Support vector machine (SVM) approach to develop the discharge prediction model for triangular labyrinth weir

C_d

coefficient of discharge

p

weir height

W

width of the channel

h/p

head to weir height ratio

F_r

froude number

L

effective length of the weir

θ

vertex angle

N

number of cycles

H

total head

W/p

vertical aspect ratio

Gentilini (Hager et al. 2015) conducted the first experiment on the sharp-crested oblique weir and triangular labyrinth weir (obtained by connecting an even number of oblique weirs) having vertex angles 60°, 90°, 120°. He changed channel width, crest length, number of oblique weirs, and orientation (normal and inverted) and plotted the graphs between the ratio of discharge coefficients (C_d/C_dn) of the labyrinth weir to the normal weir and the ratio of head over the weir to channel width (h/W), which indicate that the data points corresponding to the same side wall angle lie on the same curve irrespective of their plan shape. Hay & Taylor (1970) extensively investigated the performance of labyrinth weirs of different plan forms like triangular, rectangular, and trapezoidal. They defined the ratio of discharge over labyrinth weir Q to the discharge over normal weir Q_n operating under similar hydraulic conditions as magnification ratio. They presented the results in the form of magnification ratio (Q/Q_n) versus head-to-weir height ratio (h/p) and found that the discharge magnification decreases with an increase in h/p.

Darvas (1971) introduced an empirical discharge equation along with a trapezoidal labyrinth weir design chart. Hinchliff & Houston (1984) developed design guidelines based on the model study of the Ute dam and Hyrum dam. Lux (1984) proposed a different discharge coefficient equation, including vertical aspect ratio (w/p) and shaped constant k. Based on the experimental investigations, Tullis et al. (1995) proposed a procedure for designing a trapezoidal labyrinth weir. They also proposed the discharge coefficient equations as a fourth-order polynomial of h/p for different side wall angles. Willmore (2004) also developed equations for C_d similar to that of Tullis et al. (1995). Tullis et al. (2007) proposed a dimensionless submerged head relationship for the trapezoidal labyrinth weir.

Kumar et al. (2011) carried out an experimental study on a single-cycle sharp-crested triangular labyrinth (TL) weir having different vertex angles (θ = 30°, 60°, 90°, 120°, 150°, and 180°) and observed that the interference of nappe is more pronounced at the least apex angle. They proposed a fourth-order polynomial fit equation for C_d as a function h/p for different vertex angles. Gupta et al. (2014) extended the work of Kumar et al. (2011), including three more vertex angles (θ = 75°, 105°, 135°), and proposed another similar equation for C_d. It is worthwhile to note that using a polynomial of degree n for labyrinth weirs having m configurations (different side wall angles) requires m*(n + 1) coefficients leading to a complex form of equation for estimation of discharge coefficient.

Bijankhan & Kouchakzadeh (2017) tested a single-cycle triangular labyrinth weir having vertex angles of (74°, 90°, 120°, and 180°). They developed a criterion to distinguish between free and submerged flow conditions experimentally. They further proposed a mathematical shape for the submerged stage-discharge formula based on the dimensional analysis and self-similarity concept. They argued that the proposed rating curve may be used for free and submerged flow conditions continuously and within the transition zone. Finally, they deduced an expression for the discharge coefficient from the stage-discharge relationship. However, so many coefficients are involved in determining the discharge coefficient.

Some of the well-known predictors are summarized in Table 1. The equations developed for C_d are applicable only in the range of data covered in a particular study. Therefore, it is necessary to develop a simple and accurate equation to compute C_d covering a wide range of data available in the literature.

