3. Case studies
3.2 Real-time multi-step-ahead water level forecasting by recurrent neural networks
networks for urban flood control
In this case study, various ANNs are used to make water level forecasts for representing the behavior of the rainfall-sewer flow processes in storm events. Flood levels can be forecasted on the basis of (a) rainfall data; (b) previous water levels; and (c) a combination of both data sets. Three ANNs coupled with statistical techniques are adopted to construct real time multi-step-ahead FSP forecasting models. The
Fig. 3.7 Study flow of real-time MSA water level forecasting.
implementation procedure is shown in Fig. 3.7. The time span of rainfall affecting the rise of FSP water level is first identified by the correlation analysis. Next the GT is applied to extracting effective rainfall factors from all possible rainfall-related input
combinations. One static (BPNN) and two dynamic (Elman NN and NARX network) neural networks are proposed to construct MSA FSP water level forecasting models for two scenarios (w/ and w/o current FSP water level information). Finally, the proposed SDM with core techniques (GT and NARX network) and two other ANN models (Elman NN and BPNN) are evaluated by performance criteria. Because the main purpose of this case study is to identify the effectiveness between static and dynamic network models as well as different types of recurrent connections, the static BPNN and the dynamic Elman NN and NARX network that have different types of recurrent connections are selected for comparison purpose. These three network models are trained through batch learning algorithms. Therefore, the proposed R-RTRL NN that has the same type of recurrent connections as the Elman NN but is trained by a unique online learning algorithm is excluded in this case study.
A. Study area and dataset
Taiwan, an island located in the subtropical zone of the North Pacific Ocean, is covered with mountainous terrains and steep landforms. Taipei City, situated in the Taipei Basin of northern Taiwan, is surrounded by the Danshui River whose narrow estuary makes it difficult to discharge water effectively from the city. Consequently, the high levees along the Danshui River have been built to prevent outer flood into the city with a return period of two-hundred-year flood protection standard. Typhoons and/or
Fig. 3.8 Locations of the Yu-Cheng catchment and rainfall gauging stations.
heavy rainfall events are usually coupled with intensive rainfalls and thus easily cause urban flooding within a few hours, even within a few minutes, in Taipei City. Because of the high levees, the main threat to the city now turns out to be the floodwater inside the levee system. Therefore, pumping stations play an important role in managing internal stormwater flows for urban flood control. The Yu-Cheng catchment, located in southeastern Taipei, is selected as the study area (Fig. 3.8). There are six rainfall gauging stations (R1-R6, denoted as red dots in Fig. 3.8). Although station R2 is out of the catchment, it still belongs to a sewerage system that diverts floodwater to the
Yu-Cheng pumping station and the Keelung River. There also exist a number of water level gauging stations in this study catchment. However, the malfunctions of water level gauges caused by their collisions with unknown objects and siltation in sewerage systems raise the difficulty in the maintenance of water level gauges and their on-line monitoring. The water level data collected from the sewerage system are neither stable nor accurate, which means the FSP water level forecasting models for the Yu-Cheng Pumping Station would mostly rely on the rainfall information retrieved from its neighboring rainfall gauging stations.
The Yu-Cheng catchment occupies an area of about 1627 ha and owns the biggest drainage system in Taipei City. The Yu-Cheng Pumping Station was built in 1987 to drain or pump the internal stormwater flows into the Keelung River, a chief tributary of the Danshui River, and it was considered the most advanced and the largest pumping station in Asia in the 1980s. The pumping station is currently equipped with 11 pumps reaching a total pumping capacity of 234.1 cms, and the operation of the pumping station highly depends on the FSP water level information. If the FSP water level rises up to the warning level (1.8 m) during heavy rainfall or typhoon events, duty pumps are activated with a 3-minute warm up. Then stromwater starts to be pumped from the FSP into the Keelung River when the FSP water level reaches the start level (2.2 m). The start level is the lowest water level designed for the start of stormwater pumping as well
as for the prevention against the idle running of pumps to avoid pump damage. These 11 pump units operate independently and maintain a sequential operation according to the laddered FSP water levels during typhoon or heavy rainfall periods. This means the pumping operation begins with one pump unit, and only one pump unit, instead of all remaining pump units, will join the operation at a time if the next higher rung of the laddered water levels is reached. On the contrary, running pumps will be shut down sequentially as the FSP water level decreases to the next lower rung. The operational procedure of the Yu-Cheng Pumping Station is quite different from those of Hong Kong, Tokyo or Singapore, where all the pump units are activated at the beginning if the FSP water level exceeds the start level, then the pump units stop running as the FSP water level drops to the stop level (DSD, 2000; PUB, 2013; Tamoto et al., 2008). It suggests that the pumping operation in Taipei City is much more sensitive to the fluctuation of FSP water level than those of big cities in Monsoon Asia.
