avoid most unnecessary calculations of incorrect solutions and quickly converge to the optimal result via three states of SOS. In other words, PEOS not only provided predictions with better accuracy and robustness, but also performed better at computational efficiency. The proposed approach with these two advantages obviously outperformed other algorithms.

To illustrate the applicability of PEOS in fault detection in real-world problems, a large-scale WDN with three data collection statuses was considered as a field study to represent practical issues. The results indicated that PEOS performed well in solving the fault detection problem, considering the effects of limited observations and measurement errors in a complicated WDN. The effect of limited observations on the estimated result was not significant, but the measurement errors induced some inaccuracy. When the observations contained measurement errors, the predicted *CdLAL* and *CdBAB* had slight deviations compared to the actual ones, indicating that PEOS could achieve good results if the measurements were well collected. Moreover, the results revealed that inappropriate transient operation may not have affected the performance of PEOS in predicting head distribution and fault information.

In summary, we demonstrated via the simulations that PEOS has the ability to simultaneously detect various faults in a pipeline and pipe networks and can outperform other existing evolutionary-based algorithms. Another superiority of PEOS over competing algorithms is the small number of parameters that must be tuned. Fault information can be precisely predicted even when observations are collected with issues. The cases presented in this study were for relatively simple pipe system configurations and operations. Extending the current work from numerical simulations to solving the problems of real-world complicated WDNs would be an interesting direction for further research.

**Supplementary Materials:** The details of the SOS algorithm are available online at http://www.mdpi.com/2073- 4441/11/6/1154/s1.

**Author Contributions:** C.-C.L. designed the numerical experiment, analyzed the data, and wrote the paper. H.-D.Y. is the supervisor of the proposed research.

**Acknowledgments:** The authors would like to thank the editor and three anonymous reviewers for their valuable and constructive comments, which greatly improved the manuscript.

**Conflicts of Interest:** The authors declare no conflict of interest.
