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In this study, a support vector machine (SVM) and three optimization algorithms are used to develop a discharge coefficient (Cd) prediction model for the semi-circular side weir (SCSW). After that, we derived the input and output parameters of the model by dimensionless analysis as the ratio of the flow depth at the weir crest point upstream to the diameter (h1/D), the ratio of main channel width to diameter (B/D), the ratio of side weir height to diameter (P/D), upstream of side weir Froude number (Fr), and Cd. The sensitivity coefficients for dimensionless parameters to Cd were calculated based on Sobol's method. The research shows that SVM and Genetic Algorithm (GA-SVM) have high prediction accuracy and generalization ability; the average error and maximum error were 0.08 and 2.47%, respectively, which were about 95.72 and 60.86% lower compared with the traditional empirical model. The first-order sensitivity coefficients S1 and global sensitivity coefficients Si of h1/D, B/D, P/D, and Fr were 0.35, 0.07, 0.13, and 0.02; 0.63, 0.25, 0.30, and 0.32, respectively. h1/D has a significant effect on Cd. In particular, when h1/D < 0.24 and 0.48 < Fr < 0.58, 0.67 < Fr < 0.72, the discharge capacity of the SCSW is relatively large.

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  • We developed an effective and high-accuracy model for predicting the Cd of SCSW.

  • The importance of dimensionless parameters on Cd was quantified by Sobol's method.

  • It explored the flow characteristics of semi-circular side weir.

dimensionless parameters, discharge characteristics, intelligent model, semi-circular side weir, Sobol's method
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As one of the most common diversion structures, side weirs are used for flow control, drainage networks, irrigation, and wastewater channels (Zahiri et al. 2013). In recent years, with the change in extreme weather and a significant increase in storm floods, side weirs have been used as common equipment in sewer networks and irrigation systems to divert excess water flow from channels to other channels (Uyumaz et al. 2014). Semi-circular labyrinth side weirs are widely used due to their long overflow front length, stable overflow structure, and facilitation of sediment removal. Also, semi-circular side weir (SCSW) flow as a spatially variable flow has more parameters affecting the discharge coefficient (Cd). Therefore, it is important to accurately evaluate the influence and variation law of different factors on the Cd for the design and operation of this structure.

At present, most scholars mainly use traditional empirical methods to check the discharge capacity of SCSWs. Haghshenas & Vatankhah (2021) proposed discharge calculation equations for SCSW, in which the mean and maximum errors of the best model were 1.87 and 6.31%, respectively. Mamand & Raheem (2018) used SPSS software to fit the empirical equation of the SCSW, and the coefficients of determination (R2) in the form of multivariate linear regression and multivariate power regression were 0.8498 and 0.8584, respectively. Khalili & Honar (2017) gave the calculation equation of the Cd of the SCSW by using the model experiments and dimensional analysis. The research shows that the Cd of the SCSW was higher than that of the rectangular side weir. However, the discharge is affected by the plane position of the weir sill, the shape of the weir, the upstream and downstream flow conditions, and different flow resistances generated, resulting in different expressions of the Cd, which are not convenient for users. Also, the discharge coefficients were determined according to empirical equations, which were limited by certain datasets, effective parameter interactions, high uncertainty, numerous assumptions, and other defects (Tao et al. 2022), resulting in insufficient mining of physical properties among parameters and limited calculation accuracy.

