# Prediction of the Discharge Coefficient in Compound Broad-Crested-Weir Gate by Supervised Data Mining Techniques

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## Abstract

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_{dt}) using the computational fluid dynamics (CFD) modeling approach and soft computing models. First, CFD was applied to the experimental data and 61 compound BCW gates were numerically simulated by resolving the Reynolds-averaged Navier–Stokes equations and stress turbulence models. Then, six data-driven procedures, including M5P tree, random forest (RF), support vector machine (SVM), Gaussian process (GP), multimode ANN and multilinear regression (MLR) were used for estimating the coefficient of discharge (C

_{dt}) of the weir gates. The results showed the superlative accuracy of the SVM model compared to M5P, RF, GP and MLR in predicting the discharge coefficient. The sensitivity investigation revealed the h

_{1}/H as the most effective parameter in predicting the C

_{dt}, followed by the d/p, b/B

_{0}, B/B

_{0}and z/p. The multimode ANN model reduced the root mean square error (RMSE) of M5P, RF, GP, SVM and MLR by 37, 13, 6.9, 6.5 and 32%, respectively. The graphical inspection indicated the multimode ANN model as the most suitable for predicting the C

_{dt}of a BCW gate with minimum RMSE and maximum correlation.

## 1. Introduction

_{dt}in the compound BCW gate. All the effective parameters were changed to make different combinations of this structure. The BCW gate can be used as a flood control reservoir and a measuring device for minimizing the sediments and preventing their accumulation. An accurate prediction of the C

_{dt}in the compound BCW gate could significantly improve its operational management and, thus, water resources management.

## 2. Theoretical Background

_{1}< Z) (Figure 1b), the broad-crested weir performs like a basic weir, and there will be no flow over the compound section.

_{c}) will be lower than the central weir height (Z) and weir discharge (Q

_{w}) though the broad-crested weir can be as follows [60,61];

_{dw}denotes the weir discharge coefficient, C

_{v}denotes the weir velocity coefficient. Furthermore, h

_{1}, g, B and H

_{1}are the water height above the central weir, acceleration gravity, central weir width and head overall energy at the weir head assessment section, respectively.

_{1}) surpasses the central weir height (Z), and the weir acts as a compound weir (Figure 1c). Furthermore, the critical flow depth (y

_{c}) will be greater than the central weir height (Z). In this case, the BCW discharge (Q

_{w}) is obtained by Equation (4) [5,61]:

_{g}) can be obtained by utilizing the equation of energy, as below [62]:

_{dg}is the gate discharge coefficient and d, b and H are the gate opening height, gate breadth and level of upstream water, respectively. The investigation of the compound BCW and gate collectively for the two mentioned cases of compound BCW are elaborated below.

#### 2.1. Case 1

_{t}) is the sum of Equation (1) and Equation (7) and can be written as Equation (8). The discharge coefficient for the combined compound BCW gate (C

_{dt}) can be presented by Equation (9):

#### 2.2. Case 2

_{t}) (Figure 2c) is the sum of Equation (4) and Equation (7):

#### 2.3. Governing Equations and Numerical Method

_{x-z}, f

_{x-z}and b

_{x-z}denote body accelerations, viscous accelerations and flow losses in porous media, respectively. Furthermore, A

_{x}, A

_{y}and A

_{z}refer to the cross-sectional extent of the flow; ρ denotes the density of water, indicates the fractional volume open to flow in fractional area/volume obstacle representation (FAVOR), R

_{SOR}denotes the source term, p denotes the pressure, and the concluding terms constitute the inclusion of mass at a source signified by a geometry component. The term U

_{w}= (u

_{w}

_{,}u

_{w}, w

_{w}) is the velocity of the source component, and the term U

_{s}= (u

_{s}, v

_{s}, w

_{s}) is the velocity at the surface of the source relative to the source itself. R in Equation (12) is the coefficient that depends on the choice of coordination system.

_{e}normalization groups with the equation of stress), one equation model, large-eddy simulation (LES) and the Prandtl mixing-length model [69].

## 3. Methods

_{1}/L ≤ 0.35 (where h

_{1}and L refer to the hydraulic head on the weir’s central section and the compound weir’s length, respectively) to establish the physical and arithmetic design [70]. The flow separation was decreased by rounding with a radius of 0.065 m at the weir entrance to develop physical and numerical models of the weir [5]. Figure 3 shows the geometry of the experimental study conducted by Salmasi et al., [12].

_{K}and G

_{b}stand for the generation of turbulent kinetic energy arising because of average velocity gradients and buoyancy, separately, Y

_{M}is the fluctuating dilation in compressible turbulence, S

_{ε}and S

_{K}are the source terms set by the user, ${\alpha}_{k}$ and ${\alpha}_{\epsilon}$ are inverse effective Prandtl numbers for the turbulent kinetic energy and its dissipation, and ${c}_{1\epsilon}$, ${c}_{2\epsilon}$ and ${c}_{3\epsilon}$ are constants. Furthermore, the ${R}_{\epsilon}$ refers to the main difference between the numerical models used.

