**4. Conclusions**

Short-term water demand forecasting with the horizon ranges from sub-hourly to daily plays an important role in the field of optimal operation of pump stations and online hydraulic simulation of water distribution systems. To obtain more accurate predictions, this study proposes a hybrid framework with the error correction module which uses the chaotic time series, and investigates the performance of the framework in the short-term water demand forecasting with one day ahead and a 15-min time step. The hybrid framework is developed by integrating two modules, namely, the initial forecasting module and the error correction module. The initial forecasting model is established by the least squares support vector machines (LSSVM). In the error correction module the errors forecasting model is established by LSSVM using chaotic time series of error data from initial forecasting.

The hybrid model is implemented in the water demand forecasting of three actual district metering areas (DMAs) in Beijing, China, and the application results of the hybrid model are comparable to that of other two models including the forecasting model without error correction and the hybrid model using Fourier series for error correction. From the case study results, the following conclusions could be drawn:


In the presented study, the hybrid forecasting framework is tested by three actual DMAs in Beijing with different characteristics. Further work on other DMAs are needed to test and verify the robustness of the hybrid forecasting framework, and much more effort is needed to test the performance of chaotic methods in mining the characteristics of the disordered peak fluctuated data. This study only tested the proposed model for the 24 h forecast horizon, whereas, the hybrid forecasting framework is not limited to the forecast horizon of one day, there is a potential to implement the model to a much longer forecast horizon and frequency, such as one week ahead with a time step of 6 h. Then the feature data for model training obtained from the historical data set should be adjusted accordingly.

**Author Contributions:** Conceptualization, S.W. and B.H.; methodology, H.H. and B.H.; validation, S.W., H.H., B.H., and K.D.; formal analysis, H.H., B.H., and K.D.; investigation, H.H., B.H., and K.D.; resources, S.W., H.H., B.H., and K.D.; writing—original draft preparation, H.H. and B.H.; writing—review and editing, S.W. and K.D.; visualization, B.H. and K.D.; supervision, S.W.; project administration, S.W. and B.H.; funding acquisition, S.W. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the Major Science and Technology Program for Water Pollution Control and Treatment, grant number 2017ZX07108-002.

**Acknowledgments:** The authors would like to thank the editors and reviewers for bringing the paper to a scientific standard for inclusion in the journal.

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