Novel application of robust GWO-KELM model in predicting discharge coefficient of radial gates: a field data-based analysis

Radial gates are known as the most common elements of water control structures in use today and can be found in most irrigation networks across the world. They are constructed by the use of a curved skin plate supported by a structural steel frame and play an essential role in facilitating water delivery, maintaining river environments, and controlling flooding. These types of gates are more cost-effective and simpler to utilize and maintain in comparison to sluice gates (Clemmens et al. 2003; Shahrokhnia & Javan 2006). The performance accuracy of radial gates as check structures can vary according to the type of flow condition (i.e., free or submerged flow). Under a free-flow condition, a free hydraulic jump is developed in the downstream part of the radial gate, and the flow is just affected by the upstream flow depth. On the other hand, with increasing tail-water depth, the jet emanating from the radial gate changes to partially submerged flow with an incomplete submerged jump and totally submerged flow with no jump. The discharge coefficient of gates constitutes the basis for the determination of the opening, which is of considerable importance for accurate simulation and controlling of canal flow. Furthermore, accurate determination of discharge is crucial for informing water-saving policies and efficient operational management. Numerous studies have attempted to enhance the discharge calibration of radial gates. Early work can be attributed to Metzler (1948), who developed a series of maps to determine the discharge coefficient of radial gates under submerged flow conditions. Toch (1955) and Buyalski (1983) applied the empirical relationship method in order to illustrate the variation of the discharge coefficients for both free and submerged flow conditions. Clemmens et al. (2003) acknowledged the inefficiency of empirical relationship methods in calibrating submerged radial gates and developed the Energy-Momentum (E-M) approach in order to calibrate the discharge through a single radial gate. Using Buyalski's experimental data, Wahl (2005) introduced relative gate opening to modify the correction factor. Shahrokhnia & Javan (2006) and Zahedani et al. (2012) derived equations for quantifying the submerged flow rate on the basis of the downstream flow depth using dimensional analysis and nonlinear regression. Abdelhaleem (2016) conducted research on a discharge estimation of three submerged parallel radial gates and found that the dimensional analysis method with the incomplete self-similarity concept excels any other calibration methods. The practical implementation of the aforementioned discharge calibration methods indicated inadequate performance when subjected to highly submerged flow. More recently, Guo et al. (2021a, 2021b, 2021c) proposed new criteria in order to subdivide the submerged flow into partially submerged and totally submerged flow. Using a new discharge calibration method, expressed as the identification method, they asserted that considering the classification method can satisfactorily enhance the discharge model accuracy for totally submerged flow.

Based on the aforementioned discussion, assessment of the hydraulic operation of radial gates can be implemented following different approaches: the energy balance approach, the momentum approach, and empirical equations. The literature demonstrates that employed approaches to the analysis of radial gates are unsuccessful in providing stable performance under various hydraulic conditions and may lead to unsatisfactory results. In such cases, machine learning (ML) methods can be a suitable tool to build hydraulic models of radial gates solutions under different hydraulic conditions. The high efficiency and speed of ML methods in modeling complex hydraulic problems have made them one of the most widely used methods in predicting the discharge coefficient of different types of weirs. Among them, the strong suitability of the Artificial Neural Network (ANN) algorithm for hydraulic problems has made it one of the leading models in the prediction of the discharge coefficient of weirs (Bilhan et al. 2011; Salmasi et al. 2013; Karami et al. 2018). However, the need to define the structure of ANN before training, as well as the possibility of getting trapped in local minima in the model training process, are among the reasons that encourage researchers to use an appropriate combination of neural and fuzzy systems in the form of Adaptive Neural Fuzzy Inference Systems (ANFIS) (Shamshirband et al. 2016; Haghiabi et al. 2018). On the other hand, the use of the kernel function as the core tool of ML methods increases their efficiency in terms of reducing overfitting. In this regard, the successful application of kernel-depend Support Vector Machine (SVM) (Azamathulla et al. 2016; Roushangar et al. 2021a, 2021b) and Gaussian Process Regression (GPR) (Akbari et al. 2019; Nourani et al. 2021) were reported in the literature. Kernel-depend methods are memory intensive, and trickier to tune due to the importance of picking the right kernel. In these models, we will be able to predict the proper behavior of the system, although we will not be able to characterize its intrinsic structure and behavior. A kernel method is an algorithm that depends on the data only through dot-products. When this is the case, the dot product can be replaced by a kernel function which computes a dot product in some possibly high-dimensional feature space (Smola 1996). This has two advantages: First, the ability to generate nonlinear decision boundaries using methods designed for linear classifiers. Second, the use of kernel functions allows the user to apply a classifier to data that have no obvious fixed-dimensional vector space representation.