_{dw}.

_{dw}and can be used to simulate a combined compound BCW gate. The root mean square (RMSE) values were 0.2685 for k-w against 0.3122 for RNG, when the k-w model was used for the simulation of the combined compound BCW-gate structure by CFD code.

_{0}, L, P + Z), central weir height and width (Z and B) and gate opening height and width (d and b). For upstream water level (H), different values were considered and two types of compound weir performance (simple weir and compound weir (Figure 2) were studied. Considering the mentioned parameters, the results are presented using dimensionless parameters. The structure’s total section width (B

_{0}), the height of the structure (Z + P) and combined structure length (L) were considered as 0.25, 0.25 and 0.4, respectively. Table 2 shows the studied dimensionless parameters and values.

_{min}and X

_{max}boundary conditions. Y

_{min}, Y

_{max}and Z

_{min}were selected as boundary conditions. Furthermore, the symmetry was the Z

_{max}boundary condition. Figure 5 shows one of the used models and its boundary conditions.

_{dt}) were generated using the simulation results for the two mentioned cases. Using the generated C

_{dt}values, six soft computing-based models, including RF, M5P, SVM, GP, MLP and multimode ANN, were used to predict C

_{dt}. The mentioned dimensionless parameters were the inputs of the soft computing models.

#### 3.1. Soft Computing Models and Artificial Intelligence Techniques

#### 3.1.1. M5P Model

#### 3.1.2. Random Forest (RF)

#### 3.1.3. Gaussian Process (GP)

#### 3.1.4. Support Vector Machine (SVM)

#### 3.1.5. Multiple Linear Regression (MLR)

_{1}, x

_{2},…

_{,}x

_{n}stand for independent variables. MLR was developed in this study using XLSTAT software.

#### 3.1.6. Artificial Neural Networks (ANN)

## 4. Application of the Methods

_{0}, z/p, B/B

_{0}and h

_{1}/H whereas the C

_{dt}of the BCW gate was considered output.

## 5. Results and Discussion

_{dt}of the BCW gate using soft computing approaches are presented against the modeled (real) values in the figure to show their performance based on the best fit line (y = x).

_{dt}with CC of 0.9585, NSE of 0.8562, MAE of 0.0237, RMSE of 0.0276 and SI of 0.0337 during testing. The SVM reduced the RMSE of M5P, RF, GP and MLR by 32, 7.1, 0.4 and 27%, respectively. However, the difference between GP and SVM was minor. Figure 7 shows that the SVM-estimated C

_{dt}during training and testing was the closest to the best-fit line relative to other models. It confirmed the better performance of SVM compared to the other models. The outcomes of single-factor ANOVA presented in Table 6 show insignificant differences between theactual and predicted values for the different models.

#### 5.1. Sensitivity Investigation

_{1}/H is the most influencing parameter in predicting the C

_{dt}of BCW gate, followed by d/p, b/B

_{0}, B/B

_{0}and z/p.

#### 5.2. Results of Multimode ANN Model

_{dt}values were the output. The multimode ANN model was developed using a hit-and-miss process. Figure 8 indicates the structure of the novel multimode model in which five neurons in the input layer and four neurons in the latent layer were selected for predicting C

_{dt}of BCW gate.

_{dt}using novel multimode ANN models during the training and testing stages. The novel multimode ANN estimated C

_{dt}with CC = 0.9998 and 0.9618, RMSE of 0.0016 and 0.0258, MAE of 0.0013 and 0.0217, NSE as 0.9996 and 0.8746 and SI of 0.0020 and 0.0315 during training and testing, respectively. Overall, the performance of the novel multimode ANN was better than the RF, M5P, GP, SVM and MLR models in predicting C

_{dt}of BCW gate. Comparison with Table 5 revealed that the novel multimode ANN reduced the RMSE of M5P, RF, GP, SVM and MLR by 37, 13, 6.9, 6.5 and 32%, respectively.

_{dt}of a BCW gate.

_{dt}. Three statistical parameters involving RMSE, CC and standard deviation were used to evaluate the applied models’ accuracy in the Taylor diagram. Figure 11 indicates that the multimode ANN model attained a higher CC with minimal RMSE. The Taylor diagram also confirmed that the performance of the multimode ANN model surpassed the other applied models.