Table 1

Discharge coefficient predictors reported by various investigators

Author .	Proposed equation .	Remarks .	Weir type .	Crest Type .
Ghodsian (2009)		0.3 ≤ h/p ≤ 0.7	Triangular- 2 cycles	SC-QR-HR-FT
Kumar et al. (2011)		0	Triangular plan form	SC
Gupta et al. (2014)		0< h/p < 0.7	Triangular labyrinth weir	S.C.
Karami et al. (2017)			Triangular plan form	SC
Bonakdari et al. (2020)		y = upstream flow depth	Triangular plan form	SC

Author .	Proposed equation .	Remarks .	Weir type .	Crest Type .
Ghodsian (2009)		0.3 ≤ h/p ≤ 0.7	Triangular- 2 cycles	SC-QR-HR-FT
Kumar et al. (2011)		0	Triangular plan form	SC
Gupta et al. (2014)		0< h/p < 0.7	Triangular labyrinth weir	S.C.
Karami et al. (2017)			Triangular plan form	SC
Bonakdari et al. (2020)		y = upstream flow depth	Triangular plan form	SC

Note: S.C. – Sharp crested, Q.R. – Quarter round, H.R. – Half round, F.T. – Flat top.

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Recent advancements in soft computing (genetic programming, neural networks, fuzzy genetic programming, decision support systems, neuro-fuzzy systems, support vector machines, etc.) attracted researchers to use them in the complex flow phenomenon over a labyrinth weir and to develop discharge coefficient predictions.

Laboratory data collected by Kumar et al. (2011) for one cycle triangular labyrinth weir has been used by many researchers for the development of a discharge coefficient model based on soft computing techniques. Zaji et al. (2015) estimated the discharge coefficient employing multiple non-linear particle swarm optimization (MNLPSO), multiple linear particle swarm optimization (MLPSO), and radial basis neural network (RBNN) and found that MLPSO performs well. Zaji et al. (2016) applied a firefly optimization algorithm (FFA) to the discharge coefficient prediction model for better accuracy. Parsaie & Haghiabi (2017) reported that the SVM-based model performed better than the RBNN and MLPNN. Karami et al. (2017) showed that support vector regression (SVR)-Firefly yielded better results as compared to the classic SVR, principal component analysis (PCA), and response surface methodology (RSM). Karami et al. (2018) developed the discharge coefficient model using genetic programming (GP), artificial neural network (ANN), and extreme learning machine (ELM) and showed that the ELM model performed the best. Haghiabi et al. (2018) employed multilayer perceptron (MLP) and ANFIS to predict C_d, and they found that the structure of ANFIS model is more optimal than that of MLP model. Bonakdari et al. (2020) proposed an equation for C_d using non-linear regression (NLR) and modeled the data with gene expression programming (GEP) to predict C_d. The results of GEP were more capable than NLR. Mahmoud et al. (2021) estimated the flow rate of a sharp-crested triangular labyrinth weir as a function of head-to-weir height ratio h/p and sidewall angle (θ) through several soft computing tools such as the adaptive neuro-fuzzy inference system (ANFIS), MLP, SVR and radial basis function neural network (RBFNN). Emami et al. (2022) used the walnut optimization algorithm employing five parameters (L/h, Fr, L/p, θ, h/p) and found the model containing L/p, h/p, and θ predicts the best value of C_d. Table 2 summarizes the performance of the soft computing technique for dataset 1. The performance of the SVR (Firefly algorithm) prediction model developed by Karami et al. (2017) is the best.

Support vector machine (SVM) has also successfully employed in various fields of hydraulic engineering such as determination of discharge coefficient of weir (Haghiabi et al. 2017; Azamathulla et al. 2019), evapotranspiration (Choubin et al. 2019), sediment transport (Choubin et al. 2018; Kisi et al. 2019) and rainfall modelling and river flow prediction (Azamathulla & Wu 2011), scour prediction on grade control structures (Goel & Pal 2009).

Nonetheless, the complex three-dimensional flow pattern over the labyrinth weir makes it challenging to obtain an accurate analytical solution to the head discharge relationship (Crookston & Tullis 2012). As a result, soft computing techniques could be considered an appropriate alternative. The performance of the SVM in the discharge prediction model has been reported as the best among the other commonly used soft computing techniques.

The main objective of the present study is, therefore, to (i) assess the accuracy of SVR in the development of a discharge coefficient prediction model for a triangular labyrinth weir using a sizeable amount of laboratory data available in the literature (ii) to obtain a simple discharge coefficient equation using the same data based on the non-linear regression (NLR) approach for field application (iii) to compare the accuracy of the discharge coefficient models obtained in the present study with those reported in the literature, and (iv) to study the influence of individual parameters on discharge coefficient using sensitivity analysis.