Prediction of the discharge coefficient is required as a basis for discharge measurements in irrigation channels and plays a fundamental role in evaluating the hydraulic efficiency of radial gates. To the best of the authors’ knowledge, ML techniques have not been developed in design of the discharge coefficient of radial gates with an extensive number of field datasets. Therefore, the authors are eager to evaluate the ability of powerful kernel-depend approaches in the prediction of the discharge coefficient of radial gates. As major advantages, kernel-depend models have two main capabilities: (i) these methods can model nonlinear decision boundaries, and there are many kernels to choose from. They are also robust against overfitting, especially in high-dimensional space, (ii) in these models, we will be able to predict the proper behavior of the system, although we will not be able to characterize its intrinsic structure and behavior. Considering the advantages of kernel-depend methods, for the first time, Kernel Extreme Learning Machine (KELM), in combination with Grey Wolf Optimization (GWO), is developed for modeling the discharge coefficient of radial gates under free and submerged flow conditions. The performances of the proposed hybrid KELM-GWO model were compared with GPR and SVM that developed to this end, as well.

The rest of this paper is organized as follows. First, an extensive field database and the location of studied radial gates are presented. The development of dimensionless parameters under different hydraulic conditions is presented along with the corresponding histograms. After that, brief reviews of the theorem associated with the SVM, GPR, KELM, and GWO algorithms are presented, and consequently, the structure of the proposed KELM-GWO model and its optimization strategy is described. Next, the results are presented in terms of qualitative and quantitative performance. The conclusions are drawn in the final step of this research.

In which Q is the flow discharge, y₁ is the upstream flow depth, y₃ is the downstream flow depth, L is the channel width, w is the gate opening, is the water viscosity and ρ is the water density. Dimensional analysis may reduce the dimensions of the input matrix, so would create a low-dimensions space where the number of the studied parameters is low. Moreover, further application of experimental results can be achieved through the dimensionless groupings of variables (Granata & de Marinis 2017).

Selected input combinations for the prediction of discharge coefficient under both free and submerged conditions were extracted through a large number of trial-and-error processes and listed in Table 3. Statistical information of input and output parameters in terms of the maximum (max), minimum (min), mean and standard deviation (S.td.) values is represented in Table 4.

Table 3

Selected input combinations for prediction of discharge coefficient of radial gates

Free flow .		Submerged flow .
Model .	Input parameters .	Model .	Input parameters .
F(I)		S(I)
F(II)		S(II)
F(III)		S(III)
F(IV)		S(IV)

Free flow .		Submerged flow .
Model .	Input parameters .	Model .	Input parameters .
F(I)		S(I)
F(II)		S(II)
F(III)		S(III)
F(IV)		S(IV)

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Table 4

The statistical details of the input and output parameters

.	.	Parameter .	Min .	Max .	Mean .	St.d. .
Free flow	Input parameters		0.255	0.559	0.418	0.063
			0.065	0.788	0.442	0.152
			1.54	64.55	6.06	8.50
			0.011	0.315	0.115	0.068
	Output parameter		0.432	0.684	0.597	0.047
Submerged flow	Input parameters		−0.949	28.5	2.37	4.58
			0.568	1.76	0.776	0.299
			0.299	0.845	0.584	0.123
			1.09	37.03	5.02	4.59
			0.010	0.642	0.187	0.119
	Output parameter		0.005	6.14	0.616	0.376

.	.	Parameter .	Min .	Max .	Mean .	St.d. .
Free flow	Input parameters		0.255	0.559	0.418	0.063
			0.065	0.788	0.442	0.152
			1.54	64.55	6.06	8.50
			0.011	0.315	0.115	0.068
	Output parameter		0.432	0.684	0.597	0.047
Submerged flow	Input parameters		−0.949	28.5	2.37	4.58
			0.568	1.76	0.776	0.299
			0.299	0.845	0.584	0.123
			1.09	37.03	5.02	4.59
			0.010	0.642	0.187	0.119
	Output parameter		0.005	6.14	0.616	0.376

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SVM is known as one of the most potent kernel-depend models in the predicting field rooted in structural risk minimization (Vapnik 1999). This capability grants SVM a better generalizability to overcome the shortcomings of traditional ML approaches such as the ANN algorithm. Using the concept of kernel functions, SVM maps the complicated nonlinear problem input factors into high-dimension space to transform complicated nonlinear problems into linear problems. Therefore, the kernel function plays a significant role in the performance of SVM. The central concept of SVM is based on establishing an optimal hyperplane as the decision surface that maximizes the separation edge between the positive case and the counter-example. Introducing the kernel function neatly solves the inner product operation in the high-dimensional space and is one of the main advantages of SVM. Therefore, the kernel function plays a significant role in the performance of SVM. In the present study, the prediction performance of employed SVM methods is discussed with different kernel functions including linear, polynomial, sigmoid and Radial Basis Function (RBF). The optimum values of kernel parameters were obtained through a trial-and-error process. Furthermore, optimization of SVM hyper-parameters has been carried out by a systematic grid search of the parameters using cross-validation on the training dimensionless measures.