In recent years, many scholars have attempted to use soft computing techniques for solving the problems of large calculations and inconvenient use of empirical equations (Haghbin & Sharafati 2022; Shen et al. 2022; Gharehbaghi et al. 2023; Parsaie et al. 2023; Seyedian et al. 2023; Yarahmadi et al. 2023). Jamei et al. (2021) developed three linear models for predicting the Cd of the triangular side orifices. The research shows that the intelligent model can accurately evaluate the discharge capacity of the side orifices under free-flow conditions. Tao et al. (2022) used three machine learning models to estimate the Cd prediction models of the gate under free-flow and submerged-flow conditions. The results show that the model has higher accuracy for the free-flow condition. Ismael et al. (2021) used neural network technology for predicting the Cd of inclined cylindrical weirs with different diameters; the root mean square error (RMSE) of the radial basis function network model was reduced by 9 and 41% compared with the cascade-forward neural network and the back-propagation neural network (BPNN) in the testing stage, respectively. However, with the wide application of intelligent models in weir flow, it has been gradually discovered that this technology has problems such as overfitting and easily falling into local optimum. Therefore, researchers began to try to optimize the hyperparameters of the model through optimization algorithms to derive the best model parameters to improve the forecast accuracy and stability of the model. For example, Haghbin et al. (2022) developed a hybrid data-driven approach to evaluate the Cd of step spillways, and the optimized model improved the performance index to 86.13%. Pradeep & Samui (2022) used a neural network technology hybrid optimization algorithm to predict rock strain, and the results showed that the optimized model was better than other single models in the training and testing phases. Chen et al. (2022) aimed to predict the discharge coefficient of streamlined weirs, and the results showed that the hybrid deep data-driven algorithms provide more accurate results than the classical ones. Simsek et al. (2023) used the artificial neural network (ANN) to predict the discharge coefficient of trapezoidal broad-crested weir; the study results showed that the Froude number significantly increases the performance of the models in estimating Cd values, and the ANN method was more successful in determining Cd than other methods. Balouchi & Rakhshandehroo (2018) used the soft computing models to evaluate the discharge coefficient for combined weir-gate, and multilayer perceptron was considered superior; it had better statistical indices of RMSE, mean absolute error (MAE), and R2 (0.027, 0.022, and 0.984, respectively).

However, the prediction model needs to meet the requirements of high accuracy and stability due to the large discharge and complex physical parameters of the SCSW. According to the current literature, research shows that a high-precision SCSW Cd prediction model has not been developed yet. Therefore, it is important to develop an accurate and stable prediction model for the Cd of SCSW in this study. In addition, there is also great interest in the interaction characteristics between model inputs and outputs. Zhang et al. (2013) used Sobol's method to analyze the sensitivity of potential hydrological processes under different hydrological models and climatic conditions. Nossent et al. (2011) successfully applied the Sobol sensitivity method to the prioritization of input parameters of complex environmental models. However, most scholars pay more attention to the stability and accuracy of the weir flow prediction model, and the interactions and variation relationships between input parameters and discharge coefficients have not been explored in depth. Hence, this paper not only establishes the discharge coefficient prediction model for the SCSW but also provides a new method for the accurate calculation of the discharge of the structure. More importantly, based on predecessors, the influence of dimensionless parameters on the discharge coefficient is quantified, and this study fills the research gap in this area.

In summary, this study aims to systematically evaluate the effects of the hydraulic parameters of SCSWs on the Cd. First, the particle swarm optimization (PSO) algorithm, genetic algorithm (GA), and sparrow search algorithm (SSA) are used to optimize the hyperparameters c and γ of the support vector machine (SVM) and establish three different models for predicting the Cd of SCSWs. Then, the accuracy and generalization ability of the intelligent and traditional empirical models are compared using different performance indexes. On this basis, Sobol's method is used to explore the interaction and change process between hydraulic parameters and Cd and analyze the change law of hydraulic parameters and Cd. The sensitivity of different hydraulic parameters to Cd is quantified to provide an essential reference basis for the design and promotion of SCSWs.

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Experimental data

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In this study, the dataset was obtained from Haghshenas & Vatankhah (2021). The experimental device consisted mainly of a pumping station with a recirculation system that provided a horizontal rectangular channel 12 m long, 0.25 m wide, and 0.5 m deep. The weir upstream discharge Q1 and side weir discharge Qw were determined by triangular and rectangular weirs, respectively, and through the accuracy of ±0.5% electromagnetic flowmeter calibration. The SCSW was installed on the main rectangular channel wall 6 m away from the inlet, and the downstream and upstream water depths of the weir were measured at the centerline of the main channel using a point gauge with an accuracy of 0.1 mm. The SCSWs were made of 10-mm-thick plexiglass sheets with a crest thickness of 1 mm, and the plan layout is shown in Figure 1. Three different weir heights (P = 5, 10, and 15 cm; the weir crest height, P, varied from 5 to 15 cm for each value of weir diameter) and three different weir diameters (D = 25, 30, and 40 cm) were measured in laboratory measurements. Q1 varied from 14.7 to 42.1 L/s, Qw varied from 3.0 to 25.8 L/s, and the diverted discharge ratio was 0.14 ≤ Qw/Q1 ≤ 0.73. A total of 155 runs were carried out under free-flow conditions, and the data characteristics are shown in Table 1.
Table 1

