3.3.1. Overall Performance

Figure 4 compares the observed water demand with the forecasted water demand using the S\_LSSVM, H\_LSSVM\_Chaos, and H\_LSSVM\_FS models at 15 min steps for one day ahead. It can be seen that the predicted water demand by the three models is consistent with the trend of the observations, and the hybrid models perform better than the single forecasting models (S\_LSSVM) during the periods of water demand fluctuations. As quantified below by the model performance indicators, the H\_LSSVM\_Chaos models provide the closest estimates to the corresponding observed water demand during most of the peak periods.

**Figure 4.** Water demand forecasting for one day ahead with the time step of 15 min. (**a**) Water demand of DMA1 on 26 December; (**b**) water demand of DMA2 on 26 December; and (**c**) water demand of DMA3 on 11 August.

Table 4 gives the overall performance of the different forecasting models for the three DMAs in Beijing. It can be seen that the H\_LSSVM\_Chaos provides a higher accuracy than the other two models according to the performance indicators R2, MAE, MAPE, and RMSE. The single forecasting model S\_LSSVM is the least accurate.


**Table 4.** Performance indicators of forecasting models on testing data.

Among the three DMAs, the prediction accuracy to DMA1 is slightly worse than to DMA2 and DMA3, for example, the MAPEs of (DMA1, DMA2, DMA3) of the H\_LSSVM\_Chaos models and the H\_LSSVM\_FS models are (4.84%, 3.15%, 3.47%) and (5.44%, 3.33%, 3.72%), respectively. The reason is that the composition of the water customers in DMA1 is relatively complex, not only including residential users, but also a large number of commercial and industrial users. The statistical parameter COV of DMA1 s water demand data is 0.39, which is the largest one among the three DMAs. Larger COV indicates a high level of water demand floating and makes the demand pattern more difficult to capture. As a result, even using the error correction module, the hybrid model H\_LSSVM\_Chaos only reduced the MAPE of DMA1 from 5.64% to 4.84%, which is less than the reductions for the other DMAs. Moreover, because the water consumptions in DMA2 are mostly residential demands which thus lead to a simple water demand pattern, the prediction results for DMA2 give the highest accuracy. Therefore, as for the error correction module performance on short-term water demand forecasting, the DMAs with simple customer composition have better prediction accuracy when using error correction module.