## 6. Conclusions

_{dt}) of a combined compound rectangular broad-crested weir (BCW) gate. For this purpose, the experimental results of a simple broad-crested weir were employed for the code’s calibration, and the k-w model was selected for numerical simulation. The effective parameters of the combined structure were changed for a comprehensive investigation. The studied dimensionless parameters were: b/Bo, d/P, B/Bo, Z/P and h

_{1}/H. A total of 61 compound BCW gates were numerically simulated using different values of the dimensionless parameters. Finally, the results of the calibrated CFD code were used to develop models for the prediction of a compound BCW-gate C

_{dt}. Six data-driven algorithms, including M5P tree, RF, Gaussian process, SVM, MLR and multimode ANN, were used.

_{1}/H as the most effective parameter, followed by d/p, b/B

_{0}, B/B

_{0}and z/p, in predicting C

_{dt}using SVM. The novel multimode ANN model outperformed all other models. It reduced the RMSE by 37, 13, 6.9, 6.5 and 32% of the M5P, RF, GP, SVM and MLR, respectively. The Taylor diagram and box plot also confirmed the novel multimode ANN model as the most suitable model in predicting the C

_{dt}of a BCW gate with minimum errors and maximum correlation.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**(

**a**) Definition sketch of compound BCW: (

**b**,

**c**) two diverse flow cases through the compound BCW segment.

**Figure 2.**(

**a**) Sketch for simultaneous flow over compound BCW and below gate: (

**b**,

**c**) are two diverse flow cases through the combined compound BCW gate.

**Figure 3.**Single broad-crested weir studied by Salmasi et al., [12].

**Figure 4.**Estimated C

_{dw}using numerical models; (

**a**) Broad crested weir with B/B

_{0}= 0.24, (

**b**)Broad crested weir with B/B

_{0}= 0.32.

**Figure 5.**Combined compound BCW gate and boundary conditions used; X

_{min}: P (specified pressure); X

_{max}: O (outflow); Y

_{min}, Y

_{max}and Z

_{min}: W (wall); Z

_{max}: S (symmetry).

**Figure 7.**Performance of M5P, RF, GP, SVM and MLR models in predicting C

_{dt}of BCW gate during model calibration and validation.

**Figure 9.**Agreement plot of multimode ANN prediction with observed C

_{dt}of BCW gate during training and testing.

**Table 1.**Variables and their range used in the study of Salmasi et al., [12].

Parameters | L (cm) | B_{0} (cm) | B (cm) | p (cm) | Z (cm) |
---|---|---|---|---|---|

