An ISaDE algorithm combined with support vector regression for estimating discharge coefficient of W-planform weirs Open Access

Weirs are among the most essential components of water transmission networks, due to the necessity of determining the flow rate in channels and the allocated amount for consumers. These are essential hydraulic structures for controlling the flow and water level, which can be utilized to increase the height of the water surface level and thereby provide the required water heights to divert the flow to lateral channels. Also, these structures are utilized as flow-measuring devices in crucial applications in rivers and open channels for their safe operation (Emami et al. 2018).

The length and shape of the weir crest are among the effective parameters in the flow rates over the weir, and numerous studies have been carried out on the effect of geometric and hydraulic parameters of weirs on the discharge coefficient (C_d). The use of non-linear planar weirs with a specific shape such as triangular, trapezoidal, piano key, circular and parabolic, which are known as labyrinth weirs, is one of the effective ways to increase the flow over a specified width.

Some notable approaches recently proposed to predict the C_d of different plans of labyrinth weirs are shown in Figure 1.

An important type of triangular weir is the sharp-crested W-planform weir. A few attempts have been made to define the C_d of W-planform weirs. More discharge over them is their major advantage against other types of weirs. The first experimental study on labyrinth weirs with different plans was conducted by Taylor (Dehdar-Behbahani & Parsaie 2016). Besides the development of experimental models, numerical methods including computational fluid dynamic (CFD) techniques (Parsaie et al. 2015; Dehdar-Behbahani & Parsaie 2016), and artificial intelligent techniques have emerged as distinctive and complementary tools in the subject of hydraulic field (Parsaie et al. 2015; Khoshbin et al. 2016; Haghiabi et al. 2018; Parsaie et al. 2018a). Meta-heuristic algorithms as an important branch of artificial intelligence have been widely employed in solving various problems because of their accuracy and high performance. Aydin (2012), Aydin & Emiroglu (2013), Robertson (2014) and Crookston and Tullis (Crookston 2010) used CFD software to model labyrinth weirs and labyrinth side weirs.

Researchers have used neural networks (ANN), genetic programming (GP), support vector machine (SVM) and M5 model tree, group method of data handling (GMDH), adaptive neuro-fuzzy inference system (ANFIS), and meta-heuristic algorithms to optimize and predict the characteristics and geometry of labyrinth weirs with different plan forms including rectangular, trapezoidal, triangular, and a combination of these plans as well and their C_d (Emin Emiroglu et al. 2010; Zaji & Bonakdari 2014; Ebtehaj et al. 2015; Azamathulla et al. 2016; Najafzadeh et al. 2016; Haghiabi et al. 2017; Parsaie et al. 2018b; Bilhan et al. 2018; Roushangar et al. 2018).