The discharge prediction model for the triangular labyrinth weir was developed based on the conventional approach of regression analysis and the most popular soft computing approach using the laboratory data obtained from the literature as described herein.

Laboratory data collected under free flow conditions by Kumar et al. (2011), Gupta et al. (2014), and Bijankhan & Kouchakzadeh (2017) has been procured from the literature and grouped under four sets, as shown in Table 3. The range of the pertinent parameters is shown in Table 4.

The discharge coefficient basically depends on geometric and flow parameters. Two NLR models of the following form have been adopted in the present study in Table 5.

Table 5

Details of discharge prediction models adopted in the present study

Models .	Model details .	Remarks .
RM1		ai, bi, and ci are coefficients, and mi is the exponents. η = h/p; θ = vertex angle; β = W/P; W = channel width, P = weir height, h = total head
RM2

Models .	Model details .	Remarks .
RM1		ai, bi, and ci are coefficients, and mi is the exponents. η = h/p; θ = vertex angle; β = W/P; W = channel width, P = weir height, h = total head
RM2

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The coefficients and exponents of these models may be estimated using the observed set of data through NLR analysis.

introducing slack variables as into Equation (2) to make the constraints feasible, the problem reduces to estimate w and b that minimise the following function:

Discharge coefficient prediction models for a triangular labyrinth weir using a sizeable amount of laboratory data (260) available in the literature have been developed based on non-linear regression (Minitab Software) and SVR (DTREG Software). C_d was considered as a dependent variable against the independent variables as h/p, θ, N, and W/p for the SVR model and h/p, θ, and N in the regression model. The results of the analysis are discussed in the following sections.

The performance of these regression models for the entire data has been evaluated and shown in Table 6. This table indicates that performance of RM1 is better than that of RM2.

The predicted to observed discharge coefficients ratio (PODCR) have been compared in the box-and-whisker diagrams, as shown in Figure 2. It may be clearly observed that the scatter for the RM1 is less than that of RM2, although the mean values for both models are nearly one. This indicates that the prediction of Model 1 is more satisfactory as compared to RM2.

Different forms of discharge coefficient predictors have been reported by various investigators (Table 1). Five possible prediction models, as provided in Table 8, were considered herein to test the best model for the present set of data._. The performance of each model is assessed through statistical performance indices. Table 9 shows the performance indices of the proposed various SVR prediction models. A perusal of this table indicates that the SVR3 model containing three geometric parameters, h/p, θ, and N, produce the most accurate results. Further, it can be observed that SVR3 gives the highest accuracy with training R² = 0.997 and testing R² = 0.994 with the Hyper parameters' gamma = 15.018, C = 1.408, and epsilon = 0.001. It is further observed that the performance of SVR3, SVR4, and SVR5 are also fairly comparable.

A comparison of existing discharge coefficient predictors for dataset 1 has been made to validate the present approach. Dataset 2, containing the entire data (260), has been used for the development of a discharge coefficient prediction model based on regression and the SVR approach.

SVR and NLR models for the discharge coefficient of a TL weir have been developed in the present study using a sizeable amount of laboratory data available in the literature. The performance of these models has been assessed based on statistical performance indices such as mean absolute percentage error (MAPE) and mean square error (MSE). It was found that the SVR model performs better than the NLR model. However, a simple discharge coefficient equation obtained from the NLR approach may be used for field application. The sensitivity analysis indicates that the parameters h/p have the most, θ moderate, and W/p the least influence on the discharge coefficient. However, the influence of N is also almost in the range of W/p. The results of the accuracy assessment of various discharge coefficients prediction models for the prototype data indicate that the performance of RM1 and models proposed by Karami et al. (2017) are inaccurate; that of RM2 in the present study and Kumar et al. (2011) are well predicted, and performance of Ghodsian (2009) prediction models is reasonable on the basis of scale of the judgment of predicted accuracy (Lewis scale 1982). The inaccuracy of the present RM1 may be attributed to the involvement of insufficient size of cycle (N) of the LW in the proposed equation. A detailed investigation of the effect of N on the discharge coefficient is recommended for future research.

All relevant data are available from an online repository or repositories.

The authors declare there is no conflict.