Statistics of data characteristics

Statistical parametersB/DP/Dh1/DFrCd
Maximum 1 0.6 0.469 0.815 0.780 
Minimum 0.625 0.125 0.156 0.174 0.565 
Mean 0.799 0.299 0.304 0.433 0.663 
Middle quartile 0.833 0.250 0.305 0.420 0.652 
SD 0.155 0.136 0.083 0.153 0.056 
Statistical parametersB/DP/Dh1/DFrCd
Maximum 1 0.6 0.469 0.815 0.780 
Minimum 0.625 0.125 0.156 0.174 0.565 
Mean 0.799 0.299 0.304 0.433 0.663 
Middle quartile 0.833 0.250 0.305 0.420 0.652 
SD 0.155 0.136 0.083 0.153 0.056 
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Figure 1
Plane structure of SCSW.
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Plane structure of SCSW.

Figure 1
Plane structure of SCSW.
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Plane structure of SCSW.

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Dimensional analysis

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As can be seen from Figure 1, the variables that may affect the SCSW discharge include the following: h1 (h1 = y1 − p) is the depth of flow relative to the crest point of the upstream weir, D is the side weir diameter, P is the side weir crest height, V1 is the mean velocity at upstream of the side weir, y1 is the flow depth at the upstream end of the side weir, B is main channel width, ρ is water density, μ is water viscosity, and g is gravitational acceleration. The discharge of SCSW can be expressed as Equation (1).
(1)
According to the Buckingham-π theorem, the above parameters were dimensionally analyzed (Haghshenas & Vatankhah 2021; Saffar et al. 2021), and D, g, ρ were used as three independent variables; the dimensionless parameters that affect Cd can be expressed as Equation (2).
(2)
where Fr is the Froude number upstream of the side weir and Re is the Reynolds number. For side weirs, the flow of water is usually turbulent, and the influence of dynamic viscosity (Re) on the hydraulic characteristics of flow is negligible (Norouzi et al. 2020); the dimensionless parameters affecting the Cd can be expressed as Equation (3).
(3)

Support vector machine

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In this study, the dataset is small, the sample uncertainty is high, and the sample parameters are highly nonlinear. Therefore, a suitable large-scale, fast, and robust model is selected. Meanwhile, SVM is a powerful supervised learning technique that can provide reliable and robust predictions (Najafzadeh & Oliveto 2020). Considering that the PSO and GA belong to the traditional swarm intelligence algorithm, and the SSA belongs to the new swarm intelligence algorithm by using the same dataset to compare the hyperparameter changes between the three algorithms, the stability and reliability of the model can be better determined.

SVM is a classification technique proposed by Vapnik based on the statistical learning theory and structural risk minimization (SRM) (Cortes & Vapnik 1995) and is now widely used for high-accuracy prediction due to its advantages in solving nonlinear problems (Ahmad et al. 2014; Parsaie et al. 2019; Najafzadeh & Niazmardi 2021; Parsaie et al. 2021). Its purpose is to generate a decision boundary between two classes, which is called a hyperplane, and the separating hyperplane is determined by the orthogonal vector w and the bias b. Its direction is as far away from the nearest data points in each class, and these nearest points are called support vectors (Huang et al. 2018); its model structure is shown in Figure 2. The solution to the nonlinear problem can be achieved by mapping the data to a higher dimensional feature space with the help of kernel functions (Najafzadeh et al. 2016). There are two very important parameters C and γ in the SVM model, and the parameter C represents the penalty. The value of C affects the prediction accuracy and the value of γ affects the partitioning of the feature space; the parameter γ has a greater impact on the results than the penalty factor C. Therefore, to obtain suitable C and γ, three optimization algorithms are used in this study to optimize the values of C and γ globally to obtain the best performance prediction model for the Cd of the SCSW. The optimization process is shown in Figure 3.
Figure 2
SVM plan structure diagram.
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SVM plan structure diagram.

Figure 2
SVM plan structure diagram.
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SVM plan structure diagram.

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Figure 3
Process diagram of optimization algorithm.
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Process diagram of optimization algorithm.

Figure 3
Process diagram of optimization algorithm.
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Process diagram of optimization algorithm.

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SVM and PSO

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PSO is a classic population search algorithm, and its calculation equation is as follows (Huang & Dun 2008; Ardjani et al. 2010; Cuong-Le et al. 2022):
(4)
(5)
where xi is the position of each particle, vi is the velocity of each particle, pbest is the particle optimal value, gbest is the global optimal value, r1 and r2 are random numbers between 0 and 1, and c1 and c2 are acceleration factors.