Karami et al. (2016) predicted the C_d of the triangular labyrinth weir using SVM, ANN, and GP methods. SVM is a machine-learning technique that is widely used for classification and regression purposes by means of a separating hyperplane. The benefits of SVM, especially its significant accuracy in classification utilizing the tuning parameters, associated with its wide real-world applications has led to the ever-increasing popularity of SVM in the last decade. Artificial neural network (ANN), as a famous computing technique, is inspired by the biological neural system which is utilized in solving complicated problems. ANNs can enjoy self-learning capabilities with task-specific rules and they reach better results by more available data. Genetic programming (GP) is a subset of machine learning and like all evolutionary algorithms (EAs) works by an iterative process. EAs are generally used to discover solutions to challenging real-world problems. GP starts by setting a goal function and then uses the Darwin evolution principle to generate a set of candidate solutions. The results obtained by Karami et al. (2016) showed that the SVM method was superior to the other two methods with RMSE = 0.0059. Haghiabi et al. (2017) obtained the C_d of the triangular labyrinth weir by using the ANFIS and multi-layer perceptron (MLP) models. An adaptive neuro-fuzzy inference system (ANFIS) is based upon a Takagi–Sugeno fuzzy inference system. This model was first introduced in the early 1990s. It enjoys both neural network and fuzzy logic advantages and it is capable of capturing the benefits of both techniques in a single framework. A multilayer perceptron (MLP) is a subset of the feedforward artificial neural network and is utilized as a powerful ANN model for solving complex systems. MLP is generally used as a nonlinear mapping technique to relate the output vector to the input vector of the system. The results of the two models suggest that both models perform well in predicting the C_d. Roushangar et al. (2017) examined the C_d of the arched labyrinth weir using the SVM method and concluded that this method has high accuracy in predicting the C_d of arched weirs. Bonakdari & Zaji (2018) utilized three novel hybrid soft computing approaches, including neuro-fuzzy-differential evaluation (ANFIS-DE), neuro-fuzzy-genetic algorithm (ANFIS-GA) and neuro-fuzzy-particle swarm optimization (ANFIS-PSO), to simulate the C_d of side weirs. The used methods, as the hybrid ones, enjoy the advantages of both employed algorithms. The differential evolution (DE) algorithm, which is utilized together with ANFIS in ANFIS-DE, is a population-based direct search method that uses several vectors. The way of interaction among the vectors is determined by the evolution of living species. This heuristic global optimization algorithm is easy to apprehend, simple, and fast to implement. Parsaie et al. (2018a) applied ANN, SVM and ANFIS models to predict the C_d of a weir-gate system. Performance evaluation of the three models showed that the SVM model had the best performance (R² = 0.94 and RMSE = 0.008). Bonakdari et al. (2020) employed gene expression programming (GEP) in predicting the C_d of the triangular labyrinth weir and concluded that the proposed method is better than the nearest-living-relative (NLR) method for predicting the C_d. Gene expression programming (GEP) is a learning algorithm to find out the relationship between inputs and output of the unknown system. The main advantage of GEP is in realizing the system inter-relationship by a clear mathematical equation that needs to identify a set of pre-defined mathematical operators. Shafiei et al. (2020) applied the ANFIS-FFA method to obtain the C_d of the labyrinth weirs. A comparison of ANFIS-FFA model results with the ANFIS model showed that the proposed model has a good performance in predicting the C_d. The firefly algorithm (FFA), which is used together with ANFIS in the study, is a swarm-based meta-heuristic algorithm that mimics the flashing light behavior of fireflies. The algorithm works by mimicking how attracted a firefly is by the flashing light of any other firefly. According to the fireflies' behavior, attractiveness is specified by the brightness of the flashing light and its distance, which is the foundation of the technique.

According to the above-mentioned studies, different methods of artificial intelligence have been employed as appropriate optimization or simulation tools to determine the C_d of these weirs. In this paper, a hybrid predictive model based on SVR and improved self-adaptive evolutionary optimization algorithm (ISaDE) is proposed to study the effect of geometric parameters on the C_d of the sharp-crested W-planform labyrinth weir. The proposed method was evaluated with six different combinations of the effective parameters to determine the most appropriate combination. To train and test the proposed hybrid ISaDE-SVR model, the experimental studies of Kumar et al. (2012) and Ghodsian (2009) have been used. Also, the results of the ISaDE-SVR method are compared with the previous findings.

The remaining parts are organized as follows. Problem Definition defines the problem that this paper is focused on. Methodology describes the proposed methodology. The experiments are given in the fourth section. Discussion of results is provided in the fifth section. Finally, the sixth section concludes the paper and lists some future interesting directions.

The purpose of determining the C_d is to investigate the performance of the weir by estimating the flow characteristics passing over the weir. The C_d of the W-planform labyrinth weir is a function of six basic parameters, which are: upstream total head of flow (H_t), effective length of weir (L_e), weir height (P), weir width (W), vertex angle (θ) and flow depth (y).

Self-adaptive differential evolution (SaDE) is a well-known iterative optimization algorithm (Brest et al. 2006). SaDE is an evolutionary and multi-agent algorithm, in which each agent updates its position by the three operators of selection, mutation, and crossover. The flowchart of the SaDE algorithm is shown in Figure 2.

If the trial vector obtains a better fitness than ⁠, then is set to ⁠; otherwise the old value is maintained.

As mutation controls the algorithm exploration, it is the core operator of SaDE. Improving this parameter increases the algorithm search capability. In this study, an improved version of the mutation operator is proposed to improve the optimization ability of standard SaDE. The new algorithm is named the improved SaDE (ISaDE) algorithm. The rest of the parts are the same as in the original SaDE algorithm except for a mutation phase which is introduced here. The result is much easier to implement compared with other extensions of the SaDE algorithm.

