# Coupled Heuristic Prediction of Long Lead-Time Accumulated Total Inflow of a Reservoir during Typhoons Using Deterministic Recurrent and Fuzzy Inference-Based Neural Network

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Development of Methodology

#### 2.1. Procedures

#### 2.2. Developed Model Type of Accumulated Total Reservoir Inflow Forecast

_{1}, A

_{2}, and A

_{3}are the increasing/reducing storage of Stage I, increasing storage of Stage II, and increasing/reducing storage of Stage III, respectively; ${S}_{dam-safety}^{max}$ is maximum safety storage for the dam; ${x}_{0}^{S}$ is the initial storage; and ${S}^{dead}$ is the dead storage.

**Figure 2.**Schematic diagram of the flood operating mechanism of different stages in conjunction with the reservoir inflow.

^{total}) method, and forecasted the various delays from the current moment to the key times along the rainfall-runoff hydrograph; for example, the delay from the current time to the maximum rainfall (T

_{0-MP}), the delay to the end of the direct runoff (T

_{0-DRE}), and the delay to the end of the water retreat (T

_{0-EE}). The new method also used the observed-predicted inflow increase/decrease rate (OPIID rate) as the heuristic-type input. It is expected to be able to simulate the total reservoir inflow of the runoff hydrograph from the rainfall trend from a certain typhoon moving path in the future. A schematic diagram of hydrological key points within the rainfall-runoff hydrograph is shown in Figure 3. In this study, an original and innovative forecast model for the total reservoir inflow was developed with heuristic forecast inputs using ANFIS and RTRLNN. The model developed was analyzed and compared with the non-heuristic forecast model in which the input only included the real-time observed meteorology and hydrology information. The feasibility of the heuristic model for real-time forecast was also evaluated.

#### 2.2.1. Candidate Predictor

_{0-MP}), the delay to the end of the direct runoff (T

_{0-DRE}), and the delay to the end of the water retreat (T

_{0-EE}). The feasible alternative includes the delay to the inflection point after peak flow, which is equal to the delay to rainfall excess ending plus the time of concentration. However, it is difficult to predict the delay to rainfall excess ending in real-time across a long lead-time, leading to this alternative not being adopted.

_{0-MP},T

_{0-DRE},T

_{0-EE}, and OPIID rate). The input for the non-heuristic forecast model only included the real-time observed meteorology and hydrology information. The structure of the developed heuristic-type and non-heuristic accumulated total inflow forecast model is shown in Figure 4.

**Figure 4.**Structure of the developed coupled heuristic and non-heuristic accumulated total inflow forecast model.

#### 2.2.2. Selection of Model Inputs

_{rank}is Spearman’s rank correlation coefficient, n is the number of data, x is the candidate input of the forecast model (predictor), y is the model output also known as the predictant (accumulated total reservoir inflow during time t + 1 to t + T

_{0-EE}), and $Ran{k}_{{x}_{i}}$ and $Ran{k}_{{y}_{i}}$ are the sort values of x

_{i}and y

_{i}in their individual time-series of the variable, respectively. The most correlated candidate predictors for forecasting accumulated total inflow will be selected as optimal inputs, and the selected inputs must subject to hydrological relationships and the r

_{rank}must larger than the assigned threshold values.

#### 2.2.3. Assessment Index of Forecast Models

^{2}, and coefficient of efficiency (CE). However, RMSE and R

^{2}are respectively similar to MAE and CC, and CE cannot assess the time delay effect of the forecast. Hence, the other alternatives are not adopted. The computational equations of MAE and CC are expressed as follows:

#### 2.3. Heuristic Construction of RTRLNN

**V**and

**W**) along the negative of $\nabla $E

_{total}. The other feasible alternative is the Quasi-Newton method which is more time-consuming than the others, so the method is not adopted. Because the total error is the sum of the errors at the individual time-steps, we compute this gradient by accumulating the value of $\nabla $E for each time-step along the trajectory. The weight change for any particular weight ($\mathrm{\Delta}{v}_{kj}(t)$) can thus be written as

_{1}is the learning-rate parameter. In Equation (24), $\frac{\partial E(t)}{\partial {v}_{kj}(t)}$ can be written as

#### 2.4. Heuristic Construction of ANFIS

_{ji}and σ

_{ji}are the antecedent parameters; N is the number of inputs; and M

_{i}is the number of the fuzzy membership functions of input i.

_{p}is the weighted value; and P is the number of rules.

_{pi}represents the consequent parameters; and x

_{0}is equal to 1.