Data of FSP water levels and rainfall at stations R1-R6 were collected with a temporal resolution of 10 minutes from 13 typhoon and heavy rainfall events during 2004 and 2013. A total of 1985 datasets are used for constructing forecasting models in this study, and the numbers of datasets allocated into training, validation and testing stages are 826 (from 6 events), 651 (from 3 events) and 508 (from 4 events) accordingly.
Table 3.6 Summary statistics for FSP water levels (m) and the peaks of average rainfall datasets in consideration of the summary statistics of the 13 events shown in Table 3.6.
In addition, the weighted average rainfall (Ravg) over the Yu-Cheng catchment is computed by the Thiessen polygon method and is also considered as a potential input to the forecasting models. Furthermore, because the original FSP water level is indeed affected by the operation of pumping units, the original FSP water levels were recovered prior to model construction according to a recovery equation (provided by the Taipei City Government) that involves pumping capacity and pumping-affected area.
The summary statistics for FSP water levels and rainfall datasets are presented in Table 3.6.
Fig. 3.9 Correlation analysis between FSP water levels and rainfall gauging stations in different time steps.
*time span of rainfall affecting the rise of FSP water level
B. Identification of the time span of rainfall affecting the rise of the FSP water
level
For constructing a rainfall-sewer flow model, the first step is to identify the temporal impacts of rainfall on the rise of FSP water level. In this study, the Pearson’s correlation coefficient is applied to learning the linear relationship and the recognition of the highest correlations between FSP water level and rainfall at different time lags for each station (R1-R6) as well as the weighted average rainfall (Ravg) over the Yu-Cheng
0 1 2 3 4 5 6
0.3 0.4 0.5 0.6 0.7
Time difference (10 min)
P e a rs o n 's c o rre la ti o n c o e ffi c e n t R1
R2 R3 R4
R5 R6 Ravg
Time difference (min)
0 0 0 0 0 0
Time span*
catchment. The results shown in Fig. 3.9 indicate that it consistently takes about 40 minutes for rainfall at stations R1-R6 to cause an increase in the FSP water level, similarly for the Ravg. It is worthy to note that, in contrast to river channels, a sewerage system can be implicitly considered as a small-scale volume control system on account of the relatively small catchment with which the system was associated. The variation of the FSP water level is mainly affected by the rainfall aggregated within a short period of time in the catchment. As a result, the time span of rainfall affecting the rise of FSP
water level at the Yu-Cheng Pumping Station is set as 40 minutes. It is noted that “time span” is used in this study while “concentration time” is usually used in river channel
studies.
C. Extraction of effective rainfall factors and model construction
The Pearson’s correlation coefficients between the FSP water level and rainfall at gauging stations R1-R6 as well as Ravg are not very high but quite similar (ranging from 0.41 to 0.63), which could be due to the lumped effect of rainfall falling to the catchment and the complex interactions between rainfall and sewer flow. In order to identify effective rainfall stations that significantly affect the fluctuations of FSP water level for modeling purpose, the GT is implemented in this study. That is to say, rainfall-related inputs to the estimation models of FSP water level is determined by the GT.
In this case study, one- to six-step-ahead FSP water level forecasting models during heavy rainfall and typhoon events for the Yu-Cheng Pumping Station are constructed through the BPNN, the Elman NN and the NARX network based on the inputs determined by the GT. The practical meaning and contribution of three forecasting models will be surveyed under two scenarios: (1) the information of current FSP water level is available (denoted as scenario I hereinafter); (2) the information of current FSP water level is not available (denoted as scenario II hereinafter). The applicability and reliability of these three constructed forecasting models at different forecasting steps are evaluated by the RMSE, CC and CE.