As mentioned before, the prediction performance of the SVR with RBF kernel is highly dependent on the three parameters C, ⁠, and ⁠. In other words, selection of optimal values for these parameters improves the speed of training and increases the performance of classification as well. To find the optimal values of the parameters and optimum feature selection, we used the ISaDE optimization algorithm. Figure 4 shows the proposed ISaDE-SVR approach for SVM parameter selection and feature extraction. This algorithm provides a better training effect and improves the prediction accuracy.

A detailed description of the ISaDE-SVR follows.

The individual with a smaller value of MSE is more preferable. The idea behind using the cross-validation strategy is to prevent under-fitting or over-fitting effects in the ISaDE-SVR approach. In n-fold cross validation, the training set is divided into n equal subsets. At each iteration, one of the subsets is taken as the testing set in turn, and n − 1 subsets are considered as training sets in the SVR method, then the above procedure is iterated until each subset is validated once.

Until the termination conditions are met, three operators including mutation, crossover and selection are iteratively carried out to update the population.

To evaluate the performance of the proposed ISaDE algorithm, 14 test problems were used. These functions include nine unimodal and four multimodal functions. Table 1 lists the characteristics of the benchmark problems. A detailed description of the test functions is available in Civicioglu (2013) and Karaboga & Akay (2009).

The ISaDE algorithm is compared with five algorithms including particle swarm optimization (PSO) (Vesterstrom & Thomsen 2004), artificial bee colony (ABC) (Karaboga & Basturk 2007), SaDE (Civicioglu 2013), grey wolf optimization (GWO) (Mirjalili et al. 2014), and salp swarm optimization (SSA) (Mirjalili et al. 2017). The PSO algorithm simulates the swarm intelligence behavior of bird flocking and schooling in nature. In every iteration of PSO, the positions of search agents are updated using the local best knowledge found by each agent and the global knowledge found by the swarm. The ABC simulates the cooperation among honey bees in finding food sources. SaDE is an extension of the differential evolution algorithm that utilizes a self-adaptive approach to tune the parameters. The GWO algorithm simulates the hunting behavior of grey wolves. In GWO, there are four kinds of wolves: alpha, beta, delta and omega. The algorithm updates the positions of the wolves using three main operators: finding prey, encircling prey and attacking prey. These operators hopefully cause the greys to converge to the global optimum point in solution space. The SSA algorithm mimics the swarming mechanism of salps in foraging and navigating in oceans. This algorithm updates the positions of salps using their local and global knowledge. Table 2 shows the multi-problem-based pairwise comparison among the algorithms using the Wilcoxon signed-rank test (Derrac et al. 2011). The comparison is performed based on the mean cost values obtained over 30 simulation runs. This test is considered to statistically evaluate the performance of the algorithm when solving several benchmark functions. The results testify that the ISaDE is more successful than its counterparts in converging to the global optimum of the benchmark functions, with a significance value ⁠. Overall, the ISaDE outperforms the other algorithms in terms of convergence rate and solution quality. Due the excellent performance of the ISaDE in solving numerical optimization problems, we used it to predict the discharge coefficient of labyrinth weirs.

To examine the capabilities of the proposed method in predicting the C_d of sharp-crested W-planform weirs, Kumar et al. (2012) and Ghodsian (2009) are used. This dataset contains 223 records, each one consisting of possible values for six hydraulic and geometric parameters of the weir and a measured value for C_d. This dataset was gathered in a horizontal rectangular tilting flume made of glass walls with a width of 0.245 m, length of 5.360 m, and a depth of 0.450 m. The sharp-crested W-planform labyrinth weirs were placed 5.150 m away from the flume entrance. The experiments were done with a wide range of W-planform weir parameters including the total length of the weir (L_w), length of one cycle of the weir (L_c), the width of one cycle of the weir (W_c), vertex angle (θ) and flow depth (y). The schematic of the flume and the position of the weir used in the experiments are shown in Figure 5. The parameters of the dataset are given in Table 3. We split the dataset into two sets: training and test set. The number of 180 records is randomly selected to be in the training set and the remainder form the test set.

To investigate the most appropriate input parameters, six different input combinations were evaluated. To select the most effective input parameters, first, all input parameters are considered for the development of the ISaDE-SVR model and then one of the input parameters is removed from the inputs and the model is re-trained and tested with the same structure. Table 4 presents the different input combinations. The input parameters are non-dimensional ones that have been used in several other investigations (Parsaie & Haghiabi 2017; Haghiabi et al. 2018; Haghiabi et al. 2019).