The particle swarm regards the two parameters of C and γ of the SVM as two particle swarms and first sets the parameters of population size and iteration number for population and velocity initialization, inputs the randomly generated C and γ into the SVM model for training; the mean square error of model cross-validation (CVmse) is used as the model fitness function, the minimum fitness of the particle represents the optimal particle position at this time, and the optimization algorithm ends when the iteration number meets the set value.

SVM and GA

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The GA is an adaptive optimization method with a global search function that uses random search to efficiently guide the parameter space to encode each individual. The key technology of the algorithm consists of five elements: encoding of parameters, initialization of the population, calculation of the fitness function, layout of genetic operations, and control of the parameter arrangement (Li & Kong 2014). Therefore, through continuous evolution from generation to generation, an optimally adapted individual can eventually be obtained. It has the advantages of global optimality, implicit parallelism, high stability, and wide availability (Li & Kong 2014; Guan et al. 2021).

The basic steps of the GA:

  • (1)

    Encoding: The GA represents the solution data in the solution space as genotypic string structure data in the genetic space before searching, and the different combinations of these string structure data constitute the different points.

  • (2)

    Initial population generation: N initial string structure data are randomly generated, each string structure data is called an individual, N individuals form a population, and the GA uses these N string structure data as initial points to start evolution.

  • (3)

    Adaptability evaluation: Adaptability indicates the strengths and weaknesses of individuals or solutions. The fitness function is defined in different ways for different problems.

Finally, the optimal solution is obtained by three basic operations: selection, crossover, and variation.

SVM and SSA

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The SSA is a new intelligent optimization algorithm that simulates the foraging and anti-predation behavior of sparrows (Xue & Shen 2020; Yan et al. 2022). At present, it has been widely used in related fields. Throughout the foraging process, there are three behaviors: discoverer, joiner, and alerter. Among them, the identities of the discoverer and joiner are changed dynamically. The location update of the discoverer is shown in Equation (6).
(6)
where is the position of the ith individual in the region dimension after the tth iteration in the sparrow population; a is a uniform random number, a ∈ (0,1]; Cmax is the maximum number of iterations; R2 is a random number with a warning value of [0,1]; ST is the security threshold with an interval of [0.5, 1]; Q is a random number and obeys the standard normal distribution; and L is a 1 × d dimensional matrix. The location update of the joiner is shown in Equation (7).
(7)
where is the best position of the discoverer in the t + 1 iteration; Xw is the worst position in the current sparrow population; ⁠, A is l × d matrix with element 1 or −1. n is the population size. When i > n/2, the ith joiner with low fitness is not fed and needs to be foraged; conversely, when i ≤ n/2, the joiner will forage near the optimal position. The location update of the alerter is shown in Equation (8).
(8)
where is the global optimum position; β is a random number that conforms to the standard normal distribution. K is a uniform random number, K ∈ [−1,1]; fi, fw, and fg are the current position fitness value, the worst position fitness value, and the optimal position fitness value, respectively; ɛ is the minimum constant to the denominator is not 0. fi>fg indicates that sparrows are at the edge of the population and are vulnerable to attack; fi = fg indicates that sparrows are in the middle of the population, warning of danger, and adjust the search strategy in time to avoid attacks.

Sobol's sensitivity analysis method

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The Sobol method (Sobol 1990), as a global sensitivity analysis method based on variance decomposition, obtained the importance of the input parameters on the output results by calculating the first-order sensitivity and the global sensitivity of the input parameters (Lu et al. 2018). The objective function f (x) of the model is decomposed as the sum of 2p increasing terms:
(9)
where f0 represents the constant in the objective function, and each integral variable in the formula is 0, then the expression is
(10)
(11)
(12)
(13)
(14)
(15)
where V(Y) represents the sum of the parameters on the output results of the model objective function f(x); Vi1, i2, …, is represents the influence of the interaction of the parameter combination on the model output results. Vi represents the influence of the ith parameter on the output result of the model objective function f (x); and V∼i represents the sum of the variance caused by all parameters except the ith parameter.