D. Results and discussion
This section presents the selection result of effective rainfall factors and the forecast performance of the static (BPNN) and dynamic (Elman NN and NARX network) neural networks in two scenarios (w/ and w/o current FSP water level information). The results and discussion are addressed in details, shown as follows:
1) Identification of effective rainfall stations
For extracting effective rainfall factors, data of the antecedent 40-minute rainfall collected at six gauging stations together with the average rainfall (R1(t-4)-R6(t-4), Ravg(t-4)) are first scaled to [-1,1]. Then a total of 127 (=27-1) values corresponding
Fig. 3.10 Determination of effective rainfall stations by the GT results.
to all possible rainfall-related input combinations are calculated through the GT. The produced values are next sorted in an ascending order, in which values smaller than the 10th percentile (Γ10= 0.10) are classified as the best group (
1 0
calculated by Eq. (47) for each rainfall factors are drawn into a green dotted line.
factor score =
Therefore, effective rainfall factors can be identified as those factors that are associated with higher factor scores, and the threshold of the factor score is set as 0.5 in this study. Consequently, R2, R3and R5 are identified as the effective rainfall factors to be used in the forecasting models.
2) Performance of FSP water level forecasting in scenario I: current FSP water level is
available
In scenario I, data of the current FSP water level and rainfall of R2, R3and R5 are utilized to construct 10- to 60-min-ahead (N = 1-6) FSP water level forecasting models through three ANNs. The input-output patterns of three ANN models can be represented
as follows: the training and validation data sets, the output-memory order q for NARX networks
is 1 and all the three models are configured to have only one hidden layer with 2-4 nodes for different forecasting steps. Summarized results are presented in Tables 3.7.
Table 3.7 Model performance of one- to six-step-ahead forecasting for FSP water levels
Results indicate that the three comparative models perform rather consistently in the training and validation stages while both dynamic neural networks (the Elman NN and the NARX network) perform better than the static one (the BPNN) in the testing stages.
Besides, the NARX network outperforms the Elman NN as the forecasting step exceeds
Fig. 3.11 (a) 20, (b) 50 and (c) 60-min-ahead forecasting of the 612 heavy rainfall event for scenario I with respect to the BPNN, the Elman NN and the NARX network.
four, and it even produces a high CE value (close to 0.7) as the forecasting step reaches six (60-min-ahead forecasting). The 612 heavy rainfall event (12.4 mm/10 min; 54.1 mm/hr) with the highest peak FSP water level above 4.5 m is selected to illustrate the hydrographs of observed versus 20-, 50- and 60-min-ahead forecasted FSP water levels
0 20 40 60 80
in the testing stages of three models (Fig. 3.11). During the peak flow period (about 10 hours) of this event, 6 up to 11 pumping units were operated. Results show that the Elman NN produces the best performance for 20-min-ahead forecasting because its over-estimation between 10 and 20 time steps is comparatively less serious. However, the NARX network can significantly mitigate time-lag problems at peak values and well forecast the 50- and 60-min-ahead FSP water level, whereas the other two comparative models not only have significant time-lag phenomena but fail to well forecast 50- and 60-min-ahead water levels, in which fluctuations occur near peak values.
Fig. 3.12 (a) shows the CE of 10- to 60-min-ahead forecasting in the testing stages of three models in scenario I. The three network models perform equally well for one- to three-step-ahead forecasting, whereas significant differences among their performances are found as the forecast time step exceeds four (40 min). The reason is that the time span of rainfall affecting the rise of the FSP water level is 40 minutes such that the rainfall-water level processes at 1-4 time steps could be suitably presented by rainfall input data and FSP water level output data (Eq. (7)). In addition to the fact that the persistence of FSP water level decreases as the time step increases, the time lag of rainfall becomes significant as the forecast time step exceeds four. As a result, both conditions would cause a reduction in forecast accuracy. It clearly indicates that the NARX networks produce much higher CE values than the other two models for
Fig. 3.12 (a) CE of 10- to 60-min-ahead forecasting and (b) relationship between FSP water level forecast errors (RMSE) and forecasting steps with respect to three
forecasting models in the testing stages for scenario I.
four- to six-step-ahead forecasting, whereas the Elman NNs perform slightly better than the BPNNs.