Range of variables | 40 | 25 | 6, 8 and 12 | 10, 13 and 16 | 9 |

Parameters’ Values | Dimensionless Parameters |
---|---|

d/P | 0.36, 0.43, 0.52, 0.62 |

b/B_{0} | 0.28, 0.4 |

Z/P | 0.31, 0.56 |

B/B_{o} | 0.24, 0.32, 0.48 |

h_{1}/H | 0.15, 0.2, 0.33, 0.42, 0.45, 0.54 |

d/P | 0.36, 0.43, 0.52, 0.62 |

Range | Training Data Set | |||||
---|---|---|---|---|---|---|

d/p | b/B_{0} | Z/P | B/B_{0} | h_{1}/H | C_{dt} | |

Mean | 0.4657 | 0.3502 | 0.4722 | 0.3629 | 0.3332 | 0.7945 |

Median | 0.4375 | 0.4000 | 0.5625 | 0.3200 | 0.3333 | 0.8051 |

Standard deviation | 0.0996 | 0.0599 | 0.1203 | 0.1045 | 0.1309 | 0.0800 |

Kurtosis | −0.9827 | −1.9691 | −1.7536 | −1.7811 | −1.2383 | −0.1968 |

Skewness | 0.7604 | −0.3599 | −0.5784 | 0.0931 | 0.0572 | −0.6304 |

Minimum | 0.3684 | 0.2800 | 0.3158 | 0.2400 | 0.1579 | 0.6155 |

Maximum | 0.6250 | 0.4000 | 0.5625 | 0.4800 | 0.5429 | 0.9303 |

Confidence level (95.0%) | 0.0314 | 0.0189 | 0.0380 | 0.0330 | 0.0413 | 0.0252 |

Testing Data Set | ||||||

Mean | 0.4809 | 0.3520 | 0.4885 | 0.3600 | 0.3936 | 0.8188 |

Median | 0.4375 | 0.4000 | 0.5625 | 0.3200 | 0.4286 | 0.8329 |

Standard deviation | 0.1039 | 0.0603 | 0.1160 | 0.1054 | 0.1251 | 0.0748 |

Kurtosis | −1.4170 | −2.0180 | −1.2418 | −1.8330 | −0.5960 | −1.2048 |

Skewness | 0.5357 | −0.4421 | −0.9453 | 0.1533 | −0.4966 | −0.2163 |

Minimum | 0.3684 | 0.2800 | 0.3158 | 0.2400 | 0.1579 | 0.6915 |

Maximum | 0.6250 | 0.4000 | 0.5625 | 0.4800 | 0.5429 | 0.9235 |

Confidence level (95.0%) | 0.0486 | 0.0282 | 0.0543 | 0.0493 | 0.0586 | 0.0350 |

Approaches | Kernel Function | User-Defined Parameters |
---|---|---|

M5P | m = 4 | |

RF | m = 1 & K = 10 | |

SVM | RBF kernel | C = 2, $\gamma $ = 1 |

GP | RBF kernel | Gaussian noise = 0.01, $\gamma $ = 1 |

Approaches | CC | MAE | RMSE | Nash | SI |
---|---|---|---|---|---|

Training Data Set | |||||

M5P | 0.9550 | 0.0171 | 0.1492 | 0.9121 | 0.1878 |

RF | 0.9867 | 0.0106 | 0.0135 | 0.9706 | 0.0170 |

GP | 1.0000 | 0.0007 | 0.0009 | 0.9999 | 0.0012 |

SVM | 1.0000 | 0.0003 | 0.0005 | 1.0000 | 0.0007 |

MLR | 0.9465 | 0.0183 | 0.0255 | 0.8959 | 0.0320 |

Testing data set | |||||

M5P | 0.9148 | 0.0208 | 0.0408 | 0.8354 | 0.0498 |

RF | 0.9187 | 0.0240 | 0.0297 | 0.8339 | 0.0363 |

GP | 0.9581 | 0.0239 | 0.0277 | 0.8557 | 0.0338 |

SVM | 0.9585 | 0.0237 | 0.0276 | 0.8562 | 0.0337 |

MLR | 0.8549 | 0.0284 | 0.0380 | 0.7280 | 0.0464 |

Sr No. | Method | F | p-Value | F Crit | Variation between Groups |
---|---|---|---|---|---|

1 | M5P | 0.00483 | 0.944955 | 4.098172 | Insignificant |

2 | RF | 0.017288 | 0.896086 | 4.098172 | Insignificant |

3 | GP | 0.132998 | 0.717366 | 4.098172 | Insignificant |

4 | SVM | 0.124141 | 0.726533 | 4.098172 | Insignificant |

5 | MLR | 0.028042 | 0.867899 | 4.098172 | Insignificant |

Input Combination | Removed Parameter | Statistical Parameters (Testing Data Set) | ||||||
---|---|---|---|---|---|---|---|---|

d/p | b/B_{0} | z/p | B/B_{0} | h_{1}/H | CC | MAE | RMSE | |

None | 0.9585 | 0.0237 | 0.0276 | |||||

d/p | 0.6472 | 0.0456 | 0.0580 | |||||

b/B_{0} | 0.8553 | 0.0292 | 0.0381 | |||||

z/p | 0.9449 | 0.0212 | 0.0264 | |||||

B/B_{0} | 0.9006 | 0.0254 | 0.0330 | |||||

h_{1}/H | 0.3575 | 0.0671 | 0.0743 |

Statistic | M5P | RF | GP | SVM | MLR | Novel Multimode ANN |
---|---|---|---|---|---|---|

Minimum | −0.0460 | −0.0470 | −0.0420 | −0.0420 | −0.0468 | −0.0400 |

Maximum | 0.0960 | 0.0750 | 0.0500 | 0.0510 | 0.1130 | 0.0470 |

First Quartile | −0.0170 | −0.0295 | −0.0140 | −0.0140 | −0.0242 | −0.0118 |

Median | −0.0005 | −0.0045 | 0.0110 | 0.0095 | 0.0008 | 0.0090 |

Third Quartile | 0.0170 | 0.0153 | 0.0285 | 0.0278 | 0.0242 | 0.0263 |

Mean | 0.0016 | −0.0029 | 0.0076 | 0.0073 | 0.0037 | 0.0065 |

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## Share and Cite

**MDPI and ACS Style**

Nouri, M.; Sihag, P.; Kisi, O.; Hemmati, M.; Shahid, S.; Adnan, R.M. Prediction of the Discharge Coefficient in Compound Broad-Crested-Weir Gate by Supervised Data Mining Techniques. *Sustainability* **2023**, *15*, 433.
https://doi.org/10.3390/su15010433

**AMA Style**

Nouri M, Sihag P, Kisi O, Hemmati M, Shahid S, Adnan RM. Prediction of the Discharge Coefficient in Compound Broad-Crested-Weir Gate by Supervised Data Mining Techniques. *Sustainability*. 2023; 15(1):433.
https://doi.org/10.3390/su15010433

**Chicago/Turabian Style**

Nouri, Meysam, Parveen Sihag, Ozgur Kisi, Mohammad Hemmati, Shamsuddin Shahid, and Rana Muhammad Adnan. 2023. "Prediction of the Discharge Coefficient in Compound Broad-Crested-Weir Gate by Supervised Data Mining Techniques" *Sustainability* 15, no. 1: 433.
https://doi.org/10.3390/su15010433