Evaluation index

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This study used several statistical methods to evaluate model performance. The parameters are RMSE, correlation coefficient (R), mean absolute percentage error (MAPE), standard deviation (SD), scatter index (SI), developed discrepancy ratio (DDR), bias coefficient (Bias). Methods are defined in the following equations.
(16)
(17)
(18)
(19)
(20)
(21)
(22)
where Oi and Pi represent the experimental and predicted values of Cd, respectively, and Oa and Pa represent the mean of the experimental and predicted values, respectively.
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Model comparison

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In this study, 109 experimental datasets were selected as the training set and the remaining 46 sets were used as the testing set. h1/D, B/D, P/D, and Fr were used as model inputs and Cd as model outputs. The global optimization of the hyperparameters C and γ of SVM was performed by three optimization algorithms, PSO, GA, and SSA; the specific parameter settings of each model are shown in Table 2, and the performance indexes of all models were finally obtained as shown in Tables 3 and 4. When the SVM model is used to calculate the Cd of the SCSW, the RMSE, MAPE, SD, and R were 0.047, 0.076, 0.073, and 0.897 in the training phase, respectively. The RMSE, MAPE, SD, and R were 0.045, 0.072, 0.062, and 0.926 in the testing phase, respectively. The PSO-SVM, GA-SVM, and SSA-SVM are significantly superior in each evaluation index in the training and testing phases than SVM, indicating that all three optimization algorithms can effectively improve the performance of SVM through global optimization search.

Table 2

Parameter settings of all models

ModelParameterValueCγ
PSO-SVM Particle swarm size 20 0.1 6.72 
Number of iterations 30 
Inertia factor 0.9 
Acceleration constants 2 
Speed range [−1,1] 
GA-SVM Population size 20 4.05 4.40 
Number of iterations 30 
Crossover probability 0.5 
Mutation probability 0.1 
SSA-SVM Number of sparrows 20 0.1 4.33 
Number of iterations 30 
warning value ST 0.6 
Proportion of discoverers 0.7 
Proportion of detectors 0.2 
ModelParameterValueCγ
PSO-SVM Particle swarm size 20 0.1 6.72 
Number of iterations 30 
Inertia factor 0.9 
Acceleration constants 2 
Speed range [−1,1] 
GA-SVM Population size 20 4.05 4.40 
Number of iterations 30 
Crossover probability 0.5 
Mutation probability 0.1 
SSA-SVM Number of sparrows 20 0.1 4.33 
Number of iterations 30 
warning value ST 0.6 
Proportion of discoverers 0.7 
Proportion of detectors 0.2 
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Table 3

All model performance indexes in the training stage

ModelRMSEMAPE (%)SDRSIBias
SVM 0.047 0.076 0.073 0.897 0.071 0.0130 
PSO-SVM 0.021 0.053 0.043 0.961 0.031 0.0028 
GA-SVM 0.014 0.037 0.041 0.987 0.022 0.0008 
SSA-SVM 0.019 0.048 0.044 0.967 0.024 0.0031 
ModelRMSEMAPE (%)SDRSIBias
SVM 0.047 0.076 0.073 0.897 0.071 0.0130 
PSO-SVM 0.021 0.053 0.043 0.961 0.031 0.0028 
GA-SVM 0.014 0.037 0.041 0.987 0.022 0.0008 
SSA-SVM 0.019 0.048 0.044 0.967 0.024 0.0031 
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Table 4