_{ji}, σ

_{ji}) and consequent parameters (linear parameters: r

_{pi}), and the model structure is determined by setting the number of membership functions in the input layer and the number of nodes in the rule layer. The parameters can be solved by the steepest gradient descent method and Newton’s method, for example. However, the methods would be slow and would produce a worse convergence and drop-in local optimum if the searching problem was more complex. To decrease the time for model construction in obtaining the best network structures and parameters, this study constructs ANFIS using hybrid algorithms including subtractive clustering (SC) and a least square estimator (LSE). The input and output vectors were first classified by subtractive clustering before training the model. The number of clusters obtained from the classification was set as the number of membership functions for node fuzzification at the various input layers and the number of nodes at the rule layers. After determining the network structures, the center point and standard deviation of each cluster were taken as the initial parameters of the input layer membership functions (Gaussian function). The training data were then fed into the network with the consequent linear parameter set and the antecedent nonlinear parameter set solved by the least squares estimator and the gradient steepest descent method, respectively. The corresponding algorithm flowchart of the model construction is shown in Figure 7. The network structure significantly reduces the time required to retrieve the optimal number of fuzzy membership functions, number of rules, and network parameters; the optimal network structure and parameters can be obtained after simply setting the adjacent radius in subtractive clustering between 0 and 1 (Jang, 1993 [31]).

_{i}), and the one with the highest density (D

_{c}

_{1}) is selected as the first cluster center (x

_{c}

_{1}). The definition of the density measure is then modified to select the next cluster center. Setting that x

_{ck}is the cluster center selected at the kth round, and the corresponding density measure is D

_{ck}, the modified formula is as follows:

#### 2.5. Analysis of Temporal and Spatial Forecasted Error Feature of the Developed Long Lead-Time Models

#### 2.6. Output Sensitivity Analysis of Single or Combined Heuristic Inputs Due to Future Forecasted Uncertainty

_{0-MP}), the delay to the end of the direct runoff (T

_{0-DRE}), and the delay to the end of the water retreat in the typhoon event (T

_{0-EE}). When such heuristic information is coupled with the input of the heuristic model, it is possible that unexpected errors will be generated on the forecast output of the model. Thus, in order to evaluate the feasibility, applicability, and accuracy of the heuristic model for real-time forecasting, sensitivity analysis was conducted on the effects on the output when forecast errors exist in the heuristic input of the most optimal heuristic model. The above analysis was used to judge whether the forecast accuracy of the heuristic model was better than that of the non-heuristic model for real-time forecasting when errors exist in the input. The expression for the analysis is as shown below:

## 3. Application

#### 3.1. Study Area

^{2}. The main stream within this area is the Dahan Creek, which is the upper stream of the Tamsui River. The effective capacity is approximately 2.098 × 10

^{8}cubic meters. The annual average rainfall in the catchment area is approximately 2350 mm, with 80% of the annual rainfall concentrated in the period between May and October. Most of the rainfall is from typhoon precipitation. The annual inflow of the Shihmen Reservoir is approximately 1.510 billion tons. The study area is shown in Figure 8.

#### 3.2. Data Used in Model Construction

**Table 1.**The adopted typhoon events for model construction among the training and validation stages.

Construction Stage | Typhoon Name | Time Period | Total Reservoir Inflow (m^{3}) | Data Number | Total Data Number |
---|---|---|---|---|---|

Training | Aere | 23–26 August 2004 | 748, 936, 728 | 58 | 459 |

Matsa | 4–6 August 2005 | 541, 872, 324 | 61 | ||

Talim | 31 August 2005–2 September 2005 | 201, 308, 580 | 33 | ||

Long-Wang | 2–3 October 2005 | 68, 596, 704 | 24 | ||

Wipha | 18–20 September 2007 | 186, 601, 752 | 48 | ||

Fung-Wong | 28–29 July 2008 | 103, 422, 564 | 33 | ||

Sinlaku | 13–16 September 2008 | 554, 322, 600 | 75 | ||

Morakot | 7–10 August 2009 | 205, 435, 980 | 71 | ||

Megi | 21–22 October 2010 | 54, 991, 728 | 37 | ||

Meari | 25 June 2011 | 44, 826, 012 | 19 | ||

Validation | Haitang | 17–20 July 2005 | 237, 416, 256 | 53 | 211 |

Sepat | 18 August 2007 | 128, 935, 224 | 20 | ||

Krosa | 6–8 October 2007 | 409, 855, 824 | 53 | ||

Jangmi | 28–30 September 2008 | 220, 301, 136 | 53 | ||

Parma | 6 October 2009 | 40, 997, 340 | 18 | ||

Fanapi | 19 September 2010 | 33, 694, 956 | 14 |

#### 3.3. Results and Discussion

#### 3.3.1. Model Inputs Selection

_{rank}: 0.926), duration from current time to the end of the flood event (r

_{rank}: 0.960), duration from current time to the time of DRE (r

_{rank}: 0.751), duration from current time to the time of maximum precipitation (r