The hyper-parameters can be calculated via Equation (19) utilizing a gradient descent algorithm.

Therefore, β can be obtained byusing the formula ⁠, where stands as the pseudo-inverse of H.

Here, the GWO algorithm is employed to find the optimum values of KELM hyper-parameters (kernel parameters and regularization coefficient). The optimized parameters indicate the prey and the dynamic optimization process of the algorithm represents the grey wolves hunting.

GWO algorithm ends with the completion of the maximum number of iterations. The return value of α is the optimum value of kernel parameters and regularization coefficient.

Owing to the importance of choosing the proper kernel function in modeling accuracy, in the first step, different structures of employed kernel-depend approaches were analyzed through various kernel functions. Table 5 lists the obtained results of employed kernel-depend approaches with different kernels for models F(IV) and S(IV). Moreover, optimal parameters of employed models are listed in Tables 6–8, respectively.

The results given in Table 5 demonstrated that under free-flow conditions, the proposed KELM-GWO model with polynomial kernel function surpasses all other approaches with values of R equal to 0.927, SI = 0.023, BIAS = −0.005946, and RMSE = 0.018 for the testing phase. In order to implement the proposed KELM-GWO, the population (pack size) was applied as 300, and iteration was set to 50. The regulation coefficient parameter of ρ and polynomial kernel parameters, including the free parameter of c and polynomial degree (d) were optimized through the GWO algorithm as ρ = 1, C = 4.6017 and d = 5.1627, respectively. To tune the GRP hyper-parameters, a Standard Gradient Descent (SGD) optimizer was utilized to maximize the log marginal likelihood. In consideration of the SI values, small variations can be observed throughout the employed kernel functions. GPR performance ranges between 0.025 and 0.023 (for Exponential kernel). Using the SGD optimizer, the ideal values of Exponential kernel parameters were obtained as the length scale parameter (σ_l) equal to 10.6417 and the signal standard deviation (σ_f) equal to 0.07. For the implementation of the embedded SVM method, the best values of kernel parameters were achieved through a trial-and-error procedure. Additionally, the optimization process of the cost factor (C) and the loss function (ε) was conducted by a grid search in the C and ε parameter space using the cross-validation technique. The result proved that SVM-RBF had the best prediction accuracy (R = 0.896, SI = 0.027, BIAS = −0.001713, and RMSE = 0.021) as suggested by researchers (Komasi et al. 2018; Najafzadeh & Oliveto 2020; Amininia & Saghebian 2021; Roushangar & Shahnazi 2021). It was followed by a linear kernel function (R = 0.785, SI = 0.037, BIAS = 0.005165, and RMSE = 0.029). The results showed that using the sigmoid kernel function as a core tool of SVM led to under performance. Taking into account the obtained results, kernel functions of polynomial, Exponential, and RBF were selected as core tools of KELM-GWO, GPR, and SVM methods, respectively, to predict the discharge coefficient of radial gates under free-flow conditions. Furthermore, model S(IV) was selected in order to better understand the performance of the employed kernel-depend approaches under submerged flow conditions. Based on the results of Table 5, the SVM model with the kernel function of RBF can predict the discharge coefficient of submerged radial gates better than other employed kernel-depend methods with R = 0.940, SI = 0.106, BIAS = −0.000572, and RMSE = 0.022. It was followed by GPR-Exponential (R = 0.944, SI = 0.102, BIAS = 0.009989, and RMSE = 0.079). As can be seen, in contrast to the free-flow condition, the proposed KELM-GWO model with polynomial kernel had poor performance, and RBF kernel function performs better structure of KELM-GWO. However, Table 5 suggests that the proposed KELM-GWO model has a lower efficiency in comparison with other kernel-depend methods for the prediction of the discharge coefficient of submerged radial gates. As a result, for submerged flow conditions, KELM-GWO-RBF, GPR-Exponential, and SVM-RBF were used for the rest of the input combinations. According to the results presented in Table 5, it was found that two chief factors including the kernel type and good setting of parameters play a key role in the improving accuracy level of kernel-depend models. However, there is no method to determine how to choose an appropriate kernel and its parameters for a given dataset to achieve high generalization performance. On the other hand, coupling kernel-based methods with different optimization methods create more complex models and requires longer computation time.