The performance of the proposed hybrid method was performed with six different input scenarios to obtain the C_d of the W-planform weir. Table 5 presents the performance of the ISaDE-SVR method with the different input combinations and the optimized parameters of the SVR.

In ISaDE-SVR, the П₆ model, which includes the most effective geometric parameters (H/P, L_w/W_mc, L_c/W_c), has the best results with R² = 0.982 and RMSE = 0.006 in the testing stage. As shown in Table 5, after model П₆, models П₂ and П₄, which include the (H/W_c) parameter in addition to the mentioned geometric parameters, have surprisingly appropriate and satisfactory results. However, their results are significantly weaker than the П₆ model.

Figures 6–9 show the scatterplot of the proposed method for the П₆ model in the training and testing stages. As indicated in these figures, by using the input combinations as П₆, the performance of the ISaDE-SVR model is very close to the observed values. It can be concluded that the ISaDE-SVR method performs properly in predicting the C_d of W-planform weirs.

The numerical results obtained by the proposed method and SVR method for the П₆ combination are given in the training and testing stages based on the performance criteria in Tables 6 and 7.

In general, according to the results presented in Figures 6–9, and Tables 6 and 7, the capability of the proposed hybrid method is appropriate for predicting the C_d of W-planform weirs.

To further examine the results of the ISaDE-SVR method, the Ω index was defined as the ratio of the predicted C_d to the observed values to compare the accuracy of the results. This index helps to recognize the behavior of the models in overestimation or underestimation of the C_d. The proximity of the Ω index to unity indicates the acceptable performance of the proposed method compared with other methods. The histogram of Ω is plotted for the different input models in Figure 10.

The maximum, minimum, and average values of the Ω index for the six models are presented in Table 8. Among all the six models, the largest value of the Ω index is obtained for model П₅. Therefore, models П₂, П₄, and П₆ predict the discharge coefficient close to the observed values compared with the other combinations. The H/P, L_w/W_mc, and L_c/W_c parameters were found to be the most effective parameters in the estimation of C_d.

Given that limited studies have predicted the C_d of W-planform weirs using soft computing techniques, in this section, the results of the ISaDE-SVR model with the input combination of П₆ are compared with the prior studies of Haghiabi et al. (2017) and Haghiabi et al. (2019). Figure 11 and Table 9 show a comparison of the proposed method with the results of other mentioned studies. They show that the ISaDE-SVR method has a better performance with R² of 0.982 compared with ANFIS and GMDH models with R² of 0.97 and 0.94, respectively. The superiority of the proposed model over the others originates from two sources. The first reason is related to the good performance of the algorithms of the ISaDE and SVR models in recognition of the weight of each effective parameter, and the second reason is that the proposed model enjoys the benefits of both ISaDE and SVR models due to the hybrid property of the model.

In this study, a novel hybrid ISaDE method, namely ISaDE-SVR, was used to predict the C_d of W-planform weirs. For this purpose, six models were introduced by combining different effective input parameters to predict the C_d. Then, by analyzing the computational results, the superior model was determined. This model predicted the C_d in terms of the ratio of total head over weir crest to the weir height (H/P), the ratio of the total length of the weir to the main channel width (L_w/W_mc), and the ratio of the length of one cycle to the main channel width (L_c/W_c). Also, the H/P parameter was the most important input parameter in predicting the C_d of the W-planform weir using the proposed method. The results indicate that the proposed ISaDE-SVR model performs more efficiently than other previous studies. The capabilities of both algorithms of ISaDE and SVR models in recognition of the weight of each effective parameter and the combination of these capabilities in the proposed hybrid model are the reasons for the good performance of the model against the others. ISaDE-SVR does need parameter setting and it is a fast convergence algorithm. It outperformed the counterpart algorithms in finding the discharge coefficient. However, the performance of ISaDE-SVR is far from ideal. Therefore, interesting future work is to combine the ISaDE algorithm with artificial neural networks and neuro-fuzzy models to improve the performance of C_d prediction. Another search direction is to apply the ISaDE-SVR method on other optimization problems to evaluate its potential and disadvantages.

The manuscript is original research in the field of water resources.

We (authors) declare that we have no conflict of interest.

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