_{rank}: 0.548), and observed-predicted inflow increase/decrease rate (r

_{rank}: 0.401). These selected variables are the most correlated inputs among all heuristic candidate predictors and the r

_{rank}value of all the selected inputs must be larger than 0.4. Furthermore, the selected non-heuristic model inputs and corresponding correlation coefficients are the observed hourly basin precipitation at the current time (r

_{rank}: 0.672), hourly reservoir inflow (r

_{rank}: 0.509), typhoon central longitude (r

_{rank}: 0.610), central wind speed (r

_{rank}: 0.650), and central pressure (r

_{rank}: 0.639); these selected variables are the most correlated inputs among all non-heuristic candidate predictors and the r

_{rank}values must all be larger than 0.5. Research conducted by Lin and Chen (2005) [34] revealed that excessive model inputs could introduce additional noise into the model, therefore 10 input factors were selected as a maximum based on the correlation coefficients. Based on the above analytical results, the developed heuristic forecast model includes 10 inputs including both heuristic and non-heuristic inputs, and the non-heuristic model only includes five non-heuristic inputs. The heuristic parameters are considered to be essential for use in eliminating the forecasting uncertainty, and for characterizing future long lead-time accumulated total inflow.

#### 3.3.2. Results of Model Construction

^{3}, 30,475,270 m

^{3}, 14,429,374 m

^{3}, and 53,236,429 m

^{3}, while the CC values for the verification were 0.979, 0.876, 0.975, and 0.658, respectively. The results indicate that the respective forecast accuracy and stability of the RTRLNN-CHI and ANFIS-CHI models are significantly higher than those of the RTRLNN-NHI and ANFIS-NHI models. This shows that the proposed heuristic forecast model may be highly accurate in its forecast of the total reservoir inflow under the following conditions: (1) when the input includes key inputs such as the future accumulated total precipitation and the delays from the current moment to the key hydrology points of the hydrograph (maximum precipitation, direct runoff ending, and event recessional ending); (2) with assistance of the comprehensive simulation of the real-time observed atmospheric factors and rainfall-runoff factors of the typhoon.

Structure Parameters/Assessment Indexes | RTRLNN-CHI Model | RTRLNN-NHI Model | Structure Parameters/Assessment Indexes | ANFIS-CHI Model | ANFIS-NHI Model |
---|---|---|---|---|---|

Best node number of hidden layer | 3 | 9 | Best adjacent radius/rule number | 0.922/2 | 0.836/3 |

MAE of training (m^{3}) | 4587459 | 22430139 | MAE of training (m^{3}) | 7249160 | 59066261 |

MAE of validation (m^{3}) | 11721556 | 30475271 | MAE of validation (m^{3}) | 14429375 | 53236429 |

CC of training | 0.999 | 0.980 | CC of training | 0.998 | 0.867 |

CC of validation | 0.980 | 0.876 | CC of validation | 0.976 | 0.659 |

**Figure 10.**Training and validation results of RTRLNN-CHI-based accumulated total reservoir inflow forecast model: (

**a**) training stage; (

**b**) validation stage.

**Figure 11.**Training and validation results of ANFIS-CHI-based accumulated total reservoir inflow forecast model: (

**a**) training stage; (

**b**) validation stage.

**Figure 12.**Training and validation results of RTRLNN-NHI-based accumulated total reservoir inflow forecast model: (

**a**) training stage; (

**b**) validation stage.

#### 3.3.3. Analytical Results of Temporal and Spatial Forecasted Error Feature of the Developed Models

Forecasted Lead-Time | RTRLNN-CHI Model | RTRLNN-NHI Model | ANFIS-CHI Model | ANFIS-NHI Model |
---|---|---|---|---|

During 24 to 48 h | 4.6% | 16.7% | 7.7% | 27.1% |

During 48 to 72 h | 9.3% | 16.1% | 12.1% | 39.3% |

During 20 to 79 h | 6.3% | 15.2% | 9.2% | 31.8% |

**Figure 13.**Comparison of temporal forecasted error features of the four developed accumulated total inflow forecast models across long lead-times.