Table 9 provides the statistical indicators of the employed kernel-depend models during the testing phase. It can be seen that under free-flow conditions, using the ratio of the downstream flow depth to the gate opening (y₃/w) as the only input parameter provided the desired prediction accuracy while using only the ratio of the downstream flow depth to the radial gate radius (y₃/r) as input did not yield the appropriate prediction accuracy. This can show the importance of the gate opening on the prediction process of the discharge coefficient under free-flow conditions. According to SI values, when comparing F(II) and F(III), adding the ratio of the gate opening to the radial gate radius (w/r) improves the average prediction accuracy of employed KELM-GWO and GPR approaches by 4 and 7%, respectively. It can be seen that the parameter y₁/w is the most influential parameter for the prediction of the discharge coefficient of radial gates under free-flow conditions. Taking into consideration the statistical indices, KELM-GWO-Polynomial and GPR-Exponential models generated the most accurate results with model F(IV), where y₁/w, w/r, and α-w/r were utilized as the input parameters. As mentioned in the previous section, under free-flow conditions, the comparison results of the employed kernel-depend methods revealed that the hybridized KELM-GWO-Polynomial with model F(IV) as input combination outperformed the SVM and GPR standalone models with R = 0.927, SI = 0.023, BIAS = −0.005946, and RMSE = 0.018. It should be noticed that the prediction of discharge coefficient under field conditions for multiple gates with different openings is complicated. It may become more intractable if one gate is free-flowing and another in the transition zone.

Existing equations and related constant parameters are developed in special laboratories with specific flow conditions, therefore, these equations show acceptable results in particular conditions, but their applicability to field data with various hydraulic conditions is questionable. Another major flaw in the equations obtained from laboratory studies is that they are all based on upstream and downstream channels that are the same width and have the same floor elevation as the gate. This rarely occurs in practice. The main advantage of the kernel-depend approaches against the traditional methods is that kernel-depend models find the best solution for the problem through a high-dimensional domain by creating several candidate computer programs and choosing the best one with the highest fitness. On the other hand, proposed traditional methods are not flexible enough to present uniform results under different conditions and their applicability is limited to the specific cases of their development.

The present paper is an attempt to implement a new kernel-depend ELM GWO algorithm (KELM-GWO), GPR, and support vector machine (SVM) for modeling the discharge coefficient of radial gates. To build prediction models, an extensive field database was established, consisting of 546 data samples gathered from different types of radial gates. The optimization capability of GWO that is naturally inspired by the swarm evolutionary feature is utilized to tune the KELM model for the implemented application. The obtained results showed that the proposed KELM-GWO gives better prediction accuracy than the employed GPR and SVM approaches for the prediction of the discharge coefficient under free-flow conditions. From the current research, the following conclusions were drawn:

• •

  Among the developed input combinations, the model having input parameters of the y₁/w, w/r, and α-w/r provides the best performance. The results also showed that under submerged hydraulic conditions, the GPR model has better predicting capability with input parameters of y₁–y₃/w, y₃/r, y₁/r and w/r.

• •

  Results comparison of modeling under free-flow conditions reveals the superiority of polynomial, exponential, and RBF kernel functions as the core tool of KELM-GWO, GPR, and SVM, respectively.

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  Under submerged flow conditions, KELM-GWO model with RBF kernel function, GPR with exponential, and SVM with RBF presented better prediction accuracy.

• •

  According to the obtained results, the proposed hybrid KELM-GWO method with RBF kernel function gave the most accurate results for extremely highly submerged flow. Furthermore, the prediction performance of SVM and GPR techniques with the selected best input combinations tends to be more robust with decreasing levels of submergence.

• •

  Finally, the tested data sets of all cases were predicted by different traditional models based on dimensional analysis. Results comparison showed that employed kernel-depend methods have superior capability for prediction of discharge coefficient of radial gates under free and submerged flow conditions.

However, it is noteworthy to mention that the proposed kernel-depend techniques are data-driven, so further studies should be conducted using data ranges beyond this study and field data to confirm the high capabilities of the proposed prediction tools to model the discharge coefficient of submerged radial gates.

All relevant data are included in the paper or its Supplementary Information.

The authors declare there is no conflict.