All model performance indexes in the testing stage

ModelRMSEMAPE (%)SDRSIBias
SVM 0.045 0.072 0.062 0.926 0.069 0.0120 
PSO-SVM 0.017 0.016 0.046 0.953 0.026 0.0010 
GA-SVM 0.009 0.008 0.043 0.965 0.014 0.0004 
SSA-SVM 0.016 0.016 0.047 0.949 0.031 0.0008 
ModelRMSEMAPE (%)SDRSIBias
SVM 0.045 0.072 0.062 0.926 0.069 0.0120 
PSO-SVM 0.017 0.016 0.046 0.953 0.026 0.0010 
GA-SVM 0.009 0.008 0.043 0.965 0.014 0.0004 
SSA-SVM 0.016 0.016 0.047 0.949 0.031 0.0008 
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Figure 4 shows the scatter plot for the experimental and predicted values for all models. Larger values of R indicate the better fitting ability of the models, and the closer the predicted and experimental values are to the trend line (1:1). As can be seen from Figure 4, compared with SVM, the R of PSO-SVM, GA-SVM, and SSA-SVM increased by about 6.65, 9.11, and 7.23% in the training phase, respectively. Also, the R increased by about 2.83, 4.04, and 2.42% in the testing phase, respectively. It can be seen that among the three optimization models, GA-SVM has better generalization ability and prediction stability. Figure 5 shows the Taylor plots for all the models; the longitudinal distance from the origin represents the SD, the purple radial lines indicate the correlation coefficient (R), and the green circular arcs show the RMSE. As the circle section expanded, this parameter value increased. Moreover, the SD, R, and RMSE of the training and testing phases were specified by a single point, and the model closest to the reference point was considered the best model. It can be seen that SVM has the worst prediction effect and GA-SVM has the best prediction result, where PSO-SVM and SSA-SVM have almost the same effect. Therefore, GA-SVM can be used as the optimal intelligent prediction model for the Cd of SCSW.
Figure 4
Scatter plot of experimental and predicted values of Cd. (a) Training stage and (b) testing stage.
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Scatter plot of experimental and predicted values of Cd. (a) Training stage and (b) testing stage.

Figure 4
Scatter plot of experimental and predicted values of Cd. (a) Training stage and (b) testing stage.
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Scatter plot of experimental and predicted values of Cd. (a) Training stage and (b) testing stage.

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Figure 5
The Taylor diagram for all models. (a) Training stage and (b) testing stage.
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The Taylor diagram for all models. (a) Training stage and (b) testing stage.

Figure 5
The Taylor diagram for all models. (a) Training stage and (b) testing stage.
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The Taylor diagram for all models. (a) Training stage and (b) testing stage.

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Comparison with empirical equations

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Figure 6 shows the DDR values for all models, which can be used to evaluate the distribution of errors in detail. It can be seen that the overall error of GA-SVM is small, indicating that the model has high prediction accuracy. Meanwhile, Haghshenas & Vatankhah (2021) used the dimensional analysis technique to fit the SCSW discharge calculation model, and the average error and maximum error of the best model were 1.87 and 6.31%, respectively. The mean and maximum errors of GA-SVM were 0.08 and 2.47%, respectively, which were about 95.72 and 60.86% lower compared to the traditional empirical model, indicating that the intelligent model has higher accuracy in predicting the Cd of SCSWs. Figure 7 shows the error density plot of GA-SVM in the testing phase; 91.31% of the prediction errors were below 2%, and the errors were mainly concentrated at Cd = 0.7, indicating that GA-SVM has high accuracy and stability in predicting the Cd.
Figure 6
DDR values for all models.
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DDR values for all models.

Figure 6
DDR values for all models.
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DDR values for all models.

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Figure 7
Error density plot of GA-SVM in the testing phase.
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Error density plot of GA-SVM in the testing phase.

Figure 7
Error density plot of GA-SVM in the testing phase.
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Error density plot of GA-SVM in the testing phase.

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Quantitative analysis of parameters

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From the above analysis, it can be seen that GA-SVM can be used as a prediction model for the Cd of SCSW. Therefore, the model input parameters were further quantified and analyzed using Sobol's method. As can be seen from Figure 8, the sensitivity coefficients of the dimensionless parameters and the first-order sensitivity coefficients S1 of h1/D, B/D, P/D, and Fr were 0.35, 0.07, 0.13, and 0.02, respectively, indicating that when only the influence of a single parameter on Cd was considered, the influence of h1/D on Cd was the largest, followed by P/D, and Fr has the least effect on Cd. When a parameter interacts with other parameters, the global sensitivity coefficients Si for h1/D, B/D, P/D, and Fr were 0.63, 0.25, 0.30, and 0.32, respectively. Also, h1/D has the greatest effect on Cd, indicating that h1/D is an important parameter affecting Cd. However, the effect of Fr on Cd after interacting with other parameters was only inferior to h1/D, indicating that the ratio of flow velocity to flow depth plays an important role in the assessment of Cd under the influence of geometric parameters. Therefore, h1/D and Fr should be considered important parameters when assessing the discharge capacity of SCSW.
Figure 8
Sensitivity coefficients of dimensionless parameters.
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Sensitivity coefficients of dimensionless parameters.

Figure 8
Sensitivity coefficients of dimensionless parameters.
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Sensitivity coefficients of dimensionless parameters.