#### 3.3.4. Sensitivity Analysis Results of Output with Relation to Heuristic Inputs Due to Future Forecasted Uncertainty

**Figure 14.**Comparison of spatial forecasted error feature of the four developed accumulated total inflow forecast models for the Shihmen Reservoir with relation to the typhoon central location: (

**a**) RTRLNN-CHI model; (

**b**) RTRLNN-NHI model; (

**c**) ANFIS-CHI model; and (

**d**) ANFIS-NHI model.

**Figure 15.**Sensitivity analysis results of model output with relation to single or combined heuristic inputs due to future forecasted uncertainty.

#### 3.3.5. Construction Results of Heuristic Forecast Database for Heuristic Inputs

_{b}in Figure 16, and then the terrain factors that might affect the rainfall in the reservoir basin were identified. Then, an axis (Line M

_{1}–M

_{2}) was marked along the direction of the Snow-Capped mountain range. The second axis (Line P

_{1}–P

_{2}), was defined as being perpendicular to Line M

_{1}–M

_{2}. Using these two axes as the reference, a contour map was created of the spatial distribution of the positions of the typhoon center when the water retreated below 300 cm of the reservoir inflow for the periods when a typhoon alarm is historically issued on land until the alarm is lifted. This resulted in the elliptical distribution line (E

_{E}), from which the starting time of the forecast and the ending time of the water retreat could be determined. Regarding the model construction event, a contour map could be made for the spatial distribution of the positions of the typhoon center between the start of the rainfall and the end of the direct runoff for historical typhoon flood events. This resulted in the elliptical distribution line E

_{DRE}. Further, the elliptical distribution line E

_{MP}could be obtained from the contour map of the spatial distribution of the typhoon center when the maximum rainfall occurred in the historical typhoon flood events. The time of the maximum rainfall may be determined from this distribution line (E

_{MP}) and the contour lines of rainfall.

**Figure 16.**Construction results of heuristic forecast database for the Shihmen basin precipitation and duration characteristics curves with relation to the typhoon central location.

_{1}–P

_{2}, i.e., the perpendicular line of the Snow-Capped mountain range. Therefore, the long axes of E

_{E}and E

_{DRE}are along Line P

_{1}–P

_{2}, while the long axis of E

_{MP}was in the direction of the Snow-Capped mountain range (Line M

_{1}–M

_{2}) because whether or not strong rainfall occurred was mainly related to the angle and position between the direction from which the rain belt of typhoon entered the reservoir basin and the direction of the Snow-Capped mountain range. In addition, from a monsoon climatology point of view, Taiwan is mainly affected by southwest monsoons from mid-March to mid-September, and is predominantly under the effects of the northeast monsoon at other times. When the typhoon center was located in quadrants I and II in Figure 16, the wind field with counterclockwise rotation easily accompanied the northeast monsoon in the basin direction of the Shihmen Reservoir; when it was located in quadrants III and IV, the wind field was easily accompanied by the southwest monsoon. For typhoon invasion in quadrants I and II after mid-September and in quadrants III and IV during mid-March and mid-September, the typhoon was easily accompanied by co-existing effects of the monsoon to increase rainfall duration and precipitation. The contour lines of E

_{E}, E

_{DRE}, and E

_{MP}can be expressed with the following equations:

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Huang, C.-L.; Hsu, N.-S.; Wei, C.-C.
Coupled Heuristic Prediction of Long Lead-Time Accumulated Total Inflow of a Reservoir during Typhoons Using Deterministic Recurrent and Fuzzy Inference-Based Neural Network. *Water* **2015**, *7*, 6516-6550.
https://doi.org/10.3390/w7116516

**AMA Style**

Huang C-L, Hsu N-S, Wei C-C.
Coupled Heuristic Prediction of Long Lead-Time Accumulated Total Inflow of a Reservoir during Typhoons Using Deterministic Recurrent and Fuzzy Inference-Based Neural Network. *Water*. 2015; 7(11):6516-6550.
https://doi.org/10.3390/w7116516

**Chicago/Turabian Style**

Huang, Chien-Lin, Nien-Sheng Hsu, and Chih-Chiang Wei.
2015. "Coupled Heuristic Prediction of Long Lead-Time Accumulated Total Inflow of a Reservoir during Typhoons Using Deterministic Recurrent and Fuzzy Inference-Based Neural Network" *Water* 7, no. 11: 6516-6550.
https://doi.org/10.3390/w7116516