Finally, the relationship between forecast errors (RMSE) and forecasting steps of these three models is presented in Fig. 12 (b). The RMSE trend of the NARX network
0.2
model increases gradually as the forecasting step increases, and it becomes flat after three forecasting steps. Nevertheless the RMSE trends of the BPNN and the Elman NN models significantly increase as the forecasting step increases, and they have steeper slopes than that of the NARX network after three forecasting steps. The results provide evidence that with the feedbacks of imperfect outputs representing the information the closest to the forecasting horizon to the input layer, the NARX network can effectively adopt extra information to promote the accuracy and reliability of multi-step-ahead FSP water level forecasting.
3) Performance of FSP water level forecasting in Scenario II: current FSP water level
is unavailable
In the flood control centre of the Taipei City Government, the datasets of the current FSP water levels at sixty-five pumping stations during typhoon events are transmitted only through two channels of radio waves, and thus the backend system of the flood control centre may not successfully receive the current FSP water level information every ten minutes. Furthermore, the preliminary correlation analysis results indicate that the time span between the FSP water level and rainfall over the study catchment is about 40 minutes. An alternative to the forecasting model of scenario I is considered essential for auxiliary purposes. In this scenario (II), FSP water level
forecasting models are constructed based only on the rainfall of R2, R3and R5 to prevent any possible delay in the receipt of the current FSP water leveldue to the unstable frequency of data transmission. The input-output patterns of three ANN models
can be represented as follows:
the training and validation data sets, the output-memory order q for NARX networksis 1 and all the three models are configured to have only one hidden layer with 2-4 nodes for different forecasting steps. Summarized results are presented in Tables 3.8.
Results indicate that the NARX networks significantly outperform the other two network models in terms of lower RMSE values and higher CC and CE values in all three stages (training, validation and testing) for one- to six-step-ahead forecasting. It is noted that the performance of three ANN models in scenario II is not as good as that of scenario I. The reason is that only rainfall information is utilized as model inputs in scenario II, while another important factor, i.e., the persistent effect (auto-regression) of
Table 3.8 Model performance of one- to six-step-ahead forecasting for FSP water levels
the FSP water level, is not considered (or unavailable) in this circumstance. Under such condition, the NARX network equipped with recurrent connections from imperfect outputs can produce much more satisfactory results than the Elman NN and the BPNN.
Similar to that of scenario I, an analysis is conducted on the 612 heavy rainfall event for scenario II. The rainfall input datasets from three gauging stations and
Fig. 3.13 Rainfall input datasets from three gauging stations and 50-min-ahead forecasting of the 612 heavy rainfall event for scenario II with respect to the BPNN, the
Elman NN and the NARX network.
50-min-ahead forecasting are illustrated in Fig. 3.13. It demonstrates that the NARX network can well forecast the 50-min-ahead FSP water level and maintain the water level trail with less fluctuation than the BPNN and the Elman NN. The strong fluctuations occurring in the hydrographs associated with the BPNN and the Elman NN are mainly because these two models are driven only by the rainfall-related inputs that originally bear high variations, whereas the NARX network facilitates extra input information from the previous forecasted FSP water level to smooth the fluctuations of
0 20 40 60 80
the forecasted hydrograph.
Fig. 3.14(a) shows the CE of 10- to 60-min-ahead forecasting in the testing stages of three network models in scenario II. It clearly indicates that the NARX networks produce much higher CE values than the other two network models, while the Elman NNs perform even worse than the BPNNs, which implies the recurrent connections from the hidden layer of each Elman NNs magnify the highly variable rainfall information and thus do not increase the reliability of the Elman NNs.
Fig. 3.14(b) illustrates the relationship between forecast errors (RMSE) and forecasting steps of these three models. The results show that the NARX network produces much lower RMSE values than the other two models for one- to six-step-ahead forecasting, and the RMSE values of three models are relatively consistent (flat) for one- to four-step-ahead forecasting. For the NARX network, the RMSE value is the lowest at the 4th forecasting step and starts fast rising afterward.
This is mainly because only current rainfall information is available as the forecasting step increases to five and six steps, which significantly causes the degradation of model performance at the 5th and 6th forecasting steps. This phenomenon is consistent with the 40-min time span of rainfall affecting the rise of the FSP water level, which is determined by the correlation analysis shown in Fig 3.9.
Fig. 3.14 (a) CE of 10- to 60-min-ahead forecasting and (b) relationship between FSP
Fig. 3.14 (a) CE of 10- to 60-min-ahead forecasting and (b) relationship between FSP