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Parameter sensitivity analysis

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Figure 9 shows the variation relationship between the dimensionless parameters h1/D, B/D, P/D, and Fr and the predicted Cd for the SCSW. As can be seen from Figure 9(a), Cd decreases with the increase of h1/D. As D increases, the trend of Cd decreases more obviously; where the variation trend of D = 0.25 m and D = 0.30 m is very close. As can be seen from Figure 9(b), Cd increases with the increase of Fr, and the trend increases more slowly when B/D = 0.625. The trend increases more obviously when B/D = 0.833. For the same Fr, Cd shows a decreasing trend as B/D increases.
Figure 9
Variation of the dimensionless parameters versus predicted discharge coefficient. (a) The variation between h1/D and predicted Cd. (b) The variation between Fr and predicted Cd.
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Variation of the dimensionless parameters versus predicted discharge coefficient. (a) The variation between h1/D and predicted Cd. (b) The variation between Fr and predicted Cd.

Figure 9
Variation of the dimensionless parameters versus predicted discharge coefficient. (a) The variation between h1/D and predicted Cd. (b) The variation between Fr and predicted Cd.
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Variation of the dimensionless parameters versus predicted discharge coefficient. (a) The variation between h1/D and predicted Cd. (b) The variation between Fr and predicted Cd.

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Analysis of discharge characteristics

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Figure 10 shows the variation of the most influential input parameters against the predicted Cd. It can be seen that when h1/D < 0.24, 0.48 < Fr < 0.58, and 0.67 < Fr < 0.72, the Cd of the SCSW is larger and has a higher discharge capacity of the SCSW at this time. Also, when Fr < 0.50 and 0.40 < h1/D < 0.47, the Cd of SCSW is smaller, and the discharge capacity of SCSW is relatively small at this time.
Figure 10
Variation of the most influential inputs versus predicted Cd.
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Variation of the most influential inputs versus predicted Cd.

Figure 10
Variation of the most influential inputs versus predicted Cd.
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Variation of the most influential inputs versus predicted Cd.

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In order to achieve accurate water measurement and reasonable distribution of water resources in small channels, a semi-circular labyrinth side weir is used as an efficient and greater discharge capacity control structure. In this study, PSO-SVM, GA-SVM, and SSA-SVM optimization models were developed based on SVM. Then, Sobol's method was introduced to calculate the sensitivity coefficients of different dimensionless parameters h1/D, B/D, P/D, and Fr to Cd. This paper evaluates the effect of different factors on the discharge capacity of SCSW. The parameter variation range of various discharge capacities is proposed, and the variation law between different parameters and Cd is analyzed. The following conclusions were drawn.

  • (1)

    In the current study, GA-SVM can be used as an efficient and high-accuracy prediction model for the Cd of SCSW. In the testing phase, R = 0.987, MAPE = 0.037%, RMSE = 0.014, SD = 0.041, SI = 0.022, and Bias = 0.008, and 91.31% of the prediction errors were below 2%; the model has high generalization ability, stability, and prediction accuracy, and this model effectively solves the problems of large computational complexity and difficult coefficient correction in traditional empirical models.

  • (2)

    The quantitative analysis showed that the S1 and Si of h1/D, B/D, P/D, and Fr were 0.35, 0.07, 0.13, and 0.02; and 0.63, 0.25, 0.30, and 0.32, respectively; h1/D was the most important parameter affecting Cd, the effect of Fr on Cd after interacting with other parameters was only inferior to h1/D, and Cd decreased as h1/D increased. As D increased, Cd decreased the greater the trend. As the diameter of the side weir increases, the lateral flow will increase significantly in the subcritical flow regime.

  • (3)

    When h1/D < 0.24, 0.48 < Fr < 0.58, and 0.67 < Fr < 0.72, the Cd of SCSW is greater. Meanwhile, when Fr < 0.50 and 0.40 < h1/D < 0.47, the Cd of the SCSW is relatively small. This can provide an important reference basis for the application of SCSW in practical engineering.

In addition, in this study, the width of the main channel is constant. Therefore, it is necessary to further explore the influence of the width change of the main channel on the discharge coefficient of the SCSW.

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This study was partly supported by the National Natural Science Foundation-sponsored project (grant 52079107), the Natural Science Basic Research Project of Shaanxi Province (grant 2023-JC-QN-0395), and the General Special Scientific Research Project of Shaanxi Province (grant 22JK0470).

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

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The authors declare there is no conflict.

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